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CONTROL OF AN EPIDEMIC WITH ENDOGENOUS TREATMENT CAPABILITY UNDER POPULAR DISCONTENT AND SOCIAL FATIGUE

November 2020

Authors:

Suresh Sethi

University of Texas at Dallas

Fouad El Ouardighi

ESSEC - Ecole Supérieur des Sciences Economiques et Commerciales

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CONTROL OF AN EPIDEMIC WITH ENDOGENOUS TREATMENT

CAPABILITY UNDER POPULAR DISCONTENT AND SOCIAL FATIGUE

FOUAD EL OUARDIGHI

ESSEC Business School, Cergy Pontoise 95021, France, E-mail: elouardighi@essec.fr

EUGENE KHMELNITSKY

Tel Aviv University, Ramat Aviv 69978, Israel

SURESH P. SETHI

University of Texas at Dallas, Richardson, Texas 75080-3021, USA

Abstract. The primary issue in this paper is to determine whether mobility restrictions or securing social

interactions is most effective in countering an epidemic disease that spreads also via asymptomatic transmission.

We develop an optimal control policy model wherein i) treatment capabilities are endogenous, ii) the social loss

due to disease-related deaths is part of the tradeoff in terms of health and social welfare perspectives, iii) the

policymaker's inability to counter the disease gives rise to growing popular discontent over time, and iv) non-

therapeutic intervention policy engenders social fatigue over time. We also allow for partial immunity upon

recovery. In many ways, our model applies to the recent pandemic caused by the SARS-Cov-2 virus. In this

setup, we identify which non-therapeutic policy option between mobility restrictions or securing social

interactions most effectively minimizes both the impact of policymaker’s inability and the ensuing popular

discontent and social fatigue.

Keywords. Epidemic, Lockdown, Quarantine, Secured social interactions, Isolation, Health infrastructures,

Treatment, SARS-Cov-2, Covid-19 pandemic.

1. Introduction

The emergence of the Covid-19 disease has caused considerable worldwide human and

economic damage. While some countries (e.g., China, South Korea, Taiwan, Vietnam and

New Zealand) seem to have curbed the spread of the disease to some extent, many others are

still struggling with the pandemic. The trends of the cumulative number of confirmed infected

cases are charted for a selection of countries in Figure 1.

Asian countries in particular had learned from prior experiences with severe acute respiratory

syndrome (SARS) in 2003, Middle East respiratory syndrome (MERS) in 2015 and several

bouts of avian flu. In other countries, policymakers showed varying degrees of ability

countering the epidemic. In countries where the most devastating consequences were

observed, policymakers’ limited ability to counter the disease could be attributed to ill-

preparedness (e.g., Italy, Spain, Belgium, India), lack of benevolence (e.g., France, Great

Britain, the Netherlands), or both (e.g., Brazil, USA, Sweden). Ill-preparedness is generally

due to a recognition lag of the forthcoming danger (e.g., El Ouardighi and Grass, 2020) or to

poor economic conditions, while lack of benevolence consists in ceding a proportion of the

most fragile individuals to the disease (e.g., Denaxas et al., 2020). In a number of countries,

policymakers first deployed a cost-centered policy, seeking to accommodate the disease by

developing herd immunity against SARS-Cov-2 like with the seasonal flu, implicitly

accepting the allegedly minor deadly consequences of the disease. After realizing the

potentially destructive impact of their initial policy on their population and the consequent

unbearable pressures on their health care infrastructure due to inescapable epidemic peaks

(e.g., Lintern, 2020), and their corollary, that is, mushrooming unmanageable popular

discontent, they eventually switched to a health-centered policy that sought to suppress the

disease by imposing strict lockdowns on their population to slow the spread of the disease and

to preserve the existing health infrastructure.

Fig. 1. Cumulative number of coronavirus cases for a selection of countries

One point in common between ill-prepared policymakers and those who lack benevolence is

that, with the emergence of the epidemic, when fine-tuning and error-free health decisions are

crucial, the attempt to curb the undesirable consequences of their first move (or their initial

inaction) is subject to a hysteresis effect that undermines their credibility and exacerbates

popular discontent, which can lead to lawsuits for negligence and for endangering the lives of

others (Miles and Kaci, 2014). At the same time, growing social fatigue among the population

can affect the effectiveness of the policymakers’ non-therapeutic efforts to fight the disease

(e.g., Rozsa et al., 2020).

This paper seeks to characterize a control policy of a silent epidemic disease in a given

country. The problem can be formulated as follows: when the disease starts, in the absence of

a vaccination option, a country’s policymaker needs to 1) build up therapeutic capabilities

both by investing in health infrastructure and making treatment efforts to counter the disease

and, at the same time, 2) make non-therapeutic efforts such as prevention, isolation of infected

individuals, lockdown of the population (i.e., mobility restrictions) or neutralizing the chains

of transmission (i.e., securing social interactions), to impose a cordon sanitaire that limits the

spread of the disease among susceptible individuals. Our objective is to characterize an

optimal pattern for therapeutic and non-therapeutic policy interventions for an epidemic

disease depending on its injurious nature. The idea is to strike a proper balance giving rise to

the identification of a priority order and thus a decision sequence designed on an optimal

basis.

An important feature of the disease considered here lies with its current apparition as Covid-

19. In this context, treatment options are limited because health infrastructures, which include

scientific knowledge, hospital capacity, medical expertise and equipment stockpiles, and

which determine how effectively the infected cases are treated, are frequently in short supply

a priori due to policymakers’ usual unpreparedness to counter the disease. That is, an affected

country is generally not ready when the epidemic starts and thus there is a need to build up its

treatment capability as the disease progresses. In this regard, this article introduces a dynamic

model of control of an epidemic disease where a country's treatment capability is not a priori

available and therefore must be endogenously developed.

In the context of the endogenous development of effective treatment capability, all possible

non-therapeutic options must be considered to slow down the epidemic (Peak et al., 2017;

Flaxman et al., 2020). Among these options, the necessity of implementing the usual

preventative measures for susceptible individuals (Dalton et al., 2020) and isolating known

infected cases (Lee et al., 2010) are obvious. However, the current Covid-19 epidemic has

revealed the controversial nature of such essential non-therapeutic options as mobility

restrictions for the whole population and the securing of social interactions. While many

countries decided to totally restrict mobility for months (e.g., Italy, Spain, France, Great

Britain), others (e.g., South Korea, Taiwan, Hong Kong) instead opted to secure social

interactions by imposing physical (rather than social) distancing, general mask wearing,

disinfection of public areas, AI-based contact tracing, etc.1 The effectiveness of both options

is still a matter of public debate (e.g., Galloway, 2020; Reuters, 2020; WHO, 2020, a). In

general, the securing of social interactions appears to be a more economically and socially

less constraining option than population lockdowns (Caulkins, 2020) in that it does not seek to

drastically reduce personal interactions and movements within an entire community (Wilder-

Smith and Freedman, 2020); it aims only to mitigate the diffusion of the disease rather than to

1 Many countries opted for both options on a sequential basis, that is, first lockdown then secured social

interactions. The conditions for a sequence that placed secured social interactions first were not fulfilled, e.g.,

obligatory medical mask-wearing (Zhang, 2020; Nedelman and Yu, 2020, Ha, 2020; Nierenberg, 2020) in many

countries (e.g., France, Great-Britain, Italy, Spain). Other countries rejected both options (e.g., Sweden).

suppress it (Anderson et al., 2020). Conversely, total mobility restriction has clearly shown

significant effectiveness in reducing the number of disease-related deaths2 (Flaxman et al.,

2020; Thusnstrom et al., 2020; Zhang et al., 2020; Roux et al., 2020; Salje et al., 2020; Torry,

2020; Courtemanche et al., 2020), but it may have huge economic (Navaretti et al., 2020;

Inoue and Yasuyuki, 2020) and psychological impacts (Brooks et al., 2020).

Research

Treatment effectiveness

contingent upon health

infrastructure level

Cost of disease-related

deaths as part of

the tradeoff

Growing popular

discontent over

time

Growing social

fatigue over

time

Mobility restriction

versus secured

social interactions

Sethi (1974)

Sethi (1978)

Sethi and Staats (1978)

Behncke (2000)

Caetano and Yoneyama (2001)

Culshaw (2004)

Alexander et al. (2004)

Lenhart and Workman (2007)

Yan et al. (2007)

Yan and Zou (2008)

Arino et al. (2008)

Zaman et al. (2008)

Augusto (2009)

Blayneh et al. (2009)

Rowthorn et al. (2009)

Hansen and Day (2010)

Lee et al. (2010)

Tchuenche et al. (2011)

Lee et al. (2011)

Lee et al. (2012)

Ullah et al. (2012)

Lashari and Zaman (2012)

Okosun et al. (2013)

Okosun et al. (2014)

Di Liddo (2016)

Collins and Duffy (2018)

Takasar et al. (2019)

Bonnans and Gianatti (2020)

Caulkins et al. (2020)

Charpentier et al. (2020)

Kanyiri et al. (2020)

Kruse and Strack (2020)

Mohsin et al. (2020)

Perkins and España (2020)

This paper

Tab. 1. Basic features of research on control of epidemic disease

Growing fatigue among the population is another issue for both mobility restriction (Meichtry

and Sugden, 2020; Marcus, 2020; Rogers, 2020) and securing social interactions (e.g., Rozsa

et al., 2020). That is, the stress caused by the economic and social constraints that these two

non-therapeutic measures entail reduces their effectiveness over time. To date, no study has

2 According to Flaxman et al. (2020), strict lockdown saved about 3.1 million lives in 11 European countries.

accounted for this important feature in comparing the relative effectiveness and cost of these

two options. Our study seeks to determine whether mobility restrictions or securing social

interactions is most appropriate with regard to non-therapeutic intervention-related social

fatigue.

Another important factor is policymakers’ ability to counter the disease, which is reflected

both in its preparedness (i.e., the initial level of health infrastructures) and its reputation for

benevolence (i.e., its relative valuation of the marginal cost of disease-related loss of life). A

policymaker’s inability to counter the disease can increase stress among the population, that

is, growing popular discontent. This popular discontent might amplify the social cost of the

health and human consequences of the disease, which has to be accounted for in the

management of the disease. Our third contribution lies in the consideration of policymakers’

inability to counter the epidemic and the ensuing popular discontent and the relative impact

these factors have on the evolution of the disease.

Since the seminal work of Kermack and McKendrick (1927, 1932, 1933, reprinted in 1991, a,

b, c), dynamic models have become a mainstream approach in the epidemiological literature

(e.g., Anderson and May, 1991; Diekmann and Heesterbeek, 2000; Murray, 2002; Lenhart

and Workman, 2007). However, most of these models are used for simulations or to

determine a control via intensity parameters or other types of approximations. In this respect,

epidemiological models that use optimal control theory are rather simple (see Sharomi and

Malik, 2017). Our approach differs from these studies in that we formulate an optimal control

model that incorporates five essential, mostly unprecedented, features (see Table 1 for some

essential features of our approach not covered by the existing research on the control of

epidemic diseases), that is, we specifically consider:

- The development of treatment capability of the disease is contingent both upon treatment

efforts and the current state of health infrastructures, which itself, in turn, is endogenous. This

is in contrast to the existing literature that usually assumes that the treatment capabilities for a

disease are exogenously given as bounded control variables (e.g., Behncke, 2000; Joshi, 2002;

Hansen and Day, 2010; Lee et al., 2011; Di Liddo, 2016), or as a budget constraint (e.g.,

Rowthorn et al., 2009). We choose to assume that the treatment capabilities are built up based

on the joint deployment of health infrastructures and treatment efforts while the epidemic

progresses. An important original feature lies with the fact that the initial level of health

infrastructures reflects a policymaker’s level of preparedness to counter the epidemic disease.

- The economic and social losses incurred for each disease-related death are accounted for in a

tradeoff that seeks to determine an optimal policy against the epidemic disease. This departs

from most of the existing literature that accounts for the cost induced by all (e.g., Yan and Zu,

2008; Blayneh et al., 2009) or part (e.g., Lee et al., 2010; Hansen and Day, 2011; Lee et al.,

2011; Tchuenche et al., 2011; Lee et al., 2012; Ullah et al., 2012; Di Liddo, 2016; Collins and

Duffy, 2018) of the infected categories except deaths. Following Caulkins et al. (2020) and

Charpentier et al. (2020), wchoose here to assume that disease-related deaths cannot be

welfare-neutral. An important related novelty here lies with the possibility of an

undervaluation of the economic and social losses incurred for each disease-related death, thus

reflecting the policymaker’s lack of benevolence.

- The policymaker’s ill-preparedness and lack of benevolence jointly contribute to the

society’s inability to counter the epidemic3, engendering popular discontent. This discontent

swells over time and should be factored in by the policymaker through a time-dependent

compounding function, making disease-related social costs increase in importance over time,

and affecting the decisions made as well as the evolution of the disease over time. Most of the

literature assumes popular neutrality regarding disease-related social costs, which implies that

the policymaker’s decisions are not directly under time pressure (e.g., Behncke, 2000; Joshi,

2002; Ya and Zou, 2008; Blayneh et al., 2009; Lee et al., 2011; Hansen and Day, 2011;

Tchuenche et al., 2011; Lee et al., 2012; Ullah et al., 2012; Di Liddo, 2016; Collins and

Duffy, 2018; Caulkins et al., 2020). A few studies assume growing popular complacency

through a discounting function of the overall social cost, (e.g., Sethi, 1974; Sethi, 1978; Sethi

and Staats, 1978; Rowthorn et al., 2009; Charpentier et al., 2020), which implies that future

decisions and related outcomes are less costly than current decisions and outcomes. Instead,

we assume that the disease-related social costs compound to make them, along with

postponed decisions and their related outcomes, increasingly costly. We also assess the social

cost consequences of the case where the policymaker is unable to detect the popular

discontent caused by their own inability to counter the epidemic and thus applies a wrong

control policy, which is popular neutrality-based rather than popular discontent-based.

- Mobility restrictions versus policy options that secure social interactions are considered and

their respective effects on the evolution of the disease are compared. To the best of our

knowledge, except the simulation-based study by Ferguson et al. (2020), few papers have

compared these two non-therapeutic policy options with proper internalization of their related

costs and effects. In this paper, the tradeoff between the two options is based on their welfare

dimensions with the novelty that both options engender economic and social constraints that

3 An example of such inability can be found in Castle (2020).

give rise to growing social fatigue. To date, the consequences of this phenomenon remain

uninvestigated. Departing from the mainstream, we assume that social fatigue can prevail

through a time-dependent compounding function, making the leveraging of non-therapeutic

instruments to counter the epidemic increasingly costly over time. Finally, we compare the

case of social fatigue that arises from non-therapeutic policies with that of either relevantly or

fallaciously presumed social readiness in terms of social cost.

The inclusion of these features yields a more appropriate control policy than what is found in

the literature. In this respect, the following issues are investigated:

i) Is imposing mobility restrictions or securing social interactions more effective for the non-

therapeutic control of an epidemic from health and social welfare perspectives?

iii) How do policymakers’ inability to counter the epidemic and the related popular

discontent and non-therapeutic intervention policy-related social fatigue, respectively, affect

the patterns of decisions and the related health and social welfare consequences? Put

differently, are policymakers’ failings, with or without correlated popular discontent, or

social fatigue more detrimental to the control of an epidemic disease?

These questions will help to identify which non-therapeutic policy option between mobility

restrictions and secured social interactions is more effective in minimizing the respective

effects of policymakers’ inability to counter the epidemic and related popular discontent,

social fatigue and policymakers’ disregard of popular discontent.

To address these issues, we formulate a policy model based on a dynamic representation of a

silent epidemic disease. An important feature of our model is that, though it has the same

richness as the most common epidemiological models, that is, the susceptible-infected-

recovery (SIR) models (e.g., Lenhart and Workman, 2007), we allow for a number of

additional economic considerations. In this setup, we characterize an optimal policy

involving, alternatively, mobility restrictions or secured social interactions, along with an

optimal mix of efforts in isolating symptomatic and convalescent cases, health infrastructures

and treatment, and compare their relative effectively they impact the social cost of the

epidemic disease. This comparison sheds light on the differences between the decision

patterns associated with each policy option, contingent to the respective effects that popular

discontent and social fatigue exert on each option.

The paper is organized as follows. In the next section, we formulate an optimal control model

where a policymaker seeks to determine an optimal policy that minimizes the social cost

related to the epidemic. In Section 3, we characterize the general solution of the problem.

Section 4 investigates the model with numerical means. Section 5 concludes the paper.

2. Model formulation

2.1 Epidemic model

Let the number of individuals in a country be denoted by (0)= 1 at time = 0, all of

whom are susceptible to infection to begin with. At any time 0, the country has a number

of infected individuals that are distributed in three categories: those who have no symptoms

(asymptomatic cases), denoted by ()0, those who have symptoms (symptomatic cases),

denoted by ()0, and those who are recovering (convalescent cases), denoted by

()0. All three categories are supposed to be infectious including the convalescent cases

because the excretion of the pathogenic micro-organism lasts during the ‘shedding window’

until convalescent individuals become resilient (e.g., Chang et al., 2020; Cheung et al., 2020;

Rothe et al., 2020). Regarding the asymptomatic cases category, it has been shown that it

played a major role in the silent spread the Covid-19 disease (e.g., Mahase, 2020; Roth et al.,

2020). The reason is that the incubation period (i.e., the period between becoming infected

and showing symptoms) of Covid-19 is generally longer than the latent period (i.e., the time

between becoming infected and being able to spread the disease to others).4 This implies that

symptom-based control of the epidemic is not sufficient. The contagion process results from

the interaction between infected individuals, ()

, and the remaining susceptible

individuals, ()()

, at time 0.

In the absence of immediately available therapeutic tools, the contagion process needs to be

reduced as much as possible. Apart from standard prevention efforts (Dalton et al., 2020) and

specific isolation measures for symptomatic and convalescent cases (Yan and Zou, 2008;

Hellewell et al., 2020), two main non-therapeutic policy options are generally considered, that

is, either imposing mobility restrictions on the whole population or securing social

interactions. These options emerge mainly because asymptomatic cases are not spontaneously

detectable and cannot be subjected to imposed isolation, and thus can contribute actively to

spreading the disease in the absence of mobility restrictions or secured social interactions.

Shen et al. (2020) showed that timely lockdown has been effective in reducing about 60% of

new infections and deaths in the province of Hubei (China), and that its effect is sustained

even after its removal. Ji et al. (2020) also showed that the epidemic curve based on observed

cases in Huangshi (China) was flattened thanks to lockdown. Regarding the securing of social

interactions, it can also mitigate the contagion process (Chu et al., 2020; Prather et al., 2020)

4 The mismatch period, i.e., the period during which Covid-19 patients can spread the disease before they are

symptomatic, is 2-4 days. In fact, the 1-2 days before symptoms appear may be when they are most contagious

(Christakis, 2020).

as, for example, in Taiwan (Griffiths, 2020), South Korea and Hong Kong (Zhang and Tsang,

2020), with a broad range of measures whose effectiveness was established, such as imposing

physical distancing (Ahmed et al., 2018; Prem et al., 2020), use of medical masks by the

whole population (MacIntyre et al., 2009; Sim et al., 2014; Offeddu et al., 2017), disinfection

of public areas (Rabeneau et al., 2004; Kampf et al., 2020), AI-based contact tracing to help

uninfected individuals avoid infected individuals (Cooney et al., 2016; Peak et al., 2017).

Caulkins (2020) provides a number of additional concrete suggestions to secure social

interactions.

In the following, both the intensity of mobility restriction and the intensity of secured social

interactions are modeled as continuous control variables.

Efforts for securing social interactions, if any, are denoted by 0()1

, and are

specified as a function (1()) that apply to the social interactions between infected

cases and susceptible individuals, that is, the contagion rate. Because the symptomatic and

convalescent cases are supposed to be exposed to isolation efforts, i.e., under quarantine or

hospitalized, no additional mobility restrictions are required for them. Social interactions are

thus secured by a factor corresponding to the value of ()×100% over time.

Alternatively, mobility restriction efforts, if any, are denoted by 0()1

, where

> 0 is the marginal effectiveness parameter of mobility restriction efforts, and apply both

to the asymptomatic cases and to the susceptible individuals, respectively as (1

())() and (1())(()()

). Because we cannot distinguish between

() and (()()

), mobility restrictions are imposed on both groups. Given that the

infection rate results from the multiplication of the sizes of the two groups, the infection rate

appears to decrease as the square of (1()). Under mobility restrictions, each

individual, either asymptomatic or susceptible, should thus be mandated to have her/his

number of daily social interactions reduced by a factor corresponding to the value of

()×100% over time. Alternatively, the latter value may be interpreted as a factor of

reduction of the daily time devoted to social interactions between individuals.

On the other hand, isolation efforts, denoted by 0()1

, apply to limiting

interaction between symptomatic plus convalescent cases and susceptible individuals. This

gives us the corresponding infection rate term, (1())()+()(1

())(()()

), under mobility restriction, and (1())()+

()(()()

), under secured social interactions, where > 0 is the marginal

effectiveness parameter of the isolation effort. The specification of isolation efforts, (1

()), which can be found in, e.g., Lee et al. (2010) and Ullah et al. (2012), means that a

fraction of symptomatic and convalescent (i.e., confirmed) cases, given by the value of

×100%, are to be isolated over time.

The contagion process can be reduced through prevention efforts, which usually include

campaigns for enhanced hygiene, risk factor reduction and protective factor enhancement.

However, prevention efforts are generally supposed to be equally needed over the entire

epidemic period, regardless of their interaction with other control variables. In other words,

prevention efforts cannot be considered contingent upon the policy associated with mobility

restrictions or secured social interactions and can thus be disregarded.

Overall, the number of newly infected individuals, i.e., the contagion rate, results from direct

interaction between infected individuals and the remaining susceptible individuals, that is,

(1())(1())()+(1())()+()(1

())(()()

), where > 0 denotes a contagion rate parameter. Depending on

whether a policymaker opts for mobility restrictions, i.e., ()0, or no mobility

restrictions, i.e., ()= 0, and for secured social interactions, i.e., ()0, or no secured

social interactions, i.e., ()= 0, the contagion rate is given in Table 2.

Secured social interactions

No (= 0)

Yes (0)

Mobility

restrictions

No (= 0) ()+(1())()+

()(()()

)

(

1()

)



(



)

(

1

())()+()(()

()

)

Yes (0)

(1







())



()+

(1())()+()(1

())(()()

)

(1







())(1







())



()+

(1())()+()(1

())(()()

)

Tab. 2. Mobility restrictions vs. secured social interactions and the contagion rate

In order to compare their relative effect on the evolution of the disease, we restrict our

attention to three of the four cases in Table 2, that is, the case where the policymaker opts for

mobility restrictions only, i.e., ()0 and ()= 0, the case where the policymaker opts

for secured social interactions only, i.e., ()= 0 and ()0, and the uncontrolled case

where the policymaker opts for neither of the two previous alternatives, i.e., ()= 0 and

()= 0, and ()= 0. This last case will be used for comparative purposes.

The evolution of asymptomatic cases results from the difference between newly infected

individuals, and asymptomatic cases that either move to the category of symptomatic cases or

become resilient. The rate of transition from asymptomatic to symptomatic is (), where

> 0 is a transition rate parameter from asymptomatic to symptomatic. On the other hand,

the rate of asymptomatic cases that become resilient is (), where > 0 is a rate

parameter from an asymptomatic to a resilient state. Note that the magnitude of transition of

an asymptomatic to a resilient state is characteristic of the silent spread of the disease.

Overall, the evolution of asymptomatic cases is given by:

()=(1())(1())()+(1())()+()(1

())(()()

)(+)(), (0)=0

where either (()0, ()= 0), or (()= 0, ()0).

The evolution of symptomatic cases results from the difference between two inflows and three

outflows. The two inflows correspond to:

- Asymptomatic cases who developed symptoms, (), and

- Convalescent cases that relapse and thus move back to the category of symptomatic cases,

(), where > 0 is a relapsing rate parameter.

On the other hand, the three outflows correspond to:

- An outflow of symptomatic cases that die due to direct or indirect consequences of the

disease, that is, (), where > 0 is a lethality rate parameter.

- An outflow of symptomatic cases that recover spontaneously from the disease and thus

move to the category of convalescent cases, that is, (), where > 0 is a spontaneous

convalescence rate parameter.

- An outflow of symptomatic cases that receive a treatment to recover from the disease, and

thus move to the category of convalescent cases. The effectiveness of the treatment is

contingent both upon the treatment efforts, denoted by ()0, and the health infrastructure

regarding appropriate procedures for treatment of the disease, which include scientific

knowledge, hospital capacity and medical equipment stockpiles, denoted by ()0. That

is, in the absence of health infrastructure to deal with the disease, treatment efforts are totally

ineffective. Conversely, major health infrastructure improves the effectiveness of the

treatment that, all things being equal, then requires less effort. Put simply, the combination of

health infrastructure and treatment efforts determines treatment effectiveness. Symptomatic

cases that are treated for the disease are thus ()()(), where > 0 denotes a

treatment capability rate parameter.

Overall, the evolution of symptomatic cases is:

()=()[++()()]()+(), (0)=0

where the level of health infrastructures is given by (e.g., El Ouardighi et al., 2014):

()=(), (0)=0

where ()0 is the effort rate in the development of disease-related health infrastructure.

The evolution of convalescent cases results from the difference between the inflows of

symptomatic cases that turned convalescent either spontaneously, (), or by treatment,

()()(), and two outflows, that is:

- Convalescent cases that become resilient, that is, (), where > 0 is a resilience rate

parameter, and

- Convalescent cases that relapse, ().

Overall, the evolution of convalescent cases is:

()=+()()()(+)(), (0)=0

As noted previously, the number of overall individuals, () > 0, is time-dependent and it is

supposed to decrease over time with the instantaneous number of deaths and with a fraction of

resilient cases that developed some immunity against the disease, if any, that is:

()=()()+(), (0)= 1

where 0,1. After initial evidence to the contrary (Blanco-Melo et al., 2020), the

assumption of herd immunity has since become plausible. While it seems to prevail in post-

symptom forms (Wajnberg et al., 2020), contradictory results are reported for

asymptomatic/mild forms: Fafi-Kremer et al. (2020) observed high immunity levels in this

latter case while Long et al. (2020) finds it positively correlated with symptom intensity.

Recently, Sekine et al. (2020) observed that natural exposure or infection could prevent

recurrent episodes of severe COVID-19. We conclude that neither case can be discarded and

let the immunity rate parameter take various values between 0 and 1, i.e., ,(0,1).

2.2 Model properties and calibration

We will now explore the properties of the model. We characterize a situation with no efforts

with regards to health infrastructure, mobility restrictions, isolation, securing social

interactions and treatment over time, i.e., ()=()=()=()=()= 0, . In

this uncontrolled situation, the dynamics of the epidemic can be schematically represented in

Figure 2.

The corresponding differential system is given by:

() = ()()+(), (0)=> 0

()=()

(()()

)(+)(), (0)=0

()=()(+)()+(), (0)=0

()=()(+)(), (0)=0

The above model has several similarities with the susceptible-infected-recovered (SIR)

models (Lenhart and Workman, 2007; Sharomi and Malik, 2017). An important common

point is that it distinguishes between six possible states, that is, susceptible, asymptomatic,

symptomatic, convalescent, resilient and dead individuals. Furthermore, all three categories of

asymptomatic, symptomatic and convalescent cases are potentially infectious to susceptible

individuals. Another important similarity is that our model accounts for herd immunity.

Finally, as in Agusto (2009) in the context of tuberculosis, it includes the possibility of relapse

from convalescent to symptomatic state.5 Due to its high level of generality with a limited

number of state variables, an important feature of our model lies with its compactness.

Fig. 2. The dynamics of uncontrolled epidemic disease

Overall, the active evolution of the epidemic is summarized as:

 ()=()

(()()

)()()(),

 (0)= 0

= 1,2,3. In the above expression, the parameters relate to contagiousness, , lethality, , and

both asymptomatic and post-symptom resilience,  and . Contagiousness refers to the ease

of transmission of the disease among susceptible individuals and is usually measured with the

basic reproduction ratio (Rhodes and Anderson, 2008), , which reflects the strength of the

epidemic. Lethality is measured by the case fatality ratio which is the rate of deaths observed

among the symptomatic (confirmed) cases. Resilience indicates the rate of asymptomatic and

post-symptom cases that recover from the disease. Contagiousness and lethality together

characterize the level of severity of the disease considered (e.g., McCandless et al., 2020).

We calibrate the uncontrolled version of our model on the Covid-19 epidemic data and use a

daily frequency with =180. In the context of the Covid-19 epidemic, the basic

reproduction ratio  is usually set between 2.2 (Li et al., 2020; Riou et al., 2020) and 3.28

(Liu et al., 2020), which implies that the fraction of the population that is likely to be infected

in the absence of mitigation measures, 11

, is roughly between 54% et 70%. We make

5 Relapsing can be an important factor if the effectiveness of the treatment is not sufficient or if patients have

immunity problems, or if the disease does not give rise to herd immunity.

Asymptomatic

()

Symptomatic

()

Convalescent

()

Disease-

related deaths

Resilient with

immunity

(1)









()



(1)









Remaining susceptible

()()



an average baseline assumption that 2.74 (Wu, Leung and Leung, 2020), with 63% of

the population likely to be infected. The range of estimates of the proportion of asymptomatic

cases in the literature is wide, from 18% to over 80% (see WHO, 2020, b), but recent

evidence based on 16 studies estimated it to be 40%-45% (Oram and Topol, 2020), which

seem to make large consensus. Considering this, we posit that 40% of the current

asymptomatic cases become subsequently, yet surreptitiously, resilient, with = 0.40.

Following Li et al. (2020) and Chang et al. (2020), the average time during which

asymptomatic cases become symptomatic is 5 days, which leads to set = 1 5

. Regarding

the case fatality ratio (CFR), i.e., the ratio of reported cumulative deaths to date to cumulative

confirmed (i.e., symptomatic, in our setup) cases to date, that is, ()=()





()



, it is

generally between 2.3% (Wu and McGoogan, 2020) to 7.2% (Onder et al., 2020:).6, 7 Because

such estimates can be biased upwards by under-reporting of cases and downwards by the

failure to account for the delay from confirmation to death (Russell et al., 2020)8, it is

necessary to adjust the CFR based on the epidemiological information from a closed

population such as the quarantined Diamond Princess Cruise (NIID, 2020) by multiplying the

CFR estimates in China by 46% to obtain the corrected CFR within an interval between 1.1%

and 3.3%. Therefore, we target an average value of these values at the terminal period , that

is, ()2.2%.9 Given that the median time from the appearance of symptoms until

clinical recovery is approximately 8 days (Chang et al., 2020), the spontaneous convalescence

rate parameter of symptomatic cases is set to = 1 8

. This represents 12.5% of the outflow

of symptomatic cases which, accounting for the CFR value chosen, implies that 87.5% of

symptomatic cases, that is, 17.5% of the marginal inflow from asymptomatic cases, need

medical treatment. This estimate is close to the estimate of 20% provided by the WHO

(2020). On the other hand, the resilience of convalescent cases is usually between 8 (Chang et

al., 2020) and 20 days (Zhou et al., 2020). To be consistent with this range of values, the

resilience rate parameter is thus set as an average, that is, = 1 14

 (Salje, 2020), which

gives a total of 27 days from infection to post-symptom resilience. In percentage, this

corresponds to an evolution of the resilience rate of 7.14%.

6 The lethality rate is age-dependent (Verity et al., 2020) and contingent upon co-morbidity factors (Jordan et al.,

2020; Yang et al., 2020). It is also stage-dependent of the evolution of the epidemic, i.e., earlier stages of the

epidemic are associated with a higher lethality rate than later stages (Oke and Henean, 2020).

7 Other studies suggested that among patients admitted to hospitals, the mortality rate ranged between 11%

(Huang et al., 2020) and 15% (Chen et al., 2020).

8 See also Nishiura et al. (2009).

9 This value is similar to value of 2.3% reported from the largest study of Covid-19 from the Chinese Centers for

Disease Control and Prevention (Zhonghua et al., 2020) and the Diamond Princess Cruise (NIID, 2020).

Although the possibility of relapsing was observed in the case of Covid-19 disease (e.g., Lan

et al., 2020; Wu et al., 2020; Gousseff et al., 2020), the relapsing rate parameter is set at a

very low value, i.e., = 0.01, to include a level of under-ascertainment. Finally, based on the

above parameter values and the normalized value of the initial number of individuals, (0)=

1, the contagion rate and death rate parameter values that match with both the CFR terminal

value targeted of ()2.2% and the reproduction ratio value chosen of 2.74 (Van

den Driessche and Watmough, 2002), are estimated for the uncontrolled version of our model

as = 0.2977 and = 0.00248.10 Starting with an initial incidence rate regarding the

asymptomatic cases of = 0.001, and no symptomatic and convalescent cases, i.e., =

= 0, we seek to identify how herd immunity (or its absence) is capable of affecting its

spread. The evolution of the Covid-19 epidemic is represented with and without herd

immunity in Figure 3. It is noteworthy that a strict equivalence is observed in both cases

between our model and the SIRS model for the patterns of asymptomatic, symptomatic and

convalescent cases.

3a. Full herd immunity 3b. No herd immunity 3c. Basic reproduction ratio

Fig. 3. Evolution of the Covid-19 epidemic over time with and without herd immunity

(AC: asymptomatic cases, SC: symptomatic cases, CC: convalescent cases, CD: cumulative deaths)

In Fig. 3, the absence of herd immunity (3b) results both in delayed and higher peaks than

under full herd immunity (3a) respectively for asymptomatic, symptomatic and convalescent

cases. Given = 0.001 and = 0, the peak in the absence of herd immunity intervenes

respectively in 72, 85 and 110 days for asymptomatic, symptomatic and convalescent cases,

that is, approximately two, three and five weeks later, respectively, than under full herd

immunity. The reason lies with the fact that herd immunity reduces the contagiousness of a

disease, as suggested in Fig. 3c that illustrates the evolution of the basic reproduction ratio 

over time with various levels of the immunity. It clearly shows that contagiousness remains at

a much higher level without than with full herd immunity over time. However, after at least

50 days, the basic reproduction ratio decreases rapidly even with partial herd immunity: after

10 The estimation procedure is summarized in Appendix 1.

3 months, 60% herd immunity brings  to a value lower than 1 without any policy

intervention. On the other hand, the absence of immunity gives rise to a post-peak plateau

(Fig. 3b), while full immunity does not produce this pattern (Fig. 3a). The reason lies with the

fact that, without immunity, the value of  remains consistently high (above 2.5) after the

peak of asymptomatic, symptomatic and convalescent cases is reached, while it decreases

very rapidly under full immunity from 2 after 54 days at the peak of asymptomatic cases to 1

after 75 days at the peak of convalescent cases. The height of the plateau thus increases with

the level of contagiousness, but may be lower for a higher immunity level. Under a no-

immunity assumption, a closed loop emerges where people retread the path from Susceptibles

 Asymptomatic  Symptomatic  Convalescent  Susceptibles several times, which may

result in a cumulative number of cases much greater than the initial number of individuals.

Finally, without immunity, the number of cumulative deaths is steadily increasing (Fig. 3b),

while it is converging toward a maximum value below 1% of the initial number of individuals

in the case of full immunity (Fig. 3a).

Based on the abovementioned studies, we assume in the remainder of the paper that herd

immunity generally exists among resilient convalescent and asymptomatic cases.

Nevertheless, in order to account for the fact that an aging population (defined as people aged

65 and over), which ranges from 16.03% in the USA to 28.14% in Japan (OECD, 2020),

incurs immunosenescence (Koff and Williams, 2020), and to account for a sufficiently high

level of under-ascertainment for other immune-compromised populations, for people

experiencing homelessness (Tsai and Wilson, 2020) and other forms of vital deprivation, and

more generally for short duration seroconverters (Pollàn et al., 2020) and non-seroconverters

without co-morbidities (Staines et al., 2020), the full herd immunity assumption is ruled out

for both asymptomatic and post-symptomatic forms. In order to comply with Long et al.

(2020) and Gudbjartsson et al. (2020), who observe a positive correlation between immunity

and symptoms intensity, we set = 0.25 and = 0.5.

With such immunity levels, the overall herd immunity converges at 38%. After 4 months, the

cumulative rate of asymptomatic cases, defined as ()= 1 

()





()()

(()()

)



, reaches 

 and the proportion of resilient cases, given by

()=()()





()()

(()()

)



, tends to 95%. A peak of symptomatic cases is

reached after 70 days and the CFR ultimately tends to 2.2%. Based on the abovementioned

studies, these trends seem plausible in the context of the Covid-19 epidemic. Finally, the

infection fatality rate (IFR), defined by ()=()





()()

(()()

)



 in our

model, converge toward 0.7%, which is very similar to the point-estimate of 0.68% obtained

by Meyerowitz-Katz and Merone (2020) for the Covid-19 epidemic. These trends provide

strong support for our model and its calibration.

2.3 Objective function

We now define the objective function and consider that the policymaker aims to minimize the

overall social cost related to the disease. We assume a fixed and finite planning horizon <

 because it corresponds to a time frame beyond which the country’s economic situation

might become unmanageable for the policymaker (Brooks et al., 2020; Ollstein and Goldberg,

2020). That is, the policymaker is supposed to counter the disease in an emergency context

because, as time goes by, the health, economic and social consequences of the disease and the

constraints implied by the policymaker’s decisions impose an increasingly stressful (if not

traumatic) situation among the population (Hull, 2005; Reynolds et al., 2008; Sprang and

Silman, 2013; Jeong et al., 2016; Rubin and Wessely, 2020), which makes the policymaker

fully accountable for the negative consequences of her/his management of the disease at the

end of the planning horizon, and maybe exposing her/him to grievances and lawsuits (Miles

and Kaci, 2014).

The policymaker seeks to determine an optimal policy mix involving therapeutic efforts in

treatment and health infrastructures, and non-therapeutic efforts in the isolation of

symptomatic and convalescent cases, and either in mobility restriction of both uninfected and

infected individuals or in securing the social interactions between uninfected and infected

individuals.

The objective is to minimize the sum of the following costs:

- The disease-related costs, that is, the health, social and economic costs related to

asymptomatic, symptomatic and convalescent cases, (),(),(), with

,,> 0, and to disease-related deaths, (), with > 0. For simplicity, the

costs related to infected cases are specified linearly, i.e., (),(),()()

,

= 1,2,3. The cost incurred for a symptomatic case is greater than that for a convalescent

case due to more acute health, social and economic issues, which in turn is greater than that

for an asymptomatic case, i.e., >>> 0. Regarding the economic dimension only,

this ranking reflects the fact that a symptomatic case leads to a greater loss of labor

productivity than an asymptomatic case, which in turn leads to less productivity loss than a

convalescent case. Similarly, the cost of disease-related deaths is specified as a linear

function, ()(), where the marginal cost of a disease-related loss of a human

life is supposed to be greater than the sickness-related marginal cost, that is, >.

- The control-related costs, that is, the respective costs of therapeutic efforts in health

infrastructures, (), with > 0, and treatment, (), with > 0, and non-

therapeutic efforts in the isolation of symptomatic and convalescent cases, (), with

> 0, and either in mobility restrictions, (), with > 0, or securing social

interactions, (), with > 0. Due to decreasing returns to scale, the cost functions

of therapeutic and non-therapeutic efforts are all supposed to be increasing convex and,

respectively, specified as ()()

, > 0, ()()

, > 0,

()()

, > 0, ()()

, > 0, and ()()

, > 0. In

addition, we assume that the marginal cost parameter of mobility restriction efforts is greater

than that of social interaction securing efforts, which in turn is greater than that of isolation

efforts, that is, >>> 0. The lower value of the isolation effort cost parameter lies

with the fact that they apply to symptomatic and convalescent cases that, by definition, are

clearly identified as being ill and thus are efficiently and easily isolated. In contrast, mobility

restriction measures have the highest value cost parameter due to the huge opportunity cost

engendered by the restrained mobility of a whole population, as well as the difficulty of

convincing a whole population to comply with mobility restriction measures. The

intermediate value of cost parameter of securing social interaction reflects the lesser means

required to limit the transmission of the disease without reducing social interactions.11

- The difference between the end-of-horizon cost related to the terminal number of

asymptomatic, symptomatic and convalescent cases, (),(),(), with

,,> 0, and disease-related deaths, (), with > 0, and the salvage

value of the terminal health infrastructures level, (()), with > 0. The salvage costs of

infected cases and deaths and the salvage value of health infrastructures are all specified as

linear functions, that is, (),(),()()

, = 1,2,3, where, consistent

with the assumptions on the ranking of the corresponding instantaneous costs , >>

11 Note that securing social interactions can be environmentally harmful, as, for example, for disinfection of

public areas (Zhang et al., 2020), and epidemic-related waste (Chaudhuri, 2020). In addition, the latter exposes

garbage workers to increased risk of infection (Minter, 2020). The corresponding costs are not necessarily

negligible.

> 0, and ()(), > 0 and (())(), > 0. Here also, the

overall salvage marginal cost of a symptomatic case is +, where the marginal salvage

cost of a disease-related loss of a human life is supposed to be greater than the sickness-

related marginal salvage cost, that is, >.

The overall social cost can also be affected by growing popular discontent and social fatigue

over time. Popular discontent stems from the policymaker’s inability to counter the epidemic,

which is established by the conjunction of two characteristics: ill-preparedness and lack of

benevolence of the policymaker. The policymaker’s ill-preparedness lies with her/his

incapacity to anticipate the forthcoming danger and is reflected in the low level of health

infrastructures at the beginning of the epidemic, (0), and thus limited initial treatment

capability. As for the policymaker’s lack of benevolence, it is related to its marginal valuation

of the current and terminal cost of disease-related deaths,  and . In theory, the magnitude

of such a valuation should depend upon the economic and social losses incurred by an

additional disease-related death, but it can also be based upon the policymaker’s subjective

perception of the intrinsic value of a human life. Ideally, this perception should be consistent

with the popular valuation such that (,),, where ,> 0 reflects the popular

valuation, to ensure the policymaker has a benevolent reputation. If the policymaker’s current

and terminal valuation of a human life is too low compared to the popular perception, then the

policymaker is reputed to lack benevolence. In particular, a policymaker’s valuation of a

human life is too low if it is lower than or equal to a certain lower bound value, that is,

(,),. As a result, popular discontent engendered by both ill-preparedness and lack

of benevolence exhibited by the policymaker, i.e., policymaker’s inability-based popular

discontent, makes the disease-related costs, ()

+(), increase over time

through a time-dependent compounding function (Nordhaus, 1975; Lambertini, 2014),

denoted by  with 0. A higher (lower) level of  describes, all things being equal, a

higher level of defiance (trustfulness) of the population regarding the policymaker’s ability to

counter the epidemic.12 The case where = 0 represents popular neutrality that usually

applies to well-prepared and benevolent policymakers, and thus represents policymaker’s

ability-based popular neutrality. However, popular neutrality can also be assumed in the case

of a policymaker’s inability if the population is supposed to be docile, either relevantly, which

12 South Korea is an example of a population that trusts a policymaker’s ability to counter Covid-19

(www.koreatimes.co.kr/www/nation/2020/04/356_287965.html).

thus corresponds to policymaker’s inability-based popular neutrality, or fallaciously, which

thus characterizes policymaker’s inability-based distorted popular discontent13.

On the other hand, social fatigue emerges due to the policymaker’s leveraging of non-

therapeutic instruments to counter the epidemic. This phenomenon is common for isolation of

exposed or infected cases (Brooks et al., 2020), lockdown (e.g., Johnson et al., 2020), and

securing of social interactions (e.g., Ao, 2020), and may reflect a population’s progressive

decline in vigilance (Chow, 2020). In this regard, the social fatigue is also expressed as a

time-dependent compounding function, denoted by  with 0, that makes the non-

therapeutic control-related costs, that is, either the sum of costs of isolation and mobility

restriction efforts, (()2

+()2

), or the sum of costs of isolation and social

interaction securing efforts, (()2

+()2

), become more expensive over time.

All things being equal, a higher level of  reflects lower effectiveness of the non-therapeutic

instruments over time. Finally, the case where = 0 denotes social readiness.

The above assumptions may have important implications on the control of the epidemic both

in terms of impact and timing and should therefore be accounted for. Table 3 summarizes

various scenarios considered with a clear distinction between the policymaker’s inability

effect and the popular discontent effect.

> 0

= 0





> 0

Policymaker’s inability-based popular discontent

and Social fatigue

Policymaker’s inability-based popular discontent

and Social readiness





= 0

Policymaker’s ability-based popular neutrality

or Policymaker’s inability-based popular neutrality

and Social fatigue

Policymaker’s ability-based popular neutrality

or Policymaker’s inability-

based popular neutrality

and Social readiness

Tab 3. Popular discontent and social fatigue

To reduce the number of parameters, we introduce new control variables, that is, ()

() and ()(), and new parameters, that is, 

 and 

. The optimal

control problem then writes as:

Min(),(),(),(),() =1(()

+())+211()

+22()

+





33()

+()

+()

d+1(()

+())(),

= 1,2,3, with either (()0, ()= 0) or (()= 0, ()0), such that:

() = ()(() + ()), (0)=> 0 (1)

13 In the context of the Covid-19 epidemic, countries where ill-prepared and weakly benevolent policymakers

underestimate popular discontent or fallaciously assumes popular neutrality are, for example, Brazil (e.g.,

Phillips and Phillips, 2020) and the USA (Karson, 2020; Russonello, 2020).

()

=(1 ())[(1())() + (1())(() + ())](1 

())(()()

)(+)(), (0)=0 (2)

() = ()(++()())() + (), (0)=0 (3)

() = (+()())()(+)(), (0)=0 (4)

() = (), (0)=0 (5)

Table 4 summarizes the main variables and parameters of the model.

() > 0

Number of (susceptible and infected) individuals at t

() 0

Number of asymptomatic cases at t

() 0

Number of symptomatic cases at t

() 0

Number of convalescent cases at t

()0

Level of health infrastructures regarding effective treatment procedures of the disease at t

() 0

Health infrastructure efforts at 

()0

Mobility restriction effort toward asymptomatic cases and susceptible individuals at , ()1

()0

Isolation effort toward symptomatic and convalescent cases at , ()1

()0

Social interaction securing effort at , ()1

()0

Treatment effort at  toward symptomatic case



0

Overall social cost

Popular discontent rate parameter

0

Social fatigue rate parameter

> 0

Contagion rate parameter

> 0

Transition rate parameter from asymptomatic to resilient state

> 0

Transition rate parameter from asymptomatic to symptomatic state

> 0

Death rate parameter of symptomatic cases

> 0

Spontaneous convalescence rate parameter of symptomatic cases

> 0

Relapsing rate parameter of convalescent cases

> 0

Resilience rate parameter of convalescent cases

> 0

Fraction of resilient asymptomatic cases that reach immunity against the disease

> 0

Fraction of resilient convalescent cases that reach immunity against the disease

> 0

Marginal cost of asymptomatic cases

> 0

Marginal cost of symptomatic cases

> 0

Marginal cost of convalescent cases

> 0

Marginal cost of death of symptomatic cases

> 0

Marginal cost of health infrastructures effort

> 0

Marginal cost of mobility restriction effort

> 0

Marginal cost of isolation effort

> 0

Marginal cost of social interactions securing effort

> 0

Marginal cost of treatment effort

> 0

Marginal salvage cost asymptomatic cases at T

> 0

Marginal salvage cost of symptomatic cases at T

> 0

Marginal salvage cost of convalescent cases at T

> 0

Marginal salvage cost of the terminal number of deaths at T

> 0

Marginal salvage value of the terminal health infrastructures level at T

[0, ]

Time, <

Tab. 4. Main variables and parameters

3. Analytical derivations

Suppressing the time index for convenience, the present-value Hamiltonian writes:

=( +)

+

+





(++

)++{(1)[(1)+(1)(+)](1)( )

(+)}+[(++)+]+[(+)(+)] (6)

where (), = 1. .5, are costate variables, with either ()0 and ()= 0 or

()= 0 and ()0.

Because the numbers of asymptomatic, symptomatic and convalescent cases all have a

positive marginal influence on the social cost, their implicit price should both be non-positive

(Léonard, 1981), i.e., ,,0. Conversely, because a greater level of health

infrastructures reduces the number of symptomatic cases by moving them to the category of

convalescent cases, its implicit price should be non-negative, i.e., 0.

We first consider the case of secured social interactions, i.e., ()= 0 and ()0, and

focus on interior solutions. Necessary conditions for optimality are:

=

 (7)

=( )()( ) 

[()( )] (8)

=()( ) ( )

[()( )] (9)

=() (10)

where the costate variables are given by:

==(1)[+(1)(+)], ()= 0 (11)

==(), ()= (12)

==++{+(1)[2 +(+

)]}, ()= (13)

==(+)+(+)(+)()(1)[

2 (2 +)], ()=(+) (14)

==++()+(1)[2 

(2 +)], ()= (15)

Note that the effort in health infrastructures in (7) has a feasible value, ()0, if the

implicit benefit from a marginal increase in the health infrastructures level is non-negative,

0, as suggested above. On the other hand, the treatment efforts in (10) have a feasible

value, ()0, if the implicit cost due to a marginal increase in the number symptomatic

cases is greater than the implicit cost due to a marginal increase in the number of convalescent

cases, ()(), which is intuitive.

Lemma 1. Under secured social interactions, the Hamiltonian function attains a (local)

maximum at the point where the first order optimality conditions hold, if and only if:



(+)( )> 0

Proof. Appendix 2.

By substituting the optimal control expressions (7)-(10) into (2)-(5) and (11)-(15), we obtain

the state-costate two-point boundary value problem (TPBVP), where the state variables are

fixed at = 0 and the costate variables are fixed at = (See Appendix 2).

We now turn to the case of mobility restrictions, i.e., ()0 and ()= 0, and focus on

interior solutions. Necessary conditions for optimality are:

=

 (16)

=[()( )]( )( )

( )[()( )] (17)

=[()( )]()( )

( )[()( )] (18)

=() (19)

where the costate variables are given by:

=(1)[(1)+(1)(+)], ()= 0 (20)

=(), ()= (21)

=++{++(1)[(2)(+)(

2)(1)]}, ()= (22)

=(+)+(+)(+)()(1)[(1)(

 )(1)], ()=(+) (23)

=++()+(1)[(1)( )

(1)], ()= (24)

Lemma 2. Under mobility restrictions, the Hamiltonian function attains a (local) maximum at

the point where the first order optimality conditions hold, if and only if:

( )[2+(+)( )] > 0

Proof. Appendix 3.

The respective two-point boundary value problems involved by (1)-(5) and (11)-(15) along

with (7)-(10), on the one hand, and (20)-(24) along with (16)-(19) are highly non-linear and

cannot be solved explicitly. We therefore resort to numerical methods.

4. Numerical study

4.1. Grid of analysis

Our primary objective is to determine whether mobility restrictions or secured social

interactions are more effective for the control of an epidemic, contingent upon the respective

effects induced by growing popular discontent and social fatigue. In this regard, we conduct a

numerical analysis based on the epidemiological parameter values calibrated on the Covid-19

disease in Section 2 (Table 5).























0.2977

0.4

0.2

0.00248

0.125

0.01

0.07143

0.25

0.5

Tab. 5. Epidemiological parameter values

In addition, we use the set of values in Table 6 for the following parameters.



























100

300

200

500

1000

200

450

50000

1000

3000

2000

180

Tab. 6. Set of parameter values

The parameter values respectively used for the salvage cost of asymptomatic, symptomatic

and convalescent cases are supposed to be 10 times those used for the corresponding

instantaneous cost with, given the ranking assumed beforehand, =300 >=200 >

=100 and =3000 >=2000 >=1000. The cost parameters of efforts in

health infrastructures and treatment are respectively =500 and =50000, while those

related to efforts in mobility restriction, securing social interactions and isolation efforts are,

consistent with previous assumptions, such that: =1000 >=450 >=200. The

salvage value of health infrastructures is set to =10. Finally, the time frame beyond which

the overall economic situation could become unmanageable for the policymaker is set at =

180 days.

The remaining parameter values are varied according to the cases shown in Table 3,

depending on popular discontent and social fatigue. Popular discontent is assumed to be

caused by a policymaker’s inability to counter the disease, which is reflected in the

policymaker’s ill-preparedness and lack of benevolence. Preparedness is characterized by the

initial level of health infrastructures, that is, =(1, 5). These initial values, respectively,

correspond to the policymaker’s ill-preparedness, = 1, and well-preparedness, = 5, to

counter the disease. The policymaker’s benevolence depends on her/his valuation of the

current and terminal marginal costs of a disease-related death,  and . We assume that

(,)=(50000,500000) and (,)=(300000,3000000) respectively characterize the

policymaker’s lack of benevolence and reputation for benevolence. The gap between the low

and high values of  and , respectively, was defined as a report from 1 to 6, in order to get a

symmetric distance of  to , one from below and the other from above, that is, 300000 

0.00248/300 300/(50000 0.00248)2.4.

Both factors—policymaker’s preparedness and benevolence—are supposed to affect the

popular discontent rate, , which is set at = 0 for effective or putative popular neutrality

and = 0.002 for popular discontent. Because we use a daily time frequency, the impact of

a discontent rate value of = 0.002 would cause the current disease-related cost to double

after one year which, given the economic, social, human and psychological repercussions of

the disease, seems reasonable. Finally, the social fatigue is supposed to increase the cost of

leveraging the non-therapeutic instruments at the rate , which is set at = 0 for social

readiness and = 0.002 for social fatigue. The values of  and  respectively for popular

discontent and social fatigue are symmetrical for an assessment of comparable magnitude of

the relative impact of the two effects on the management of the disease.

Overall, seven scenarios are compared (Table 7) for each of the two non-therapeutic policy

options, i.e., mobility restrictions and secured social interactions, among which the six

scenarios defined in Table 3, as well as the case of an uncontrolled epidemic that combines

ill-preparedness, proper valuation of disease-related costs and popular discontent (UC). We

consider two additional pre-defined scenarios wherein an ill-prepared and weakly benevolent

policymaker underestimates the popular discontent and deploys a control policy fallaciously

based on popular neutrality. In these scenarios, which characterize policymaker’s inability-

based distorted popular discontent and involve either social fatigue (PI/DPD-SF) or social

readiness (PI/DPD-SR), the parameter  does not explicitly affect the state equations. As a

consequence, a control policy fallaciously based on popular neutrality only increases the

overall social cost, whose value explicitly depends on . A comparison of these scenarios

with the seven abovementioned ones will determine whether mobility restrictions or secured

social interactions better mitigates the distortion effect due to a policymaker’s underestimation

of the popular discontent.

Scenarios









Policymaker’s inability-based popular discontent and social fatigue (PI/PD-SF)

50000

500000

0.002

Policymaker’s inability-based popular discontent and social readiness (PI/PD-SR)

50000

500000

0.002

0.0

Policymaker’s inability-based popular neutrality and social fatigue (PI/PN-SF)

50000

500000

0.0

0.002

Policymaker’s inability-based popular neutrality and social readiness (PI/PN-SR)

50000

500000

0.0

Policymaker’s ability-based popular neutrality and social fatigue (PA/PN-SF)

300000

3000000

0.0

0.002

Policymaker’s ability-based popular neutrality and social readiness (PA/PN-SR)

300000

3000000

0.0

Uncontrolled case (UC)

300000

3000000

0.002

0.0

Tab 7. Discontent vs. inability effect and social fatigue

For the numerical resolution, a time decomposition approach is used (Maimon et al., 1998).

The results are computed with C++ and the feasibility of the solutions generated was not

affected by variations from the baseline values.

4.2. Results

Given the above scenarios, we first investigate the respective effects of policymakers’

inability to counter the epidemic, the related popular discontent and social fatigue for each

separate policy option, i.e., mobility restriction and secured social interactions. We compare

the influence of the previously cited effects on therapeutic and non-therapeutic effort patterns

and on the evolution of the epidemic. We then establish a ranking related to the magnitude of

the policymaker’s inability to counter the epidemic, popular discontent and social fatigue

effects on each separate policy option. Next, using performance criterions of effectiveness and

efficiency, we compare the two policy options, that is, mobility restriction and secured social

interactions, to determine which of them best mitigates policymakers’ inability to control the

epidemic, popular discontent and social fatigue, as well as the distortion effect due to

policymakers’ underestimation of popular discontent.

a. Mobility restriction b. Secured social interactions

Fig. 5. Treatment capability over time under mobility restriction and secured social interactions policy options

In Figure 5, the evolution of treatment capability is shown to increase rapidly at first and then

to decrease slowly, with a peak that coincides with the peak of asymptomatic cases, as shown

in Figures 9a-b. That is, treatment capability and the number of new infections are mutually

complementary. Regardless of the non-therapeutic policy option chosen, significantly greater

treatment capability is developed with a well-prepared and highly benevolent policymaker

than an ill-prepared and weakly benevolent policymaker. That is, the inability effect, i.e.,

without popular discontent, has a predominantly negative impact on the treatment capability

level. On the other hand, the discontent effect leads to a slightly greater development of

therapeutic interventions. In contrast, social fatigue leads to the development of less treatment

capability than social readiness during an initial relatively short time interval, then an increase

over approximately half of the remaining time horizon, and finally less afterward. The fatigue

effect is clearly less obvious than the inability and discontent effects, though it is more

marked with a well-prepared and highly benevolent policymaker than an ill-prepared and

weakly benevolent policymaker, in particular under the secured social interactions policy

option. The question is how the rate of cumulative treated cases, defined as ()=

()()()



()



, is affected by the above patterns.

a. Mobility restriction b. Secured social interactions

Fig. 6. Rate of treated cases over time under mobility restriction and secured social interactions policy options

Figure 6 shows that the rate of treated cases is substantially higher with a well-prepared and

highly benevolent policymaker than with ill-prepared and weakly benevolent policymaker.

Here also, the policymaker’s inability effect reduces the rate of treated cases to the lowest

level, but popular discontent slightly mitigates this tendency. Therefore, popular discontent is

to some extent therapeutically beneficial to symptomatic cases. In comparison, the social

fatigue effect is much less perceptible, except with a well-prepared and highly benevolent

policymaker under the secured social interactions policy option. In this case, social fatigue

leads to the treatment of slightly fewer symptomatic cases than social readiness during a first

relatively short time interval, then a little bit more during the whole remaining time horizon.

a. Instantaneous mobility restriction efforts b. Instantaneous secured social interactions efforts

c. Cumulative mobility restriction efforts d. Cumulative secured social interactions efforts

Fig. 7. Mobility restriction and secured social interactions efforts over time

As for non-therapeutic instruments (Figs 7-8), a straightforward interpretation of the mobility

restriction efforts in Fig 7a is that () corresponds to the proportion of susceptible

individuals that are protected from contagion at day  thanks to the fact that they have no

mutual interaction with a similar proportion of asymptomatic cases. In comparison, the effort

to secure social interactions () in Fig 7b is interpreted as the factor of contagion risk

reduction throughout the interaction between infected cases and susceptible individuals at day

. Regardless of the policy option chosen, the non-therapeutic instruments are more heavily

leveraged by well-prepared and highly benevolent policymakers than by ill-prepared and

weakly benevolent policymakers, especially under social readiness. The inability effect is by

and large dominant, and even more so under secured social interactions than the mobility

restriction policy option (Figs 7c-d and 8c-d). In contrast, the discontent effect is also

significant but less under secured social interactions than the mobility restriction policy option

(Figs 7c-d and 8c-d). Mobility restriction as secured social interaction efforts are initially high

but rapidly kept at a lower level for one third of the time interval then increased and finally

slowly lowered during the remaining time horizon (Figs 7a-b). In contrast, from an initial low

level, isolation efforts are rapidly set at a higher though decreasing level for one third of the

time interval, then steadily increased and finally decreased during the remaining time horizon

(Figs 8a-b).

a. Instantaneous isolation efforts under MR b. Instantaneous isolation efforts under SSI

c. Cumulative isolation efforts under MR d. Cumulative isolation efforts under SSI

Fig. 8. Isolation efforts over time under mobility restriction (MR) and secured social interactions (SSI) policy

options

To some extent, these cyclical paths seem dampened for mobility restriction and secured

social interaction efforts and amplified for isolation efforts. All non-therapeutic instruments

follow an increasing trend at the peak of asymptomatic cases (Figs 9a-b), that is, the decrease

of the number of asymptomatic cases over time is due to heavier non-therapeutic instrument

leveraging. An important result here is that mobility restriction should never be set at its

maximum value (Fig. 7a); only partial mobility restriction prevails, but over most of the

planning horizon. Similarly, the securing of social interactions can never be complete over

time (Fig. 7b). Finally, it is not recommended to isolate all but a fraction of just the confirmed

cases over time. In contrast with therapeutic effort patterns, social fatigue has now a

significant negative impact both under the mobility restriction and secured social interactions

policy options, but more in the latter case. Put differently, social readiness can mitigate the

effect of popular discontent on mobility restriction efforts in the policymaker’s inability

scenario (Fig. 7c), and even more notably on secured social interaction efforts (Fig. 7d) and

isolation efforts (Figs 8c-d). In the latter cases, social fatigue has a greater influence than

popular discontent.

PA/PN-SR

PA/PN-SF

PI/PD-SR

PI/PD-SF

PI/PN-SR

PI/PN-SF

 









()

2d





 ()

d





7.51% 5.23% 4.96% 3.96% 3.70% 2.71%

SSI 13.56% 7.91% 7.52% 5.77% 6.48% 4.63%

 ()

2d



 MR

13.79% 9.07% 9.60% 7.26% 9.90% 7.14%

SSI

16.98%

9.74%

10.03%

7.47%

10.42%

7.37%

()

2d



 MR

36.20% 39.35% 44.91% 46.64% 45.15% 47.09%

SSI 32.03% 37.86% 43.36% 45.60% 43.43% 45.97%

()

2d



 MR

42.50% 46.34% 40.53% 42.14% 41.26% 43.06%

SSI 37.42% 44.49% 39.10% 41.16% 39.67% 42.03%

Total

100%

SSI

100%

Tab. 8. Structure of control-related costs under mobility restriction (MR) and secured social interaction (SSI)

policy options

Table 8 presents the structure of control-related cumulative costs under mobility restriction

and secured social interactions policy options, that is, respectively, the cumulative non-

therapeutic costs of mobility restriction, ()

d



, secured social interactions,

()

d



, isolation of symptomatic and convalescent cases, ()

d



, and the

cumulative therapeutic costs to health infrastructures, ()

d



, and treatment, ()

d



.

Among the therapeutic instruments, the policymaker’s ability scenario leads them to put

relatively more effort into treatment than in health infrastructure while the policymaker’s

inability scenario does exactly the opposite. That is, the inability effect leads to privileging

health infrastructure to the detriment of treatment efforts to make up for the policymaker’s ill-

preparedness. Neither the discontent nor the fatigue effect mitigates this tendency. Relatedly,

the policymaker’s inability scenario leads to less preventive and more curative policy than the

policymaker’s ability scenario, and vice versa. That is, the inability effect reflects less use of

non-therapeutic efforts and a greater leveraging of therapeutic efforts to compensate for the

policymaker’s ill-preparedness. Though it is slightly more prevention-oriented, popular

discontent does not significantly affect these policy differences. In contrast, social fatigue

leads to a less preventive and more curative policy in all scenarios. Among the non-

therapeutic instruments, isolation efforts is systematically more intensively activated than

either mobility restriction or secured social interactions, regardless of inability, discontent and

fatigue effects. Isolation of manifestly infectious cases should be treated as the main non-

therapeutic action.

a. Asymptomatic cases under MR b. Asymptomatic cases under SSI

c. Active cases under MR d. Active cases under SSI

Fig. 9. Asymptomatic and active cases over time under mobility restriction (MR) and secured social interaction

(SSI) policy options

As a consequence of the policies characterized above, the evolution of the epidemic over time

is represented in terms of the number of asymptomatic and active cases in Figure 9.

Compared to the uncontrolled epidemic, the peak of asymptomatic cases is delayed by more

than two weeks and flattened in the best case (i.e., social readiness) by almost 40% in the

policymaker’s ability-based popular neutrality scenario (Figs 9a-b). The peak in this scenario

comes about 10 days later and is 30% lower than in the policymaker’s inability-based popular

discontent scenario. The inability effect on asymptomatic cases is clearly detrimental. By

comparison, the discontent effect is also detrimental but less discernibly. Finally, the fatigue

effect is more visibly negative under secured social interactions than under a mobility

restriction policy, especially in the policymaker’s ability-based popular neutrality scenario.

Similar trends are observed for the number of active cases (Figs 9c-d) where the magnitude of

the inability, discontent and fatigue effects is lower.

Regardless of the policy option, the rate of cumulative infections, defined by ()=

(13())(11())1()+(12())2()+3()(11())(()()

)



0, is

significantly lower in the policymaker’s ability-based popular neutrality scenario than in the

uncontrolled model (Figs 10a-b). Also, the inability effect is prevalent while the discontent

effect seems rather limited. The fatigue effect is mostly prominent in the policymaker’s

ability-based popular neutrality scenario, in particular under the secured social interactions

policy option. In the other scenarios, the discontent effect is slightly more perceptible than the

fatigue effect. These results are somehow similar to the previous ones.

a. Cumulative infection rate b. Cumulative infection rate

Fig. 10. Cumulative infection rate over time under mobility restriction and secured social interactions policy

options

To assess the mortality rate of the epidemic, the cumulative deaths, measured as ()=

()



, and the case fatality ratio, given by ()=()





()



, are represented in

Figure 11. Under both policy options, the cumulative deaths observed during the last period of

the uncontrolled epidemic are reduced by approximately three quarters in a scenario with a

well-prepared and highly benevolent policymaker, and by almost two thirds in a scenario with

an ill-prepared and weakly benevolent policymaker. Regarding the CFR, the last period

mortality observed in an uncontrolled epidemic (2.2%) is reduced by almost three quarters in

the scenario with a well-prepared and highly benevolent policymaker with a CFR of

approximately 0.6%, and by more than one half in a scenario with an ill-prepared and weakly

benevolent policymaker with a CFR of approximately 1%. Though not extreme, the inability

effect on mortality is strong with a 70% increase in the CFR. On the other hand, the discontent

effect slightly reduces mortality. In other words, the discontent effect slightly mitigates the

inability effect on mortality. This is due to the greater treatment capability popular discontent

leads to, compared to popular neutrality. Popular discontent reveals itself to be somehow

salutary in terms of mortality. Finally, the fatigue effect operates negatively mainly for the

cumulative deaths in the policymaker-ability scenario under the secured social interactions

policy option. The reason lies with the far more effective leveraging of non-therapeutic

instruments with this policy option. In the other scenarios, social fatigue does not have a

particularly perceptible effect.

a. Cumulative deaths under MR b. Cumulative deaths under SSI

c. Case fatality ratio under MR d. Case fatality ratio under SSI

Fig. 11. Cumulative deaths and case fatality ratio over time under mobility restriction (MR) and secured social

interactions (SSI) policy options

Regarding the overall social cost, it is represented in Figure 12, however the salvage value is

not included. The lowest control policy-driven path is related to the policymaker’s inability-

based popular neutrality scenario, while the higher control policy driven path is that of the

policymaker’s ability scenario. In the latter case, the reason clearly lies with the greater

control-related costs compared to the other scenarios, which makes the inability effect have,

in absolute value, a cost-reducing property. Nevertheless, the discontent effect considerably

reduces the gap between the overall social costs respectively induced by the policymaker’s

inability and the policymaker’s ability scenarios. That is, the discontent effect significantly

mitigates the inability effect on the overall social cost. In other words, a policymaker’s

preparedness and benevolence are not economically excessively expensive. Even if the

population can reasonably be supposed to be docile, i.e., popular neutrality, a policymaker’s

ill-preparedness and lack of benevolence can hardly be motivated by economic reasons, given

the discrepancy in terms of mortality with the policymaker’s ability scenario. Finally, the

fatigue effect on the overall social cost is almost imperceptible.

a. Mobility restriction b. Secured social interactions

Fig. 12. Overall social cost over time under mobility restriction and secured social interactions policy options

To better visualize the differences of relative magnitude between the inability, popular and

fatigue effects, we represent the cumulative rate of prevented infections, defined as ()=

1(())(())()(())()()(())(()()

)



()()()(()()

)



, and the related

cost, that is, the cost of prevented infections, given by ()=()

() in Figure 13. The first

indicator characterizes the degree of effectiveness in reducing the disease impact associated

with each scenario, while the second indicator reflects the corresponding degree of efficiency.

The policymaker’s ability scenario is clearly the most effective and the most efficient in

preventing infections (Figs 13a-c and 13b-d). Then the policymaker’s inability scenario

follows with an important discrepancy: the inability effect leads the prevention of fewer

infections at a higher cost. Though the discontent effect generates prominent transient

ineffectiveness, it finally vanishes during the last third of the time horizon (Figs 13a-b).

However, it results in a greater cost per infection prevented (Figs 13c-d). Overall, the

discontent effect on prevention is neutral on effectiveness but generates more inefficiency.

Finally, the fatigue effect lowers effectiveness, notably in the policymaker’s ability scenario

with the secured social interactions policy option. In addition, it gives rise to less efficiency.

In the policymaker’s inability scenario, the fatigue effect even outperforms the discontent

effect. Finally, the fatigue effect results in both less effectiveness and efficiency.

a. Rate of prevented infections under MR b. Rate of prevented infections under SSI

c. Cost of prevented infections under MR d. Cost of prevented infections under SSI

Fig. 13. Rate and cost of prevented infections over time under mobility restriction (MR) and secured social

interactions (SSI) policy options

Figure 14 shows the cumulative rate of saved lives, defined as ()= 1 

()



 

()



 , and the associated cost, the cost of saved lives, given by ()=

()

(). The inability effect on saved lives has a doubly detrimental impact, that is, it’s

substantially detrimental to effectiveness and modestly detrimental to efficiency (Figs 14a-b

and 14c-d). On the other hand, the discontent effect on saved lives slightly promotes

effectiveness during the second half of the planning horizon, but also, more significantly,

inefficiency. Finally, the fatigue effect on saved lives results in both lower effectiveness and

efficiency, though more notably in the policymaker’s ability scenario under secured social

interactions policy.

a. Rate of saved lives under MR b. Rate of saved lives under SSI

c. Cost of saved lives under MR d. Cost of saved lives under SSI

Fig. 14. Rate and cost of saved lives over time under mobility restriction (MR) and secured social interaction

(SSI) policy options

Table 8 summarizes the impact of inability, discontent and fatigue effects on policy

instruments and outcomes under mobility restriction and secured social interactions policy

options.

Inability effect

Inability & discontent effect

Fatigue effect

Therapeutic instruments

Mobility restriction

---

Secured social interactions

---

Non-therapeutic instruments

Mobility restriction

---

Secured social interactions

---

Epidemic activity

Mobility restriction

Secured social interactions

Mortality

Mobility restriction

+++

Secured social interactions

+++

Social cost

Mobility restriction

Secured social interactions

Prevented infections

Mobility restriction

Secured social interactions

---

Saved lives

Mobility restriction

---

Secured social interactions

---

Cost of prevented infections

Mobility restriction

+++

Secured social interactions

+++

Cost of saved lives

Mobility restriction

Secured social interactions

Tab 8. Impact of inability, discontent and fatigue effects on policy instruments and outcomes under mobility

restriction and secured social interactions policy options

An important observation that emerges is that a policymaker’s inability has the single most

detrimental impact. Popular discontent both mitigates the ineffectiveness and amplifies the

inefficiency engendered by the policymaker’s inability to save lives. On the other hand, social

fatigue generates inefficiency in saving lives similar to the results of a policymaker’s

inability. Overall, a policymaker’s ill-preparedness and lack of benevolence has the most

detrimental effects on the control of an epidemic. To some extent, these effects can

nevertheless be mitigated by the popular discontent induced by the policymaker’s inability,

but at a higher cost. The social fatigue that the leveraging of non-therapeutic instruments

entails is also detrimental but much less than a policymaker’s inability. In other words, a

policymaker’s inability is more problematic than social fatigue in the control of an epidemic.

However, social fatigue is less apparent in the policymaker’s inability scenario than in the

policymaker’s ability scenario.

We now turn to the comparison between mobility restrictions and secured social interactions

policy options to determine which most effectively mitigates a policymaker’s inability,

popular discontent, social fatigue and the distortion effect due to a policymaker’s

underestimation of popular discontent

a. Policymaker’s ability-based popular neutrality b. Policymaker’s inability-based popular discontent

c. Policymaker’s inability-based popular neutrality d. Policymaker’s inability-based distorted popular discontent

Fig. 13. Overall social cost over time under mobility restriction and secured social interactions policy options

In terms of overall social cost, the difference between the two policy options is tiny (Figs 13a-

b-c-d), though a bit less in the policymaker’s ability scenario with social readiness, in which

case securing social interactions is briefly less costly. Overall, mobility restriction and

secured social interactions policy options give rise to similar economic performance,

regardless of the presence of social fatigue or not.

a. Policymaker’s ability-based popular neutrality b. Policymaker’s inability-based popular discontent

c. Policymaker’s inability-based popular neutrality. d. Policymaker’s inability-based distorted popular

discontent

Fig. 14. Prevented infections over time under mobility restriction and secured social interactions policy options

In terms of prevented infections, the secured social interactions policy option combined with

social readiness is the most effective of all scenarios (Figs 14a-b-c-d). Though the differences

between the two policy options are strongly contingent upon the presence or absence of

popular discontent and social fatigue, the secured social interactions policy option slightly

surpasses the mobility restriction policy option for the prevention of infections.

a. Policymaker’s ability-based popular neutrality b. Policymaker’s inability-based popular discontent

c. Policymaker’s inability-based popular neutrality. d. Policymaker’s inability-based distorted popular

discontent

Fig. 15. Cost of prevented infections over time under mobility restriction and secured social interactions policy

options

A similar result to the previous one is obtained regarding the related cost criterion: the

secured social interactions policy is generally more efficient than the mobility restriction

policy in preventing infections, including in the scenario where the policymaker

underestimates popular discontent (Figs 15a-b-c-d). That is, the secured social interactions

policy option maintains its superior efficiency in the presence of popular discontent and social

fatigue.

a. Policymaker’s ability-based popular neutrality b. Policymaker’s inability-based popular discontent

c. Policymaker’s inability-based popular neutrality. d. Policymaker’s inability-based distorted popular

discontent

Fig. 16. Rate of saved lives over time under mobility restriction and secured social interactions policy options

In terms of cumulative saved lives, the differences between the two policy options are not

significant, except transiently in the policymaker ability scenario (Figs 16a-b-c-d). That is, the

secured social interaction and mobility restriction policy options exhibit similar effectiveness

for the prevention of deaths.

a. Policymaker’s ability-based popular neutrality b. Policymaker’s inability-based popular discontent

c. Policymaker’s inability-based popular neutrality. d. Policymaker’s inability-based distorted popular

discontent

Fig. 17. Cost of saved lives over time under mobility restriction and secured social interactions policy options

In Figure 17, the secured social interactions policy and the mobility restriction policy options

show similar efficiency in the prevention of deaths in general (Fig. 17a-b-c-d), except in the

policymaker’s ability scenario. In this scenario, the superior efficiency of the secured social

interactions policy is cancelled by the fatigue effect.

Overall, the mobility restriction and secured social interactions policy options perform

similarly in terms of overall social cost and prevention of mortality. However, the secured

social interactions policy is both more effective and efficient in preventing infections,

regardless of the presence or absence of the policymaker’s inability, popular discontent and

social fatigue. These conclusions remain valid for a more widespread epidemic, that is, for an

initially significant incidence rate regarding the asymptomatic cases, i.e., = 0.25, and the

symptomatic cases, i.e., = 0.2.

5. Conclusion

The current Covid-19 crisis has shown that the control of an epidemic disease is dependent

upon three main interfering factors: the policymaker’s ability reflected in its degree of

preparedness and benevolence, the induced popular discontent, and the non-therapeutic

interventions-related social fatigue. In this paper, we formulate a dynamic model that

describes the progression of an epidemic disease and the possible means to counter it with

therapeutic and non-therapeutic instruments over a finite time horizon in the presence of these

interfering factors. This model considers therapeutic instruments, that is, treatment capability,

to be a combination of current treatment efforts and past health infrastructures that combine to

form a cumulative effort, and considers non-therapeutic policy options to be either mobility

restriction or securing social interactions, along with the isolation of manifestly infected

cases.

Based on a methodical calibration of the epidemiological parameters of the model and an

extensive numerical analysis, we compared the health and welfare implications of mobility

restrictions and secured social interactions policy options to determine which was the most

robust with respect to policymaker’s inability, related popular discontent, social fatigue and

policymaker’s underestimation of popular discontent.

It appears that, regardless of the policy option chosen, a policymaker’s ill-preparedness and

lack of benevolence reduces, in absolute value, both therapeutic and non-therapeutic

interventions. In relative terms, policymaker’s inability to handle an epidemic is recognizable

in that it mainly manifests as the building up of health infrastructure to make up for being ill-

prepared. Compared to a well-prepared and highly benevolent policymaker, the disease-

related lethality is considerably higher (by more than two thirds) and only leads to modest

savings in terms of overall social cost (less than 7%). Considering the human stakes, ill-

preparedness and lack of benevolence are not a substantial source of savings and should thus

not be promoted as economically viable approaches in the setup of a control policy of an

epidemic disease.

The popular discontent induced by a policymaker’s inability mitigates her/his effect on

therapeutic interventions and also slightly mitigates the disease-related lethality. In this

regard, as time goes by, popular discontent acts more and more as a substitute for a

policymaker’s preparedness and high benevolence. As for social fatigue, it mostly negatively

affects the non-therapeutic interventions, but has no visible impact on mortality rates.

Compared to an uncontrolled epidemic, the policymaker’s inability exhibits both less

effectiveness and efficiency in preventing infections and saving lives. Popular discontent

considerably improves the lack of effectiveness caused by the policymaker’s inability but not

the lack of efficiency. Expectedly, social fatigue causes both ineffectiveness and inefficiency

in preventing infections and saving lives, but in a more apparent way with a well-prepared

and highly benevolent policymaker than the converse case. Nevertheless, a policymaker’s

inability is more problematic than social fatigue in the control of an epidemic.

Though the mobility restriction and secured social interactions policy options result in quite

similar social costs, the latter more effectively prevents infections. However, the secured

social interaction and mobility restriction policy options are similarly effective and efficient in

saving lives both for emerging and more widespread epidemics. Because it more effectively

minimizes the impact of a policymaker's inability with or without correlated popular

discontent, social fatigue, as well as the loss caused by a policymaker’s disregarding of

popular discontent, the secured social interactions policy option is preferable to the mobility

restriction policy option to counter an epidemic.

Our model could help to give some valuable insights into how to prevent the disaster that is

being observed in some countries. The reality is very complicated, so we are not able to

develop a model that is simple enough to be analytically tractable. Taking that into account,

we have developed a model that can nevertheless be considered a simplified representation of

the current situation that also involves significant international interactions, including

coordination or hostility between countries, unmentioned in the current model. An extension

of this work to multiple countries would significantly raise the number of states and control

variables and make the resolution procedure even more cumbersome, but would provide

valuable insights on the global management of a pandemic.

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