Effect of Awareness, Quarantine and Vaccination as Control Strategies on COVID-19 with Co-Morbidity and Re-Infection

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Title: Effect of Awareness, Quarantine, Vaccination and Non-Pharmaceutical Interventions as Control
Strategies on COVID-19 with Co-Morbidity and Re-Infection
1st Author and Corresponding Author: Amit Kumar Saha
Given Name: Amit Kumar
Family Name: Saha
Affiliation: Department of Mathematics, University of Dhaka,
Dhaka 1000, Bangladesh
E-mail: amit92.du@gmail.com
2nd Author: Shikha Saha
Given Name: Shikha
Family Name: Saha
Affiliation: Department of Mathematics, Bangladesh University of Engineering and Technology (BUET)
Dhaka 1000, Bangladesh
E-mail: shikhadumath58@gmail.com
3rd Author: Chandra Nath Podder
Given Name: Chandra Nath
Family Name: Podder
Affiliation: Department of Mathematics, University of Dhaka,
Dhaka 1000, Bangladesh
E-mail: cpodder@du.ac.bd
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Effect of Awareness, Quarantine and Vaccination as Control Strategies on COVID-19 with Co-Morbidity and
Re-Infection
Amit Kumar Saha ID ∗1, Shikha Saha2, and Chandra Nath Podder3
1Department of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh
2Department of Mathematics, Bangladesh University of Engineering and Technology (BUET), Bangladesh
3Department of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh
1. Introduction
The novel corona virus (COVID-19) caused by SARS-CoV-2 became a global public health concern during 2020 and 2021 and
is still posing a health and economic threat throughout the world CDC (a). Almost all the countries in the world are trying to deal with this
new contagious disease and getting rid of it has now become the most important challenge for all the countries. It first appeared in China in
December 2019 and due to its high infectiousness, it spreads very fast all over the world, putting the world at extreme global crisis Bubar
et al. (2021); Wu et al. (2020). It becomes more dangerous for people of any age with certain medical issues including cardiovascular disease,
diabetes, high blood pressure and cancer etc CDC (c). A report from a survey of 138 COVID-19 infected individuals confirms it by showing
that more than 45% of the infected individuals had one or more co-morbidities and that infected individuals who admitted to the intensive care
unit (ICU) had a higher number of co-morbidities (72.2%) compared to the infected individuals who didn’t admit to the ICU (37.3%) Jain and
Yuan (2020). As of June 20, 2022, 539928791 people infected with the Covid-19 and 6320448 people died worldwide WHO (a).
On the one hand, its high infectious rate, and on the other hand, the frequent emergence of new variants have made
the control of COVID-19 even more challenging. In these circumstances, invention of effective vaccine is not the only way to address this
challenge. Hence non pharmaceutical interventions should also be maintained. At the beginning of 2020, the genetic sequence of SARS-CoV-2
was published. After that, corporations, governments, international health organizations, and university research groups started to work for
developing vaccines against COVID-19 WHO (c); Le et al. (2020). After the initial development and three-stage clinical trials for safety and
effectiveness the following vaccines obtained World Health Organization’s EUL (Emergency Use Listing): The Pfizer-BioNTech Comirnaty
vaccine on 31 December 2020, the SII/COVISHIELD and AstraZeneca/AZD1222 vaccines on 16 February 2021, the Janssen/Ad26.COV 2.S
vaccine on 12 March 2021, the Moderna COVID-19 vaccine (mRNA 1273) on 30 April 2021, the Sinopharm COVID-19 vaccine on 7 May
2021, the Sinovac-CoronaVac vaccine on 1 June 2021, the Bharat Biotech BBV152 COVAXIN vaccine on 3 November 2021, the Covovax
(NVX-CoV2373) vaccine on 17 December 2021, the Nuvaxovid (NVX-CoV2373) vaccine on 20 December 2021 WHO (b). The Pfizer-
BioNTech COVID-19, the Moderna, and the Johnson and Johnson’s Janssen vaccines are fully approved by FAD for people 18 years of age
and older and only the Pfizer-BioNTech COVID-19 has approbation for emergency use for children ages 5 years and older CDC (b). More than
529 million vaccine doses have been administered in the United States from 14 December 2020 through 18 January 2022. During this period,
the mortality rate received by Vaccine Adverse Event Reporting System (VAERS) was 0.0022% among the people who received a COVID-19
vaccine CDC (b). Globally a total of 9,571,502,663 vaccine doses have been administered by 18 January 2022 WHO (a).
Besides the use of effective vaccine and medical research, mathematical models can be a powerful means in getting
insight into the dynamics of any infectious disease like COVID-19 which can help decision makers take necessary decisions to prevent the
spread of COVID-19. It also helps assess the impact of vaccines and the use of NPIs in controlling the spread of the pandemic and mitigating
its life-threatening effects. A significant number of mathematical models have already been developed and used to study the transmission
dynamics of COVID-19 and also to control the disease burden (some of them are given here Atangana (2020); Ferguson et al. (2020); Gumel
et al. (2021); Ivorra et al. (2020); Khan and Atangana (2020); Kucharski et al. (2020); Mancuso et al. (2021); Mizumoto and Chowell (2020);
Ngonghala et al. (2020); Okuonghae and Omame (2020); Saha et al. (2022). But in this paper we have formulated a new mathematical model
based on the model Saha et al. (2022). It is novel in the sense that in this paper we have considered vaccination class. It is also novel in the sense
that unlike many scholars who didn’t consider vaccination and re-infection of the recovered individuals, we have considered vaccination and
re-infection of the recovered individuals. In the case of optimal control measures, besides the control measures (isolation, detection, awareness
and speed of vaccination) considered in the previous literature, we have considered a new control measure to ensure better treatment and better
care for hospitalized individuals. The aim of this research is to assess the impact of vaccination and non-pharmaceutical interventions (NPIs) on
the spread of COVID-19. Our aim is also to highlight the effect of co-morbidity and re-infection on the transmission dynamics of COVID-19.
The sensitivity analysis of the parameter of our model with respect to some response functions is performed to detect which parameters have
greater impact on the transmission of COVID-19. In addition, optimal control theory has been applied and analyzed numerically to assess the
impact of all combination of the control strategies considered in this model on the prevention of COVID-19.
The entire paper is decorated in the following manners. In section 2, the formulation of the COVID-19 model is presented
and non-negativity and boundedness of the model solutions are proved. Section 3 is engaged with the rigorous theoretical analysis of the model
to discuss about the stability of equilibrium. In section 5, the model is extended based on optimal control theory and analyzed mathematically
∗corresponding author: e-mail:amit92.du@gmail.com
1
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to prove the existence of an optimal control using the Pontryagin’s maximum principle. Numerical simulations are presented in section 6.
Section 7 is devoted to the discussion and conclusion about the findings.
2. Model formulation
We develop the model by dividing the total human population at time t, denoted by N(t), into: susceptible (S(t)), susceptible with
co-morbidity (Sc(t)), vaccinated (Sv(t)), exposed in early stage (E1(t)), pre-symptomatic (E2(t)), asymptomatic infected without
co-morbidity (Ia(t)), symptomatic infected without co-morbidity (Is(t)), asymptomatic infected with co-morbidity (Iac(t)), symptomatic
infected with co-morbidity (Isc(t)), quarantined (Q(t)), hospitalized (H(t)) and recovered (R(t)) twelve mutually exclusive classes, so that
N(t) = S(t) + Sc(t)+ Sv(t) + E1(t) + E2(t) + Ia(t)+ Is(t) + Iac(t) + Isc(t) + Q(t) + H(t) + R(t).
To formulate the model we consider the following assumptions:
•Birth rate is not considered.
•Exposed individuals in early stage are asymptotically infected and unable to infect others.
•Pre-symptomatic infectious individuals are shedding viruses and can infect others.
•Quarantine and hospitalization are perfect and individuals belonging to these classes can not infect others.
•Individuals recovered from COVID-19 may again return to exposed in early stage class at a lower rate.
Susceptible individuals acquire infection with COVID-19 upon contacting with individuals in the E2,Ia,Is,Iac and Isc classes, at a rate λ,
where
λ=(1−em)β(ηeE2+ηaIa+Is+T1Iac +T2Isc)
N−(Q+H),(1)
where, βrepresents the contact rate for effective transmission of COVID-19. 0 <m≤1 represents the percentage of mask coverage and
0<e≤1 indicates face masks efficacy. It is assumed that pre-symptomatic individuals (E2class) and asymptomatic infected individuals (Ia
class) infect others at a lower rate, ηeβandηaβ, respectively with 0 <ηe,ηa<1. Furthermore modification parameter T1,T2>1 indicate
individuals in Iac and Isc classes can transmit COVID-19 at an increased rate , T1βandT2β, respectively. The equations for the transmission
Fig. 1. Schematic diagram of the COVID-19 model (2).
dynamics of COVID-19 with co-morbidity in the presence of vaccination is given by the following system of non-linear differential equations
(the schematic diagram of the model is shown in Fig. 1 and the parameters are described in details in Table 1).
˙
S=Λ−λS−(Ω+ξs+µ)S,
˙
Sc=ΩS−T3λSc−(θ1ξs+µ)Sc,
˙
Sv=ξsS+θ1ξsSc+θ2ξsR−(1−ε)λSv−µSv,
˙
E1=λS+T3λSc+ (1−ε)λSv+α λ R−(σ1+µ)E1,
˙
E2=σ1E1−(σ2+δe+µ)E2,
˙
Ia=d1σ2E2−(ψa+δa+µ)Ia,
˙
Is=d2σ2E2−(σs+ψs+φs+δs+µ)Is,
2
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˙
Iac =d3σ2E2−(ψac +δac +µ)Iac,
˙
Isc = (1−d)σ2E2−(σsc +ψsc +φsc +δsc +µ)Isc,
˙
Q=σsIs+σsc Isc −(ψq+φq+µ)Q,
˙
H=φsIs+φsc Isc +φqQ−(ψh+δh+µ)H,
˙
R=ψaIa+ψsIs+ψac Iac +ψsc Isc +ψqQ+ψhH−α λ R−(θ2ξs+µ)R,(2)
where, d=d1+d2+d3.
Λis the recruitment rate of susceptible humans into the population. Ωrepresents the ratio of susceptible individuals who have co-morbidity. It
is assumed that susceptible individuals having co-morbidity are more susceptible to COVID-19 infection (T3λwith T3>1) than susceptible
individuals having no co-morbidity.
Table 1
Model parameters with description
Parameter Description
ΛRecruitment rate
βEffective contact rate for COVID-19 transmission
ξsVaccination rate for susceptible individuals
εVaccine efficacy
mProportion of individuals who use masks
eFace mask efficacy
T1,T2Relative risk of high infectiousness of individuals in Iac and Isc classes
compared to individuals in Isclass
T3Modification parameter accounting for increased susceptibility
to COVID-19 infection by co-morbid susceptible
ηe,ηaRelative risk of low infectiousness of individuals in E1and Iaclasses
compared to individuals in Isclass
ΩProportion of co-morbid susceptible individuals
αRe-infection rate of recovered individuals
σ1Progression rate of early exposed individuals (E1)to pre-symptomatic (E2)class
σ2Rate of progression of pre-symptomatic (E2)individuals to infectious classes
(Ia,Is,Iac and Isc,respectively)
d1,d2and d3Fraction of pre-symptomatic individuals who progress to the Ia,Isand Iac classes, respectively
(d1+d2+d3≤1)
1−(d1+d2+d3)Fraction of individuals move from E2class to Isc class
σsand σsc Transmission rate from Isand Isc classes to Qclass, respectively
φs,φsc and φqTransition rate from Is,Isc and Qclasses to Hclass, respectively
ψa,ψs,ψac,ψsc ,ψqand ψhRecovery rate of individuals from Ia,Is,Iac,Isc,Qand Hclasses, respectively
δe,δa,δs,δac,δsc and δhDisease related death rate for individuals in the E2,Ia,Is,Iac,Isc and Hclasses, respectively
µNatural death rate
θ1modification parameter (θ1>1)implying high vaccination rate provided to the co-morbid susceptible
individuals
θ2modification parameter (0<θ2<1)implying low vaccination rate provided to the recovered individuals
3. Theoretical analysis
3.1. Fundamental properties
3.1.1. Non-negativity of the solutions
To show the non-negativity of the solutions we prove the following theorem.
Theorem 1. The solutions of the model (2), with initial conditions S(0)>0,Sc(0)≥0,Sv(0)≥0,E1(0)≥0,E2(0)≥0,Ia(0)≥
0,Is(0)≥0,Iac(0)≥0,Isc(0)≥0,Q(0)≥0,H(0)≥0,and R(0)≥0are positive for all time t >0.
3
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Proof. Let nS,Sc,Sv,E1,E2,Ia,Is,Iac,Isc,Q,Hand Robe the set of solutions of the model (2). From the first equation of the model (2) we
can write
d
dt "S(t)exp(Zt
0
λ(u)du +k1t)# =Λ"exp(Zt
0
λ(u)du +k1t)#,(3)
where, k1=Ω+ξs+µ.
From (3),
S(t)exp(Zt
0
λ(u)du +k1t)−S(0) = Zt
0
Λ"exp(Zx
0
λ(u)du +k1t)#dx.
Hence,
S(t) = S(0)exp(−Zt
0
λ(u)du +k1t)+exp(−Zt
0
λ(u)du +k1t)Zt
0
Λ"exp(Zx
0
λ(u)du +k1t)#dx >0.
Proceeding in the same way, it can be shown that
Sc≥0,Sv≥0,E1≥0,E2≥0,Ia≥0,Is≥0,Iac ≥0,Isc ≥0,Q≥0,H≥0,and R≥0 for all t≥0.
3.1.2. Boundedness of the solution
Adding all the equations of the model (2), we get
dN
dt =Λ−µN−δeE2−δaIa−δsIs−δac Iac −δsc Isc −δhH.(4)
It is obvious that 0 <E2≤N, 0 <Ia≤N, 0 <Is≤N, 0 <Iac ≤N, 0 <Isc ≤N, 0 <H≤N.
It follows that
Λ−(µ+δe+δa+δs+δac +δsc +δh)N≤dN
dt <Λ−µN,(5)
Thus, Λ
µ+δe+δa+δs+δac +δsc +δh
≤liminf
t→∞N≤limsup
t→∞
N≤Λ
µ.
This implies limsup
t→∞
N≤Λ
µ.
3.1.3. Invariant regions
Now let us consider the region D=((S,Sc,Sv,E1,E2,Ia,Is,Iac,Isc,Q,H,R)∈R12
+:N≤Λ
µ).
From equations (4) and (5) we can write
dN
dt ≤Λ−µN.(6)
Solving this and using a comparison theorem as described in Lakshmikantham et al. (1989) we have N(t)≤N(0)e−µt+Λ
µ1−e−µt.
Particularly, it can be shown that N(t)≤Λ
µif N(0)≤Λ
µ. This implies all the solutions of the system (2) with initial conditions in Dremains
in Dfor all time t>0. Thus, the region Dis positive invariant and attracting Hethcote (2000).
3.2. Local asymptotic stability of the DFE
From the COVID-19 model (2), the disease-free equilibrium, E0, is obtained as
E0= (S∗,S∗
c,S∗
v,E∗
1,E∗
2,I∗
a,I∗
s,I∗
ac,I∗
sc,Q∗,H∗,R∗) = Λ
k1
,Ω Λ
k1k2
,Λξs(k2+Ωθ1)
k1k2k3
0,0,0,0,0,0,0,0,0!.(7)
To establish the condition for local asymptotic stability (LAS) of the DFE, the next generation operator method described in Diekmann
et al. (1990); Van den Driessche and Watmough (2002) is used. The next generation matrices for the new infection terms and remaining transfer
terms, denoted by F and V respectively, are given by
4
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F=













0FeFaFsFac Fsc 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0













,V=















k40 0 0 0 0 0 0
−σ1k50 0 0 0 0 0
0−d1σ2k60 0 0 0 0
0−d2σ20k70 0 0 0
0−d3σ20 0 k80 0 0
0−(1−d)σ20 0 0 k90 0
0 0 0 −σs0−σsc k10 0
0 0 0 −φs0−φsc 0k11















,
where,
Fe= (1−em)β ηe
S∗+
τ
3S∗
c+ (1−ε)S∗
v+αR∗
N∗∗ ,Fa= (1−em)β ηa
S∗+
τ
3S∗
c+ (1−ε)S∗
v+αR∗
N∗∗ ,
Fs= (1−em)βS∗+
τ
3S∗
c+ (1−ε)S∗
v+αR∗
N∗∗ ,Fac = (1−em)β
τ
1
S∗+
τ
3S∗
c+ (1−ε)S∗
v+αR∗
N∗∗ ,
Fsc = (1−em)β
τ
2
S∗+
τ
3S∗
c+ (1−ε)S∗
v+αR∗
N∗∗ ,N∗∗ =N−(Q+H),
k1=Ω+ξs+µ,k2=θ1ξs+µ,k3=µ,k4=σ1+µ,k5=σ2+δe+µ,k6=ψa+δa+µ,k7=σs+ψs+φs+δs+µ,k8=ψac +δac +µ,
k9=σsc +ψsc +φsc +δsc +µ,k10 =ψq+φq+µ,k11 =ψh+δh+µ,k12 =θ2ξs+µand d=d1+d2+d3.
Following the approach described in Chavez et al. (2002); Hethcote (2000), it can be shown that the basic reproduction number, denoted by
Rc,is given by
Rc=ρ(F V −1) = Re+Ra+Rs+Rac +Rsc,(8)
where, ρrepresents the spectral radius of the next generation matrix F V −1and
Re=FeBe,Ra=FaBa,Rs=FsBs,Rac =Fac Bac ,Rsc =Fsc Bsc,
with,
Be=σ1
k4k5
,Ba=σ1σ2d1
k4k5k6
,Bs=σ1σ2d2
k4k5k7
,Bac =σ1σ2d3
k4k5k8
,Bsc =σ1σ2(1−d)
k4k5k9
.
Consequently, using Theorem 2 of Van den Driessche and Watmough (2002) the following result can be established.
Lemma 1. The DFE of the COVID-19 model given by (2), is locally-asymptotically stable (LAS) if Rc<1, and unstable if Rc>1.
3.3. Endemic equilibrium point (EEP)
Let E1= (S∗,S∗
c,S∗
v,E∗
1,E∗
2,I∗
a,I∗
s,I∗
ac,I∗
sc,Q∗,H∗,R∗)be any equilibrium point of the model (2) and let
λ∗=β(1−em)(ηeE∗
2+ηaI∗
a+I∗
s+
τ
1I∗
ac +
τ
2I∗
sc)
N∗−(Q∗+H∗)(9)
Now setting the left hand side of each equation of the system (2) to zero and solving for the variables we have,
S∗
c=ΩS∗
(
τ
3λ∗+k2),
S∗
v=A7λ∗3+A8λ∗2+A9λ∗+A10
n(1−ε)λ∗+k3o(
τ
3λ∗+k2)(A4λ∗2+A5λ∗+A6)
S∗,
E∗
1=λ(A1λ∗2+A2λ∗+A3) (α λ +k12 )
(A4λ∗2+A5λ∗+A6)(
τ
3λ∗+k2)S∗,
E∗
2=WeE∗
1,I∗
a=WaE∗
1,I∗
s=WsE∗
1,I∗
ac =Wac E∗
1,I∗
sc =Wsc E∗
1,Q∗=WqE∗
1,H∗=WhE∗
1,
R∗=Wr
α λ∗+k12
E∗
1,(10)
where,
We=σ1
k5
,Wa=d1σ1σ2
k5k6
,Ws=d2σ1σ2
k5k7
,Wac =d3σ1σ2
k5k8
,Wsc =(1−d)σ1σ2
k5k9
,Wq=(σsWs+σsc Wsc)
k10
,
Wh=φsWs+φsc Wsc +φqWq
k11
,Wr=ψaWa+ψsWs+ψacWac +ψsc Wsc +ψqWq+ψhWh,
A1= (1−ε)
τ
3,A2= (1−ε)k2+ (1−ε)
τ
3Ω+{k3+ (1−ε)ξs}
τ
3,A3={k3+ (1−ε)ξs}k2+{k3
τ
3+ (1−ε)θ1ξs}Ω,
A4=α(1−ε)k4−(1−ε)αWr,A5= (1−ε)k4k12 +k4k3α− {αk3+ (1−ε)θ2ξs}Wr,A6=k3k4k12,A7=A4
τ
3ξs,
A8=ξs(
τ
3A5+A4k2) + A4Ωθ1ξs+A1θ2ξsWr,A9=ξs(
τ
3A6+A5k2) + A5Ωθ1ξs+A2θ2ξsWr,
A10 =A6Ωθ1ξs+A3θ2ξsBr.
5
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Substituting (10) into (9) gives
λ∗=β(1−em)(ηeWe+ηaWa+Ws+
τ
1Wac +
τ
2Wsc)E∗
1
S∗+Ω
(
τ
3λ∗+k2)S∗+A7λ∗3+A8λ∗2+A9λ∗+A10
n(1−ε)λ∗+k3o(
τ
3λ∗+k2)(A4λ∗2+A5λ∗+A6)
S∗+Wc+Wr
α λ ∗+K11 E∗
1
,
(11)
where,
Wc=1+We+Wa+Ws+Wac +Wsc.
After some algebraic calculation we get the following equation in terms of λ∗
λ∗nP
5λ∗5+P
4λ∗4+P
3λ∗3+P
2λ∗2+P
1λ∗+P
0o=0,(12)
where,
P
5=Wcα(1−ε)A1,
P
4=A1Wcαk3+ (1−ε)A1(Wck12 +Wr) + A2Wcα(1−ε) + A4
τ
3(1−ε)−A1(1−ε)α,
P
3=A4
τ
3k3+ (1−ε)A4k2+ (1−ε)A5
τ
3+A7+A1k3(Wck12 +Wr) + A2Wcαk3+ (1−ε)A2(Wck12 +Wr)+
A3Wcα(1−ε)−A1(1−ε)k12 −A1αk3−A2(1−ε)α,
P
2=A4k2k3+A6
τ
3(1−ε) + A5
τ
3k3+A5(1−ε)k2+A8+A3Wcαk3+ (1−ε)A3(Wck12 +Wr)+
A2k3(Bck12 +Wr)−A1k3k12 +A3(1−ε)α+A2(1−ε)k12 +A2αk3,
P
1=A5k2k3+A6
τ
3k3+ (1−ε)k2A6+A9+A3k3Wck12 +A3k3Wr−A2k3k12 −A3(1−ε)k12 −A3αk3,
P
0=A6k2k3+A10 −A3k3k12.
Out of the six roots, the root λ∗=0, of (12), corresponds to the DFE E0. Equation (12) says that the non-zero equilibria of the model satisfy
f(λ∗) = P
5λ∗5+P
4λ∗4+P
3λ∗3+P
2λ∗2+P
1λ∗+P
0=0.(13)
Using the parameter values as given in Table 2, it can be shown that Rc<1 and out of the five roots, two roots are real positive, one is real
negative and other two roots are complex. Thus there exists two positive endemic equilibria of the system (2) which implies the possibility of
the presence of backward bifurcation phenomena.
3.4. Backward Bifurcation Analysis:
Here we will discuss about the possibility of having backward bifurcation phenomena. To explore this phenomena we will use the center
manifold theory described in Carr (2012); Van den Driessche and Watmough (2002) and apply change of variable formula. For this, let S=x1,
Sc=x2,Sv=x3,E1=x4,E2=x5,Ia=x6,Is=x7,Iac =x8,Isc =x9,Q=x10,H=x11 and R=x12 , and hence model (2) can be written as
dX
dt = ( f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12 )T, where X= (x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11 ,x12)Tand then we have
d x1
d t =f1=Λ−λx1−k1x1,
d x2
d t =f2=Ωx1−T3λx2−k2x2,
d x3
d t =f3=ξsx1+θ1ξsx2+θ2ξsx12 −(1−ε)λx3−k3x3,
d x4
d t =f4=λx1+T3λx2+ (1−ε)λx3+α λ x12 −k4x4,
d x5
d t =f5=σ1x4−k5x5,
d x6
d t =f6=d1σ2x5−k6x6,
d x7
d t =f7=d2σ2x5−k7x7,
d x8
d t =f8=d3σ2x5−k8x8,
d x9
d t =f9= (1−d)σ2x5−k9x9,
d x10
d t =f10 =σsx7+σsc x9−k10 x10,
d x11
d t =f11 =φsx7+φsc x9+φqx10 −k11 x11,
d x12
d t =f12 =ψax6+ψsx7+ψac x8+ψsc x9+ψqx10 +ψhx11 −α λ x12 −k12 x12 ,(14)
6
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λ=(1−em)β(ηex5+ηax6+x7+T1x8+T2x9)
N−(x10 +x11).
The Jacobian of the system (14) is given by:
J(E0) =

























−k10 0 0 −ηeJ1−ηaJ1−J1−
τ
1J1−
τ
2J10 0 0
Ω−k20 0 −ηeJ2−ηaJ2−J2−
τ
1J2−
τ
2J20 0 0
ξsθ1ξs−k30−ηeJ3−ηaJ3−J3−
τ
1J3−
τ
2J30 0 θ2ξs
0 0 0 −k4ηeJ4ηaJ4J4
τ
1J4
τ
2J40 0 0
0 0 0 σ1−k50 0 0 0 0 0 0
0 0 0 0 d1σ2−k60 0 0 0 0 0
0 0 0 0 d2σ20−k70 0 0 0 0
0 0 0 0 d3σ20 0 −k80 0 0 0
0 0 0 0 (1−d)σ20 0 0 −k90 0 0
0 0 0 0 0 0 σs0σsc −k10 0 0
0 0 0 0 0 0 φs0φsc φq−k11 0
0 0 0 0 0 ψaψsψac ψsc ψqψh−k12

























,
where,
J1=(1−em)βk2k3
k1k2
,J2=(1−em)βk3
τ
3Ω
k1k2
,J3=(1−em)β(1−ε)ξs(k2+θ1Ω)
k1k2
,
and J4=
(1−em)βnk2k3+
τ
3Ωk3+ (1−ε)ξs(k2+θ1Ω)o
k1k2
.
Now consider Rc=1 and β=β∗is a bifurcation parameter. Thus we get
β=β∗=1
(1−em)nS∗+
τ
3S∗
c+ (1−ε)S∗
v+αR∗o(ηeBe+ηaBa+Bs+T1Bac +T2Bsc)
.
The Jacobian J(E0)of (14) with β=β∗(β∗calculated at the DFE, E0), denoted by Jβ∗, has a simple zero eigenvalue (with all other
eigenvalues having negative real part). Hence, center manifold theory Carr (2012); Castillo-Chavez and Song (2004), can be applied.
Eigenvectors of Jβ∗=J(E0)β=β∗:
When Rc=1,a right eigenvector corresponding to the zero eigenvalue of the jacobian (Jβ∗) is given by
w= [w1,w2,w3,w4,w5,w6,w7,w8,w9,w10,w11 ,w12]T, where,
w1=−(ηeJ1w5+ηaJ1w6+J1w7+
τ
1J1w8+
τ
2J1w9)
k1
,
w2=Ωw1−(ηeJ2w5+ηaJ2w6+J2w7+
τ
1J2w8+
τ
2J2w9)
k2
,
w3=ξsw1+θ1ξsw2+θ2ξsw12 −(ηeJ3w5+ηaJ3w6+J3w7+
τ
1J3w8+
τ
2J3w9)
k3
,
w4=k5w5
σ1
,w5=w5,w6=d1σ2w5
k6
,w7=d2σ2w5
k7
,w8=d3σ2w5
k8
,w9=(1−d)σ2w5
k9
,
w10 =σsw7+σsc w9
k10
,w11 =φsw7+φsc w9+φqw10
k11
,w12 =ψaw6+ψsw7+ψac w8+ψsc w9+ψqw10 +ψhw11
k12
.
Further, a left eigenvector of Jβ∗corresponding to the zero eigenvalue is given by v= [v1,v2,v3,v4,v5,v6,v7,v8,v9,v10,v11 ,v12]
where,
v1=v2=v3=v10 =v11 =v12 =0,v4=v4,
v5=ηeJ4v4+d1σ2v6+d2σ2v7+d3σ2v8+ (1−d)σ2v9
k5
,v6=ηaJ4v4
k6
,v7=J4v4
k7
,v8=
τ
1J4v4
k8
,v9=
τ
2J4v4
k9
.
Computations of aand b:
The expression for aand bfrom Carr (2012); Castillo-Chavez and Song (2004) is:
a=
n
∑
k,i,j=1
vkwiwj
∂2fk
∂xi∂xj
(0,β∗),
and b=
n
∑
k,i=1
vkwi
∂2fk
∂xi∂ β (0,β∗),
which becomes
7
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a=1
Λ(Ωk3+Ωθ1ξs+k2k3+k2ξs)2n2(1−em)βk1k2k3(ηaw6+ηew5+
τ
1w8+
τ
2w9+w7)(Ωv3w1θ1ξs+Ωv3w2θ1ξs−Ωv3w3k3
+Ωv3w4ξc+Ωv3w5θ1ξs+Ωv3w6θ1ξs+Ωv3w8θ1ξs+Ωv3w9θ1ξs+Ωv3w12 θ1ξs+Ωv4w1k3−Ωv4w2θ1ξs+Ωv4w3k3−Ωv4w4θ1ξs
−Ωv4w5θ1ξs−Ωv4w6θ1ξs+Ωεv4w7θ1ξs−Ωk3
τ
3v4w7+εk2v4w7ξs−Ωεv3w7θ1ξs−εk2v3w7ξs+v2w7k3
τ
3Ω+εv3w3k2k3
−εv3w4ξsk2−εv3w5ξsk2−εv3w6ξsk2−εv3w8ξsk2−εv3w9ξsk2−εv3w12 ξsk2+εv4w1ξsk2+εv4w2ξsk2−εv4w3k2k3
+εv4w4ξsk2+εv4w5ξsk2+εv4w6ξsk2+εv4w8ξsk2+εv4w9ξsk2+εv4w12 ξsk2+v4w2ξsk2
τ
3+v4w2k2k3
τ
3−w2ξsk2
τ
3v2
−w2k2k3
τ
3v2+Ωαv4w12 ξc+Ωαv4w12 k3−Ωαv12 w12 θ1ξs−Ωαv12 w12 k3−Ωεv3w1θ1ξs−Ωεv3w2θ1ξs+Ωεv3w3k3
−Ωεv3w4θ1ξs−Ωεv3w5θ1ξs−Ωεv3w6θ1ξs−Ωεv3w8θ1ξs−Ωεv3w9θ1ξs−Ωεv3w12 θ1ξs+Ωεv4w1θ1ξs+Ωεv4w2θ1ξs
−Ωεv4w3k3+Ωεv4w4θ1ξs+Ωεv4w6θ1ξs+Ωεv4w8θ1ξs+Ωεv4w9θ1ξs+Ωεv4w12 θ1ξs−Ωv4w1k3
τ
3+Ωv4w2θ1ξs
τ
3
−Ωv4w3k3
τ
3−Ωv4w4k3
τ
3−Ωv4w5k3
τ
3−Ωv4w6k3
τ
3−Ωv4w8k3
τ
3−Ωv4w9k3
τ
3−Ωv4w12 k3
τ
3+Ωw1k3
τ
3v2
−Ωw2θ1ξs
τ
3v2+Ωw3k3
τ
3v2+Ωw4k3
τ
3v2+Ωw5k3
τ
3v2+Ωw6k3
τ
3v2+Ωw8k3
τ
3v2+Ωw9k3
τ
3v2+Ωw12 k3
τ
3v2
+αv4w12 ξsk2+αv4w12 k2k3−αv12 w12 ξsk2−αv12 w12 k2k3−εv3w1ξsk2−εv3w2ξsk2−Ωv4w8θ1ξs−Ωv4w9θ1ξs
−Ωv4w12 ξc−Ωw1θ1ξsv1−Ωw1k3v1+v3w1ξsk2+v3w2ξsk2−v3w3k2k3+v3w4ξsk2+v3w5ξsk2
+v3w6ξsk2+v3w8ξsk2+v3w9ξsk2+v3w12 ξsk2−v4w2ξsk2−v4w2k2k3−v4w4ξsk2−v4w4k2k3
−v4w5ξsk2−v4w5k2k3−v4w6ξsk2−v4w6k2k3−v4w8ξsk2−v4w8k2k3−v4w9ξsk2−v4w9k2k3−v4w12 ξsk2
−v4w12 k2k3−w1ξsk2v1+w2k2k3v1+w3k2k3v1+w4k2k3v1+w5k2k3v1+w6k2k3v1
+w8k2k3v1+w9k2k3v1+w12 k2k3v1+Ωεv4w5θ1ξs+Ωv3w7θ1ξs+k2v3w7ξs+v1w7k2k3−Ωv4w7θ1ξs−k2k3v4w7−k2k3w7ξs)o,
b=(1−em)
(Ωk3+Ωθ1ξs+k2k3+k2ξs)(ηaw6+ηew5+
τ
1w8+
τ
2w9+w7)(Ωεv3θ1ξs−Ωεv4θ1ξs−Ωk3
τ
3v2+Ωk3
τ
3v4+εk2v3ξs
−εk2v4ξs−Ωv3θ1ξs+Ωv4θ1ξs−k2k3v1+k2k3v4−k2v3ξs+k2v4ξs).
Hence according to the Theorem 4.1 of Castillo-Chavez and Song (2004), it follows that model (2) will exhibit backward bifurcation at
Rc=1 whenever a>0 and b>0.In this case with α=10, we have a=0.00002083149795 >0 and b=0.4997487034 >0. Thus
backward bifurcation phenomenon occurs at Rc=1. This is shown in the figure (Fig. 2) below.
Fig. 2. Backward Bifurcation Diagram of Model (2)
4. Dynamics of the model considering no re-infection:
Now we will discuss the cases when there is no re-infection.
4.1. Global Stability of DFE with α=0
Theorem 2. The DFE of the COVID-19 model (2)with α=0, given by E0, is globally asymptotically stable (GAS) whenever Rc≤1.
Proof. We consider the following linear Lyapunov function:
L=`1E1+`2E2+`3Ia+`4Is+`5Iac +`6Isc,
where,
`1=σ1
k4k5k6k7k8k9hηek6k7k8k9+k7k8k9d1σ2ηa+k6k8k9d2σ2ηa+k6k7k9d3σ2T1+k6k7k8(1−d)σ2T2i,
8
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`2=k4
σ1
`1, `3=ηak9
k6T2
, `4=k9
k7T2
, `5=T1k9
k8T2
, `6=1.
Differentiating the above Lyapunov function we have the following
˙
L=`1˙
E1+`2˙
E2+`3˙
Ia+`4˙
Is+`5˙
Iac +`6˙
Isc
=`1nλS+T3λSc+ (1−εv)λSv+α λ R−k4E1o+`2(σ1E1−k5E2) + `3(d1σ2E2−k6Ia)+
`4(d2σ2E2−k7Is) + `5(d3σ2E2−k8Iac) + `6n(1−d)σ2E2−k9Isco
= (−`1k4+`2σ1)E1+n`1ηe
β(1−em) (S∗+
τ
3S∗
c+ (1−ε)S∗
v)
N∗∗ −`2k5+`3d1σ2+`4d2σ2+`5d3σ2+`6(1−d)σ2oE2+
n`1ηa
β(1−em) (S∗+
τ
3S∗
c+ (1−ε)S∗
v)
N∗∗ −`3k6oIa+n`1
β(1−em) (S∗+
τ
3S∗
c+ (1−ε)S∗
v)
N∗∗ −`4k7oIs+
n`1
τ
1
β(1−em) (S∗+
τ
3S∗
c+ (1−ε)S∗
v)
N∗∗ −`5k8oIac +n`1
τ
2
β(1−em) (S∗+
τ
3S∗
c+ (1−ε)S∗
v)
N∗∗ −`6k9oIsc
After some rigorous calculation it can be shown that
˙
L≤ηek9
τ
2
τ
2
k9
Rc−1!E2+ηak9
τ
2
τ
2
k9
Rc−1!Ia+k9
τ
2
τ
2
k9
Rc−1!Is+
τ
1k9
τ
2
τ
2
k9
Rc−1!Iac +k9
τ
2
k9
Rc−1!Isc.
Thus
˙
L≤k9
τ
2
τ
2
k9
Rc−1!(ηeE2+ηaIa+Is+T1Iac +T2Isc)
=λN∗∗ k9
β(1−em)
τ
2
τ
2
k9
Rc−1!
<0 for Rc≤k9
τ
2
<1.
Also ˙
L=0 if and only if E2=Ia=Is=Iac =Isc =0. Hence ˙
L≤0. Therefore, Lis a Lyapunov function on Dand thus it follows by the
LaSalle’s invariance principle LaSalle (1976) that, the DFE of the model (2) is globally asymptotic stable whenever Rc≤1.
4.2. Endemic Equilibrium Point (EEP) with α=0:
When α=0 equation (12) reduces to
f(λ∗) = M4λ∗4+M3λ∗3+M2λ∗2+M1λ∗+M0=0.(15)
where,
M4= (1−ε)A1(Wck12 +Wr)+ A4
τ
3(1−ε),
M3=A4
τ
3k3+ (1−ε)A4k2+ (1−ε)A5
τ
3+A7+A1k3(Wck12 +Wr)+(1−ε)A2(Wck12 +Wr) + A4Ω(1−ε)
−β(1−em)(ηeWe+ηaWa+Ws+
τ
1Wac +
τ
2Wsc)A1(1−ε)k12,
M2=A4k2k3+A6
τ
3(1−ε) + A5
τ
3k3+A5(1−ε)k2+A8+ (1−ε)A3(Wck12 +Wr) + A2k3(Wck12 +Wr) + A4Ωk3+A5Ω(1−ε)
−β(1−em)(ηeWe+ηaWa+Ws+
τ
1Wac +
τ
2Wsc)(A1k3k12 +A2(1−ε)k12),
M1=A5k2k3+A6
τ
3k3+ (1−ε)k2A6+A9+A3k3Wck12 +A3k3Wr+ΩA5k3+ΩA6(1−ε)
−β(1−em)(ηeWe+ηaWa+Ws+
τ
1Wac +
τ
2Wsc)(A2k3k12 −A3(1−ε)k12),
M0=ΩA6k3+A6k2k3+A10 −A3β(1−em)(ηeWe+ηaWa+Ws+
τ
1Wac +
τ
2Wsc)k3k12,
where Ai’s, i=1,2,3,....., 10 are the expressions from subsection 3.3 with α=0.
4.2.1. Local Asymptotic Stability of Endemic Equilibrium Point (EEP) with α=0
Using the parameter values as given in Table 2 with α=0, it can be shown that Rc>1 and out of the four roots, one root is real positive,
one root is real negative and other two roots are complex. So there exists a unique endemic equilibrium of the system (2). Again using the
same parameter values in the expression of aand b, we get a=−0.000001860809076 <0 and b=0.4997487034 >0.Thus according to the
Center Manifold Theorem Castillo-Chavez and Song (2004), this unique endemic equilibrium is locally asymptotically stable when Rc>1.
4.2.2. Global Asymptotic Stability of EEP with α=0
Theorem 3. The EEP of the model (2)with no re-infection (α=0)is globally asymptotically stable (GAS) whenever Rc>1.
The graph-theoretic approach discussed in Shuai and Driessche (2013) will be used to construct a Lyapunov function and to prove
this theorem. Using Theorem 3.3, Theorem 3.4 and Theorem 3.5 of Shuai and Driessche (2013), the Lyapunov function can be constructed as
follows:
9
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Proof. The following Lyapunov function is considered:
L1= S−S∗−S∗ln S
S∗!+ Sc−S∗
c−S∗
cln Sc
S∗
c!+ Sv−S∗
v−S∗
vln Sv
S∗
v!+ E1−E∗
1−E∗
1ln E1
E∗
1!,
L2=E2−E∗
2−E∗
2ln E2
E∗
2
,L3=Ia−I∗
a−I∗
aln Ia
I∗
a
,L4=Is−I∗
s−I∗
sln Is
I∗
s
,L5=Iac −I∗
ac −I∗
ac ln Iac
I∗
ac
,
L6=Isc −I∗
sc −I∗
sc ln Isc
I∗
sc
.
Differentiating with respect to twe get
L0
1≤
β ηe(1−em)E∗
2nS∗+
τ
3S∗
c+ (1−ε)S∗
vo
N∗ E2
E∗
2
−ln E2
E∗
2
−E1
E∗
1
+ln E1
E∗
1!=:a12 G12
+
β ηa(1−em)I∗
anS∗+
τ
3S∗
c+ (1−ε)S∗
vo
N∗ Ia
I∗
a
−ln Ia
I∗
a
−E1
E∗
1
+ln E1
E∗
1!=:a13 G13
+
β(1−em)I∗
snS∗+
τ
3S∗
c+ (1−ε)S∗
vo
N∗ Is
I∗
s
−ln Is
I∗
s
−E1
E∗
1
+ln E1
E∗
1!=:a14 G14
+
β
τ
1(1−em)I∗
ac nS∗+
τ
3S∗
c+ (1−ε)S∗
vo
N∗ Iac
I∗
ac
−ln Iac
I∗
ac
−E1
E∗
1
+ln E1
E∗
1!=:a15 G15
+
β
τ
2(1−em)I∗
sc nS∗+
τ
3S∗
c+ (1−ε)S∗
vo
N∗ Isc
I∗
sc
−ln Isc
I∗
sc
−E1
E∗
1
+ln E1
E∗
1!=:a16 G16,
L0
2≤σ1E∗
1 E1
E∗
1
−ln E1
E∗
1
−E2
E∗
2
+ln E2
E∗
2!=:a21 G21,
L0
3≤d1σ2E∗
2 E2
E∗
2
−ln E2
E∗
2
−Ia
I∗
a
+ln Ia
I∗
a!=:a31 G31,
L0
4≤d2σ2E∗
2 E2
E∗
2
−ln E2
E∗
2
−Is
I∗
s
+ln Is
I∗
s!=:a41 G41,
L0
5≤d3σ2E∗
2 E2
E∗
2
−ln E2
E∗
2
−Iac
I∗
ac
+ln Iac
I∗
ac !=:a51 G51,
L0
6≤(1−d)σ2E∗ E2
E∗
2
−ln E2
E∗
2
−Isc
I∗
sc
+ln Isc
I∗
sc !=:a61 G61,
where,
a12 =
β ηe(1−em)E∗
2nS∗+
τ
3S∗
c+ (1−ε)S∗
vo
N∗,a13 =
β ηa(1−em)I∗
anS∗+
τ
3S∗
c+ (1−ε)S∗
vo
N∗,
a14 =
β(1−em)I∗
snS∗+
τ
3S∗
c+ (1−ε)S∗
vo
N∗,a15 =
β
τ
1(1−em)I∗
ac nS∗+
τ
3S∗
c+ (1−ε)S∗
vo
N∗,
a16 =
β
τ
2(1−em)I∗
sc nS∗+
τ
3S∗
c+ (1−ε)S∗
vo
N∗,
a21 =σ1E∗
1,a31 =d1σ2E∗
2,a41 =d2σ1E∗
2,a51 =d3σ2E∗
2,a61 = (1−d)σ2E∗
2.
With the constants aij and A= [ai j ], the following directed graph (Fig. 3) can be constructed. ∑Gi j =0 along each of the cycles on the graph;
for instances, G41 +G14 =0,G61 +G16 =0,and so on. Then by Theorem 3.5, there exist constants ci,i=1,2, ...., 6 such that L=
6
∑
i=1
ciLi
is a Lyapunov function for equation (2). To find the constants ciwe use Theorem 3.3 and Theorem 3.4. d+(2) = 1 we have c2a21 =c1a12 .
Hence setting c1=1 we get c2=
β(1−em)nS∗+
τ
3S∗
c+ (1−ε)S∗
voηeE∗
2
σ1E∗
1N∗.d+(3) = 1 implies c3a31 =c1a13.
Hence setting c1=1 we get c3=
β(1−em)nS∗+
τ
3S∗
c+ (1−ε)S∗
voηaI∗
a
d1σ2E∗
2N∗.d+(4) = 1 implies c4a41 =c1a14.
Hence setting c1=1 we get c4=
β(1−em)nS∗+
τ
3S∗
c+ (1−ε)S∗
voI∗
s
d2σ2E∗
2N∗.d+(5) = 1 implies c5a51 =c1a15.
Hence setting c1=1 we get c5=
β(1−em)nS∗+
τ
3S∗
c+ (1−ε)S∗
vo
τ
1I∗
ac
d3σ2E∗
2N∗.d+(6) = 1 implies c6a61 =c1a16.
Hence setting c1=1 we get c6=
β(1−em)nS∗+
τ
3S∗
c+ (1−ε)S∗
vo
τ
2I∗
sc
(1−d)σ2E∗
2N∗.
10
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Therefore with the functions Li, constants cigiven above and X=
β(1−em)nS∗+
τ
3S∗
c+ (1−ε)S∗
vo
N∗,
L=L1+XηeE∗
2
σ1E∗
1
L2+XηaI∗
a
d1σ2E∗
2
L3+XI∗
s
d2σ2E∗
2
L4+X
τ
1I∗
ac
d3σ2E∗
2
L5+X
τ
2I∗
sc
(1−d)σ2E∗
2
L6is a Lyapunov function for (2). One
can easily verify that for the system (2) with this Lyapunov function and with L0=0 the largest invariant set will be the set E1. Hence, using
LaSalle’s invariance principle LaSalle (1976), we can say that E1is globally asymptotically stable in the interior of D.
Fig. 3. Directed graph of the system (2)
5. Optimal Control
In this section, to control the spread of Covid-19, we reconsider the model (1) and formulate an optimal control problem with four control
variables u1(t),u2(t),u3(t)and u4(t). The control u1(t)aims the efforts to increase awareness towards preventing COVID-19 infections by
susceptible individuals (S), co-morbid susceptible humans (Sc), vaccination individuals (Sv)and recovered individuals (R)through various
awareness program. Control u2(t)ensures the implementation of continuous vaccination, increase of vaccination rate and spread of vaccination
program nationwide. u3(t)is COVID-19 detection control that represents the fraction of symptomatic individuals (Isand Isc)that are identified
and quarantined for prevention of contacts with susceptible individuals. u4(t)represents the control that ensures better treatment for hospitalized
individuals. Thus the revised model becomes:
˙
S=Λ−(1−u1)λS−ΩS−ξs(1+u2)S−µS,
˙
Sc=ΩS−(1−u1)T3λSc−θ1ξs(1+u2)Sc−µSc,
˙
Sv=ξs(1+u2)S+θ1ξs(1+u2)Sc+θ2ξs(1+u2)R−(1−u1)(1−ε)λSv−k3Sv,
˙
E1= (1−u1)λS+T3Sc+ (1−ε)Sv+αR−k4E1,
˙
E2=σ1E1−k5E2,
˙
Ia=d1σ2E2−k6Ia,
˙
Is=d2σ2E2−σs(1+u3)Is−ψs+φs+δs+µIs,
˙
Iac =d3σ2E2−k8Iac,
˙
Isc = (1−d)σ2E2−σsc (1+u3)Isc −ψsc +φsc +δsc +µIsc,
˙
Q=σs(1+u3)Is+σsc (1+u3)Isc −k10 Q,
˙
H=φsIs+φsc Isc +φqQ−ψh(1+u4)H−(δh+µ)H,
˙
R=ψaIa+ψsIs+ψac Iac +ψsc Isc +ψqQ+ψh(1+u4)H−(1−u1)α λ R−θ2ξs(1+u2)R−µR.(16)
The objective of optimal control system is to find the controls that minimize the total infected individuals and the cost of implementing the
controls, that is, to find the minimal values of u1,u2,u3and u4subject to the state system (16). For this, we consider a quadratic objective
functional of the following form:
J(u1,u2,u3,u4) = ZT
0hD1E2+D2Ia+D3Is+D4Iac +D5Isc +1
2F1u2
1+F2u2
2+F3u2
3+F4u2
4idt (17)
11
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The positive coefficients D1,D2,D3,D4,F1,F2,F3and F4are balancing weight parameters, while the controls u1,u2,u3and u4are bounded,
Lebesgue integrable functions.
Theorem 1. Let the set of controls for problem (16)be Lebesgue integrable functions (instead of just piecewise continuous functions) on t0≤
t≤t1with values in R. Then there exists an optimal control u∗= (u∗
1,u∗
2,u∗
3,u∗
4∈U)such that J(u∗
1,u∗
2,u∗
3,u∗
4) = min nJ(u1,u2,u3,u4):
u1(t),u2(t),u3(t),u4(t)∈Uo,
where U=n(u1,u2,u3,u4):ui(t)is measurable on [0,T],0≤ui(t)≤1,i=1,2,3,4ois the closed set subject to the control system if the
following conditions are satisfied Fleming and Rishel (2012)
1. The set of state variables and controls is non-empty.
2. The control and state variables are non-negative values.
3. The control set Uis convex and closed.
4. The integrand of the objective functional is convex on U.
5. Successful responses on [0,T]satisfy an a priori bound :
|x(t;x0,u(.))| ≤ α,f or al l u(.)∈U(T),0<t≤t1
where α=α(T)is a constant depending only on T. This condition is implied by the followings:
(a) |g(t,x1,u)| ≤ C1(1+|x|+|u|)
(b) |g(t,x1,u)−g(t,x,u)| ≤ C2|x1−x|(1+|u|)
6. There exists constants C3,C4>0and C5such that L(t;u1;u2;u3;u4)satisfies
L(t;u1;u2;u3;u4)≥C3+C4|u1|2+|u2|2+|u3|2+|u4|2
C5
2
Proof. Let us consider the following basic optimal control problem in the form of ordinary differential equation
˙x=g(t,x(t),u(t)),x(0) = x0,u(.)∈Umwith associated cost C[u(.)] = ZT1
0
f(t,x(t),u(t))dt,
where x(t)represents state variable and Urepresents control and f,gare given continuous functions with values in Rnand R.
1. Let Ube the class of all admissible controls in time t1,0<t1≤T.Obviously for some T,U(T)is non-empty, U(T)6=/0 , since we
can’t have an optimal control without at least one successful control. To prove that the set of controls is nonempty, we will use a
simplified version of an existence theorem (Theorem 7.1.1) from Boyce and DiPrima (2020). Consider S=x1,Sc=x2,Sv=x3,
E1=x4,E2=x5,Ia=x6,Is=x7,Iac =x8,Isc =x9,Q=x10,H=x11 , and R=x12 , and thus in vector notation the system (16)
becomes dX
dt =F(t;X), where X= (x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11 ,x12)Tand
F= ( f1,f2,f3,f4,f5,f6,f7,f8,f9,f10 ,f11,f12 )T. Let u1,u2,u3,and u4are some constants. Since all parameters are constants and
all xi’s are continuous, then all fi’s are also continuous (i=1,2,....., 12). Additionally, the partial derivatives ∂fi
∂xi,i=1,2,....., 12 are
also continuous. Therefore, there exists a unique solution (S,Sc,Sv,E1,E2,Ia,Is,Iac,Isc,Q,H,R)that satisfies the initial conditions.
Thus, the set of controls and the corresponding state variables is nonempty and hence condition 1 is satisfied.
2. It is obvious that the set of state variables and controls are non-negative.
3. Let u,v∈Uand r∈[0,1], then obviously r u + (1−r)v≥0.Again r u ≤rand (1−r)v≤(1−r)
Thus, r u + (1−r)v≤r+ (1−r) = 1. Hence we have 0 ≤r u + (1−r)v≤1.
Thus the control space
U=n(u1,u2,u3,u4):(u1,u2,u3,u4)is measurable and 0 ≤u1min ≤u1(t)≤u1max ≤1,
0≤u2min ≤u2(t)≤u2max ≤1,0≤u3min ≤u3(t)≤u3max ≤1,0≤u4min ≤u4(t)≤u4max ≤1ois convex.
4. The integrand of the objective functional is given by
L(t;u1;u2;u3;u4) = D1E2+D2Ia+D3Is+D4Iac +D5Isc +1
2(F1u2
1+F2u2
2+F3u2
3+F4u2
4)
Here Lis a twice differentiable function of many variables on the convex set Uand let Hdenotes the Hessian of L. We can determine
the (strict) convexity of Lby determining whether the Hessian is positive (definite) semi-definite. The second partial derivatives of Lare
Lu1u1=F1,Lu1u2=0,Lu1u3=0,Lu1u4=0,
Lu2u1=0,Lu2u2=F2,Lu2u3=0,Lu2u4=0,
Lu3u1=0,Lu3u2=0,Lu3u3=F3,Lu3u4=0,
Lu4u1=0,Lu4u2=0,Lu4u3=0,Lu4u4=F4,
So its Hessian is
12
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H=




F10 0 0
0F20 0
0 0 F30
0 0 0 F4





,
The Hessian is positive definite and hence Lis strictly convex.
5. Consider g(t,x,u) = α(t,x) + uβ(t,x)and assume that g(t,x,u)is of class C1and
|g(t,0,0)| ≤ C,|gx(t,x,u)| ≤ C(1+|u|),|gu(t,x,u)| ≤ Cfor some constant C.
Applying Mean Value Theorem we get
g(t,x1,u)−g(t,x,u)
x1−x=gx(t,x,u) =⇒g(t,x1,u)−g(t,x,u) = (x1−x)gx(t,x,u)
|g(t,x1,u)−g(t,x,u)|=|x1−x||gx(t,x,u)| ≤ |x1−x|C(1+|u|)
Therefore, |g(t,x1,u)−g(t,x,u)| ≤ C|x1−x|(1+|u|)
g(t,x,0)−g(t,0,0)
x=gx(t,x,0) =⇒g(t,x,0)−g(t,0,0) = x gx(t,x,0)
|g(t,x,0)−g(t,0,0)|=|x gx(t,x,0)|
|g(t,x,0)|−|g(t,0,0)| ≤ |x||gx(t,x,0)|as |g(t,x,0)|−|g(t,0,0)| ≤ |g(t,x,0)−g(t,0,0)|
|g(t,x,0)| −C≤C|x|=⇒ |g(t,x,0)| ≤ C|x|+C=⇒ |g(t,x,0)| ≤ C(1+|x|)
Now
g(t,x,u)−g(t,x,0)
u=gu(t,x,u) =⇒g(t,x,u)−g(t,x,0) = u gu(t,x,u)
|g(t,x,u)−g(t,x,0)|=|u gx(t,x,u)|=⇒ |g(t,x,u)|−|g(t,x,0)| ≤ |u| |gx(t,x,u)|
|g(t,x,u)| −C(1+|x|)≤C|u|=⇒ |g(t,x,u)| ≤ C|u|+C(1+|x|)
Therefore, |g(t,x,u)| ≤ C(1+|x|+|u|).
6. The state variables being bounded,
let C3=min(D1E2+D2Ia+D3Is+D4Iac +D5Isc),C4=min F1
2+F2
2+F3
2+F4
2,and C5=2.Then it follows that
L(t;u1;u2;u3;u4)satisfies
L(t;u1;u2;u3;u4)≥C3+C4|u1|2+|u2|2+|u3|2+|u4|2
C5
2for all twith t0≤t≤t1,x,x1,uin R.
After establishing the existence of an optimal control, to obtain the necessary conditions for the optimal solution, we applied
Pontryagin’s maximum principle Pontryagin (1987) to the Hamiltonian Hdefined by
H=D1E2+D2Ia+D3Is+D4Iac +D5Isc +1
2(F1u2
1+F2u2
2+F3u2
3+F4u2
4)
+g1hΛ−(1−u1)λS−ΩS−ξs(1+u2)S−µSi
+g2hΩS−(1−u1)T3λSc−θ1ξs(1+u2)Sc−µSci
+g3hξs(1+u2)S+θ1ξs(1+u2)Sc+θ2ξs(1+u2)R−(1−u1)(1−ε)λSv−µSvi
+g4h(1−u1)λ(S+T3Sc+ (1−ε)Sv+αR)−k4E1i
+g5hσ1E1−k5E2i
+g6hd1σ2E2−k6Iai
+g7hd2σ2E2−σs(1+u3)Is−(ψs+φs+δs+µ)Isi
+g8hd3σ2E2−k8Iaci
+g9h(1−d)σ2E2−σsc (1+u3)Isc −(ψsc +φsc +δsc +µ)Isci
+g10 hσs(1+u3)Is+σsc (1+u3)Isc −k10 Qi
+g11 hφsIs+φsc Isc +φqQ−ψh(1+u4)H−(δh+µ)Hi
+g12 hψaIa+ψsIs+ψac Iac +ψsc Isc +ψqQ+ψh(1+u4)H−(1−u1)α λ R−θ2ξs(1+u2)R−µRi,(18)
where gi,i=1,2,......, 12 are the adjoint variables.
Theorem 2. Given an optimal control (u∗
1,u∗
2,u∗
3,u∗
4)and corresponding state solutions S1=S,S2=Sc,S3=Sv,S4=E1,S5=E2,S6=
Ia,S7=Is,S8=Iac,S9=Isc,S10 =Q,S11 =H,S12 =R of the corresponding state system (16), there exists adjoint variables, gi,i=
13
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1,2,......, 12 satisfying
d gi
d t =−∂H
∂Si
with transversality conditions gi(T) = 0, where, i =1,2, ......, 12 and control set (u∗
1,u∗
2,u∗
3,u∗
4)characterized by
u∗
1=maxn0,min1,(g4−g1)λS+ (g4−g2)
τ
3λSc+ (g4−g3)(1−ε)λSv+ (g4−g12)α λ R
F1o,
u∗
2=maxn0,min1,(g1−g3)ξsS+ (g2−g3)θ1ξsSc+ (g12 −g3)θ2ξsR
F2o,
u∗
3=maxn0,min1,(g7−g10)σsIs+ (g9−g10)σsc Isc
F3o,
u∗
4=maxn0,min1,(g11 −g12)ψhH
F4o.
(19)
Proof.
d g1
d t =−∂H
∂S=g1n(1−u1)λ+ (Ω+ξs(1+u2)) + (1−u1)S−λ
N∗o−g2nΩ−(1−u1)
τ
3Sc
−λ
N∗o,
−g3n(ξs(1+u2)−(1−u1)(1−ε)Sv
−λ
N∗o−g4n(1−u1)λ−(1−u1)(S+T3Sc+ (1−ε)Sv+αR)−λ
N∗o,
−g12 n(1−u1)αRλ
N∗o,
d g2
d t =−∂H
∂Sc
=−g1n(1−u1)S−λ
N∗o+g2n(1−u1)
τ
3λ+θ1ξs(1+u2)−(1−u1)