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Published in Philippine Science Letters, Vo. 14, No. 1, pp. 69 - 78

A pivotal restructuring of modelling the control of COVID-19 during and

after massive vaccination for the next few years

Jose B. Cruz, Jr.1, Tirso A. Ronquillo*2, Ralph Gerard B. Sangalang2, Albertson D. Amante2,

Divina Gracia D. Ronquillo2, Janice F. Peralta2, Antonette V. Chua2,

Oliver Lexter July A. Jose2, Raynell A. Inojosa2

1National Academy of Science and Technology, Taguig City, Philippines

2Batangas State University, Rizal Avenue, Batangas City, Philippines

ABSTRACT

This paper presents a new mathematical feedback model to demonstrate how direct

observations of the epidemiological compartments of population could be mapped to inputs,

such that the social spread of the disease is asymptotically subdued. Details of the

stabilization and robustness are included. This is a pivotal restructuring of modelling the

control of corona virus from the current models in use world-wide which do not utilize

feedback of functions of epidemiological compartments of population to construct the inputs.

Although several vaccines have received Emergency Use Authorization (EUA) massive

vaccination would take several years to reach herd immunity in most countries. Furthermore,

the period of efficacy of the vaccination may be approximately one year only resulting in an

unending vaccination. Even during the vaccination, there would be an urgent need to control

the spread of the virus. When herd immunity is reached without feedback control and

vaccination is discontinued, there could be new surges of the disease. These surges of disease

could be prevented in appropriately designed stable feedback models. Moreover, extensive

testing, contact tracing, and medical treatment of those found infected, must be maintained.

KEYWORDS: COVID-19, SARS-CoV-2, pandemic, social spread of disease, feedback control

of corona virus, closed loop models, interventions for spread of corona virus, Batangas State

University engineered closed loop model

1.0 Introduction and background

Starting with a very small outbreak of a coronavirus disease in Wuhan, China in December

2019, the World Health Organization (World Health Organization 2020a), on February 11,

2020, labeled the virus as Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-

2), and the corresponding disease as Corona Virus Disease 2019 (COVID-19). By the same

day, COVID-19 has grown to 43,103 confirmed cases world-wide, 42,708 in China, 395 in

24 other countries, 1,017 deaths in China, and one death for the rest of the world (World

Health Organization 2020b). On March 11, 2020, WHO declared COVID-19 as a pandemic.

By March 5, 2021, the pandemic has reached 115,289,961 confirmed cases and 2,564,560

deaths world-wide; 28,468,736 confirmed cases and 515,013 deaths in the United States; and

535,207 confirmed cases and 12,423 deaths in the Philippines (World Health Organization

2020c).. There are other sources of data (Dong et al. 2020, C3ai DTI 2020a, C3ai DTI

2020b). As of December 1, 2020, there have been new surges of COVID-19 infection, for a

variety of reasons, in many parts of the world, and the pandemic is still raging (Dong et al.

2020).

Several interventions have been employed to mitigate the exponential growth of COVID-19

such as quarantine or lockdown, wearing of masks, physical distancing, and frequent washing

of hands. However, these interventions are unevenly practiced in some countries, and in

many others, they are lifted soon after they “flatten the curve”, leading to unstable behavior

again.

Recently, several pharmaceutical companies have received Emergency Use Authorization

(EUA) for use of vaccines in the last two (2) weeks of 2020, and more companies and more

vaccines are now available in 2021. Initial availability would be limited to health care

providers and other high-risk segments of the population. How should the vaccines be

allocated? (Emmanuel et al. 2020). Allocations can be optimized (Sy et al. 2020). Each

political jurisdiction will decide on prioritization and optimization. For the rest of the global

population, some pharmaceutical companies have announced that more vaccines will be

available by the middle of 2021. These companies are still investigating the duration of

efficacy of the vaccines. For the near term, there is a need to continue the interventions, or

innovative ways to modify the interventions so they will become more effective. It is

expected that massive vaccination will be available later this year and in the next two (2)

years.

Is there further need for modelling the trend of COVID-19? Yes.

(a) If 100,000 persons can be vaccinated everyday, it will take two (2) years to

vaccinate 80% of the Philippine population, assuming vaccinations that require

only one shot.

(b) The duration of efficacy of the various vaccines is not known at this time. If the

efficacy duration is one (1) year, massive vaccination is needed for the foreseeable

future with no end in sight.

(c) During the vaccination period of several years, can the current models used

worldwide be used to mitigate the growth of disease spread? After more than a

year of COVID-19 pandemic, the Philippines and the entire world are still

suffering from the raging disease. A new type of intervention is needed, a pivotal

restructuring of modelling the control of corona virus.

2.0 Review of science models of disease spread

The first mathematical model to describe the spread of a disease was published in 1927

(Kermack and McKendrick 1927). The basic idea in the model is to divide the population into

epidemiological compartments. In the Kermack and McKendrick model, the compartments of

population are: Susceptible ( ), Infected ( ), and Removed ( ). The rates of change with

respect to time of and , are modeled as functions of and , respectively. This

model has been adopted for modeling the spread of COVID-19, with several layers of sub-

societies, and each layer with more compartments beyond and , such as Exposed ( ),

Hospitalized ( ), and further subdivision of into Symptomatic and Asymptomatic (Chen et

al. 2020). Others have further analyzed the Wuhan outbreak (Kucharski et al. 2020, Ndairou

et al. 2020, Saad-Roy et al. 2020, Britton et al. 2020). SARS-CoV-2 is known to mutate, and

mutation of viruses during propagation has been studied (Eletreby et al. 2019). There are

updates available for the progression of COVID-19 (World Health Organization 2020c)

including a science model with the updates (Dong et al. 2020). The effect of virus mutation,

on therapeutics for COVID-19, has been studied also (Hou et al. 2020), and it appears that the

mutation D614G propagates faster than the original SARS-CoV-2. They state that “current

vaccine approaches directed against the WT (wild-type) spike should be effective against the

D614G strains.” (Hou et al. 2020).

In the only paradigm in use today, interventions are deployed to change the environment,

resulting in changing direction of observations, and subsequent revised mathematical models.

When the interventions are relaxed or removed altogether, subsequent observations will show

that the disease spread will increase exponentially and the process repeats.

The main purpose of this paper is to provide a new engineered closed loop model for

stabilizing systems such as a model for the social spread of disease. As in existing science

models, the engineered closed loop model in this paper addresses the macro-scale level of

modeling the spread of a disease, not necessarily COVID-19. The simplest science model is

considered to modify into an engineered closed loop feedback system to focus on the

advantages of closed loop models.

2.1 Review of SIR science model

The starting point in the development of an engineered closed loop model with feedback of

observation in Section 3 is the Kermack and McKendrick model (1927). Figure 1 shows a

diagram of the Science Model. There are three differential equations but only two are

independent because of the constraint , where = Susceptible, = Infected,

= Removed and = Population.

Normalizing the compartments of , , , , the constraint becomes

. The rate of change with respect to time of s and the rate of change of

with respect to time are the postulated models

(1)

(2)

where is the infectivity rate and is the recovery rate.

The rate of change of with respect to time follows from the constraint,

(3)

(4)

2.2 Review of the SIFD science model

In the SIR model, the compartment is divided into two sub-compartments, and , where

is the number of persons that are Free of Virus but possessing antibodies, and is the

number of persons that are Deceased. As in the SIR model, and , respectively, are

normalized by dividing each by , yielding

(5)

Figure 1 still represents the science model. The rate equations for and , respectively,

are the proposed behavioral postulates as in (Kermack and McKendrick 1927)

(6)

(7)

.

(8)

The rate equation for is obtained from the constraint:

(9)

yielding Equation (4).

Observations are used to calibrate the parameters and . Analysis of the mathematical

models leads to obtaining conditions that cause exponential growth, suggesting how

instability can be avoided. A typical intervention in science models is to change the

environment, such as modification in the use of (a) quarantine or lock down, (b) face masks,

(c) physical distancing, and (d) contact tracing. These modifications in interventions can

change the calibrated values of and so that stability can be achieved, after several

cycles of modifications in the environment. It should be noted that the modifications in

environment are carried out first and then the science model is constructed. However,

experimental assessment of the effectiveness of the model is carried out after it is

constructed. Thus by increasing interventions the resulting system could become stable, but

removing the intervention would eventually make the system unstable.

3.0 Pivoting from science model to Batangas State University engineered feedback model

There are two steps in the transition. The first step is to create explicit inputs to the science

model. The second step is to construct a mapping from the outputs to the inputs of the science

model.

3.1 Creating the Batangas State University input-output model

The first step in pivoting from a science model to an engineered feedback model is to

explicitly create inputs to the science model as indicated in Figure 2. In the science model,

the environment could be regarded as input but it is not treated as an external signal. There

are intrinsic benefits to creating models with inputs and outputs (Tan et al. 2018). In this

paper we create two inputs to the SIFD science model.

As in the science model without inputs, the rate equations, for and , are postulated. The

rate equation for is derived from the constraint in Equation (9) where the quantities and

are inputs.

(10)

(11)

(12)

and Equation (8). Equation (10) is a modification of the Equation (6) and it is one of the

equations for the input – output model. Equation (12) is a modification of Equation (7) and it

is a second equation for the input – output model. Equation (8) of the SIFD model is retained

as a third equation for the input – output model. The fourth equation for the model, Equation

(11), is obtained from Equation (9),

and Equations (8), (10) and (12). Thus, Equations (8), (10) and (12) are modelling statements

similar to the science model, Equations (6), (7) and (8), except that in the Batangas State

University model, inputs z1 and z2 have been added.

The input , when scaled up to *N, is the number of persons per day that we plan to

change the decrease in the susceptible rate per day by. It has a simultaneous partial effect

of opposite change in the infected per day.

The input when scaled up to *N is the additional increase in the number of persons per

day that are free of virus and have antibodies in them, per day. It has a simultaneous partial

effect of reducing the infected per day.

The Batangas State University input-output model is dynamic and nonlinear.

3.2 Creating the Batangas State University pivotal engineered feedback model

Figure 3 shows the diagram of the closed loop system. One of the key advantages of adding

feedback to an input – output system is the capability to stabilize the entire system (Åström

and Murray 2020, Albertos and Mareels 2010, Cruz 1971, Dorf and Bishop 2004 and Kuo

and Golnaraghi 2002). The design of the two inputs of the Batangas State University input-

output model in two-stages. The input is divided into two parts:

(13)

where

The value of in the science model is usually uncertain or variable but Ba is chosen to match

a specific value, . The value of Ba may be chosen adaptively based on a sequence of

estimates of β (Kanellakopoulos et al. 1991) but for simplicity, the value of β* is chosen only

once. In the simulations, it is assumed that there are four different values of : 0.20, 0.23,

0.25 and 0.28. Choosing Ba = β* = 0.20 and , the engineered model for

becomes linear as indicated in Equations (8), (14), (15), (16)

(14)

(15)

(16)

The quantities and are represented by , and

The quantities , and are desired constant asymptotes where is

always set equal to

The first three linear differential equations can be written in matrix form as

(17)

(18)

(19)

It can be readily verified that any value of the state is reachable and is controllable

(Åström and Murray 2020). Denoting the vector input by

(20)

where

.

(21)

The three eigenvalues of can be specified arbitrarily. In particular, the eigenvalues

can be set to be real and negative.

For illustrative purposes, the three eigenvalues are chosen as

, the parameter Ba = 0.20 and the six coefficients in the

matrix K as stated earlier. For simulation, the values for are: 0.20, 0.23, 0.25 and 0.28.

The matrix, K, is calculated in MATLAB Control Toolbox using the command

K = place(A,B,p).

(22)

Based on Equations 13, 20, 21, z1N=Basi and Ba= β* the final structure of the Batangas State

University engineered nonlinear closed loop model equations are

(23)

(24)

(25)

(26)

It is assumed that β remains in a certain range that includes β*. Summarizing, the Batangas

State University feedback control is

(27)

where Ba = β* and K is the matrix defined in Eq 21 under the condition that β = β*. β*is a

design parameter chosen as one of the possible values of β such as the minimum value of β.

Earlier, β* was set to 0.20.

3.2.1 Setting the target asymptotic values, system parameters and initial conditions

Using the Philippine population: 109,581,078 and data from the Philippine Department of

Health, =32,031, =10,136, = 467,720, as of January 22, 2021 (COVID 19

Update2021) and the constraint: Initial values: s0 = 0.995347,

i0 = 2.923 x 10-4 , f0 = 0.004268, d0 = 9.249772 x 10-5. The initial values for and

are , respectively.

Note that the constraint is given by If vaccines are available, set to be

quite high, for example . Set , = 1.095 x 10-4. Thus, = 0.4 - 1.095 x

10-4 = 0.39989. Based on available data, the other system parameters were estimated to have

values of γ = 0.20 and δ = 0.02.

3.3. Simulations

Figures 4, 5, 6, and 7 display results when feedback is present.

In Figures 4 and 6, the plots for S and F , respectively, it is not possible to visually distinguish

the curves for four different values of , displaying not only stability, but also robustness as

well. In Figure 5, the plot of the infected compartment of Philippine population, , shows the

four projections for different values of . Note that the scale of and D, in Figures 5 and 7 is

in ten thousands. Thus, and hardly affect and because , where

and are in ten million.

Figures 8 and 9 show the feedback z1*N and z2*N with initial values of 3.2 million people

per day. The simulation is for illustrative purpose only and in an actual application the

eigenvalues for pole placement may have to be increased resulting in a slower decay to match

the capacity for massive vaccination. Figures 10 and 11 provides details of the components

of z1*N.

Figures 12 and 13, show plots of I and D respectively, when feedback is not present, z1 = 0

and z2 = 0. Notice that in Figure 12, when β = 0.28, the maximum value of I is 4.8 million out

of 108 million population. For the other values of β, the values of I in Figure 12 are also

unacceptably large, except for β = 2.0. Figure 13 shows the graphs of mortality for different

values of β which are also unacceptably large, except for β = 2.0. In contrast, when feedback

control is applied (z1 and z2 are present), Figures 4 to 11 indicate dramatically better results.

3.4 Relationship of controls to interventions

Vaccination intervention is related to control z2. The number of vaccinations need to be

counted and related to the desired increase provided by z2. If z2 is greater than the maximum

capacity for vaccinations the maximum capacity would be used otherwise the eigenvalues

need to be adjusted. The control z1 is related to the intervention of wearing mask, social

distancing and quarantine. If the control z1 is greater than the maximum requirements of

wearing mask, social distancing and quarantine, the maximum could be utilized or the

eigenvalues could be adjusted. In an actual application, iterative adjustments may be needed.

3.5 Stabilization and robustness properties of Batangas State University engineered closed

loop model

The principal advantage of utilizing a closed loop model that includes a feedback mechanism

for mapping the output to the input of the systems is the possibility of stabilization (Cruz

1971, Åström and Murray 2020, Åström and Kumar 2014), and increasing the stability

margin. The Batangas State University engineered closed loop model is designed to be stable.

There is extensive literature on stabilization available (Albertos and Mareels 2010, Dorf and

Bishop 2004 and Kuo and Golnaraghi 2002), for example. Simulations show that the model

remains stable for a range of values of . Furthermore, the simulations show that the outputs

are robust against variations in the value of (Cruz and Perkins 1964, Cruz et al. 1981a,

Cruz et al. 1981b, Freudenberg et al. 1982).

4.0 Comparison of interventions in science models and engineered closed loop models

4.1 Science models

The prevailing method for the study of disease propagation, such as the spread of COVID-19

is through the construction of mathematical models. Typically, these models consist of sets of

differential equations with parameters that are chosen so that computer simulations using the

models would show results that closely approximate of real observations. If the observations

indicate that the disease keeps growing, the model can be helpful in identifying what

environmental conditions influence the growth. For example, the mathematical model could

determine that some transmissivity parameters might have values that are too high, and

interventions such as increased isolation of infected persons or increased use of face masks

are warranted. With the interventions, as suggested by the model, new observations might

indicate that growth has stopped or even reversed, and revised models would show that the

new situations are stable. If the interventions are removed, there might be resurgence of

growth in the spread of the disease. If the fixed intervention is not removed, it would be

maintained at the latest level until a resurgence occurs for whatever reason. Note that the

latest model is obtained after the latest change in the environment. However, experimental

assessment of the effectiveness of the latest model is conducted after it is constructed.

4.2 Engineered closed loop models

The principal reason for using an engineered closed loop mathematical model, that adds a

feedback mechanism of measured epidemiological compartments of population, to construct

the inputs of the input-output model, is to provide a capability to stabilize the composite

model (Åström and Murray 2020, Cruz 1971). Parameters of the input-output model are

determined as in Section 4.1, using observations without feedback. However, experimental

assessment of the effectiveness of the closed loop model cannot be accomplished until the

processes of the closed loop model are in place. This is the same situation as in the science

model because the effectiveness of the most recent science model can only be assessed after

new observations are made and compared with the predicted outputs of the science model.

Thus, policy makers need to allow that the interventions be modulated in accordance with the

process of the closed loop model. As in the science model, new observations can be gathered

to verify if the computer calculations of the closed loop model are reasonable approximations

of new real observations. One major capability of using a closed loop model is that even if

the input-output model is unstable, potentially, the entire system with feedback can be stable.

When the closed loop model is designed to be stable, then the epidemiological compartments

of population would tend to the designed asymptotic values, but the closed loop structure

needs to be in place indefinitely. Testing will remain and if there is occasional infection, the

infected persons will receive medical treatment, as appropriate, and contact tracing will be

conducted. A second advantage of using a closed loop model is that it can be robust against

variations in the parameters, such as the infectivity rate, β (Cruz and Perkins 1964, Cruz et al.

1981a, Cruz et al. 1981b, Freudenberg et al. 1982).

4.3 Other closed loop models

The only closed loop model in the literature was proposed by a team from the University of

Montreal (Stewart et al. 2020). They used an input-output model proposed by the University

of Notre Dame (Kantor 2020).

4.3.1 Notre Dame University input-output model

The University of Notre Dame model (Kantor 2020) added an input to the Kermack and

McKendrick SIR Science Model (Kermack and McKendrick 1927). A control input is

introduced by multiplying β by (1-u) in the science model

(28)

(29)

together with Equation (2).

In Kantor (2020), simulations were performed for various fixed values of , where 0 < < 1,

to demonstrate mitigation of the propagation of COVID-19.

4.3.2 University of Montreal closed loop model

A University of Montreal team (Stewart et al. 2020) took the University of Notre Dame

model and used the control input to propose a closed loop function of the number of

hospital beds available in a city. Several scenarios were discussed, including use of

alternating fixed and output-dependent interventions. Their goal was to use feedback control

to stabilize the system. However, the paper does not provide details of how the number of

available hospital beds is related to the state variables of the model. No details are provided

in the paper as to what the mathematical expression of the control is, and no details are

provided as to how their chosen stabilizes the system.

4.3.3 Comparing the Notre Dame/Montreal model with the Batangas State University

engineered closed loop model

The Notre Dame model introduced a multiplier wherever appears. It has the

conceptual effect of reducing the value of , very similar to the science model of changing

the environment to alter the value of . In the Montreal closed loop model, their team

proposed to use of the Notre Dame model as a feedback function of the available hospital

beds. The ultimate impact of this feedback function on stability would be quite complicated,

without having a detailed analysis, not available in the literature. The Batangas State

engineered closed loop model introduces the intervention at a broader level, without

committing to reducing the value of . The inputs and directly influence the rate of

change of , and Furthermore, the input implies utilizing vaccination, without

increasing the infected compartment. This paper is the first to incorporate the effect of

vaccination on input-output modeling of the spread of a disease. Furthermore, this paper is

first to consider the use of a vaccine in a closed loop model incorporating a feedback

mechanism to relate observations of epidemiological compartments of population, to the two

modulated inputs or interventions.

5.0 Concluding remarks

The major contribution of this paper was the design of the Batangas State University

engineered feedback model that utilizes the outputs in tempering the inputs to control the

spread of COVID-19. The model was designed to be stable and robust against variations in

the values of transmissivity, . The Batangas State University engineered closed loop model

introduced an input ( ) utilizing feedback, that led to a substantial increase in the number of

epidemiological compartment of population that is Free of virus (F), without significant

increase in the number of Infected (I). Even when the spread is reduced to almost zero,

testing, contact tracing, and medical treatment of those tested positive, must continue as part

of the closed loop process, to avoid new surges of infection, and further reduce mortality. The

specific design using specific eigenvalues is for illustrative purpose. Before applying to an

actual situation, various capacities need to be determined and the eigenvalues need to be

chosen so that the values of the controls do not exceed the capacities.

6.0 Acknowledgments

Jose B Cruz Jr. acknowledges support from the National Academy of Science and

Technology for a Research Fellowship. The authors would also like to acknowledge Batangas

State University for all the support and opportunity to complete this project.

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https://covid19.who.int

Figure 1. Science model showing outputs or observations

Figure 2. Creating an input-output model by adding inputs to the science model

Figure 3. Mapping the outputs or observations to the inputs via a controller

Figure 4. Predicted number of Susceptible (S) compartment of Philippine population for

different values of

Figure 5. Predicted number of Infected (I) compartment of Philippine population for

different values of

Figure 6. Predicted number of Free of Virus (F) compartment of Philippine population for

different values of

Figure 7. Predicted number of Deceased (D) compartment of Philippine population for

different values of β

Figure 8 Feedback z1*N for different values of β

Figure 9 Feedback z2*N for different values of β

Figure 10 Feedback z1L*N for different values of β

Figure 11 Feedback z1N*N for different values of β

Figure 12. Predicted number of Infected (I) compartment of Philippine population for

different values of when z1 and z2 are zero

Figure 13. Predicted number of Death (D) compartment of Philippine population for

different values of when z1 and z2 are zero