Optimal Vaccination and Treatment Strategies in Reduction of COVID-19 Burden
Bishal Chhetri, D.k.k. Vamsi, S Balasubramanian, Carani B Sanjeevi
OOptimal Vaccination and Treatment Strategies in Reduction of COVID-19 Burden
Bishal Chhetri ,a , D. K. K. Vamsi ∗ ,a , S Balasubramanian b , Carani B Sanjeevi c,d a,b Department of Mathematics and Computer Science, Sri Sathya Sai Institute of Higher Learning, Prasanthi Nilayam , Puttaparthi, Anantapur District - 515134, Andhra Pradesh, India c Vice-Chancellor, Sri Sathya Sai Institute of Higher Learning - SSSIHL, India d Department of Medicine, Karolinska Institute, Stockholm, [email protected], [email protected] ∗ ,[email protected], [email protected] First Author , ∗ Corresponding Author
Abstract
In this study, we formulate a mathematical model incorporating age specific transmission dynamics ofCOVID-19 to evaluate the role of vaccination and treatment strategies in reducing the size of COVID-19burden. Initially, we establish the positivity and boundedness of the solutions of the model and calculate thebasic reproduction number. We then formulate an optimal control problem with vaccination and treatmentas control variables. Optimal vaccination and treatment policies are analysed for different values of the weightconstant associated with the cost of vaccination and different transmissibility levels. Findings from thesesuggested that the combined strategies(vaccination and treatment) worked best in minimizing the infectionand disease induced mortality. In order to reduce COVID-19 infection and COVID-19 induced deaths tomaximum, it was observed that optimal control strategy should be prioritized to population with age greaterthan 40 years. Not much difference was found between individual strategies and combined strategies in caseof mild epidemic ( R ∈ (0 , R ( R ∈ (2 , Mathematical modeling of infectious diseases such as COVID-19, influenza, dengue, HIV/AIDS etc. is one of themost important research areas today. Mathematical epidemiology has contributed to a better understandingof the dynamical behavior of these infectious diseases, its impacts, and possible future predictions about itsspreading. Mathematical models are used in comparing, planning, implementing, evaluating, and optimizingvarious detection, prevention, therapy, and control programs. COVID-19 is one such contagious respiratory andvascular disease that has shaken the world today. It is caused by severe acute respiratory syndrome coronavirus2 (SARS-CoV-2). On 30 january it was declared as a Public Health Emergency of International Concern. Asof latest statistics(on 24 January 2021) of COVDI-19, around 96.2 million cases has been reported and around2 million have died worldwide. Several mathematical models has been developed to understand the dynamicsof the disease. In [6] a basic within host model is developed to determine the crucial inflammatory mediatorsand the role of combined drug therapy in the treatment of COVID-19. A SAIU compartmental mathematicalmodel that explains the transmission dynamics of COVID-19 is developed in [23]. The role of some of thecontrol policies such as treatment, quarantine, isolation, screening, etc. are also applied to control the spreadof infectious diseases [9, 18, 3]. COVID-19 has caused the most severe health issues for adults over the age of60 with particularly fatal results for those 80 years and older. This is due to the number of underlying healthconditions present in older population [1]. A mathematical model for estimating the age-specific transmissibilityof a novel coronavirus is developed in [27]. In this study the age age-specific SEIARW model was fitted with thereported data well by dividing the population into four age groups and the results from this study suggestedthat the highest transmissibility occurred from age group 1 −
14 to 15 − a r X i v : . [ q - b i o . P E ] F e b harat Biotech’s Covaxin [2]. Several mathematical models are developed to study the role of vaccination andtreatments in reducing the disease burden. In [5] a mathematical model is used to compare five age-stratifiedprioritization strategies. A highly effective transmission-blocking vaccine prioritized to adults ages 20-49 yearswas found to minimize the cumulative incidence, whereas mortality and years of life lost were minimized in mostscenarios when the vaccine was prioritized to adults over 60 years old. Reports from Israel suggested that onedose of Pfizer vaccine could be less effective than expected [19]. A two-dose regimen of BNT162b2 conferred 95%protection against Covid-19 in persons 16 years of age or older. Safety over a median of 2 months was similar tothat of other viral vaccines [22]. The mRNA-1273 vaccine showed 94.1% efficacy at preventing Covid-19 illness,including severe disease [4]. The combination vaccines for protection against multiple diseases began with thecombination of individual diphtheria, tetanus, and pertussis (DTP) vaccines into a single product; this combinedvaccine was first to be used to vaccinate infants and children in 1948. Over the years we have seen the additionof other vaccines to the combination and the replacement of components to improve its reactogenicity profile[25]. The addition of inactivated polio, Haemophilus influenzae, and hepatitis B vaccines into the combinationhas facilitated the introduction of these vaccines into recommended immunization schedules by reducing thenumber of injections required and has therefore increased immunization compliance [25].To reflect the real behavior of some infectious diseases and to make models more realistic, many researchershave proposed and analyzed more realistic models including delays to model different mechanisms in the dy-namics of epidemics like latent period, temporary immunity and length of infection [15, 26]. An optimal controlproblem with time delay in both the state variable and control variable is studied in [11].Motivated by the above, in this study, we consider a nine compartment age structured model to study therole of individual vaccines, combination vaccines and treatment in reducing the COVID-19 infection. In themodel we incorporate time delay in both the control and state variables.The paper is organised as follows: In section 2 we formulate a mathematical model explaining the detailsof the parameters and variables used and establish the positivity and boundedness of the solutions. In section3 we formulate an optimal control problem to evaluate the role of vaccination and treatment in reducing thecumulative infection and disease induced mortality. Numerical simulation is presented in section 4 followed bydiscussion and conclusion in section 5. 2 Model Formulation
Various mathematical models has been developed and studied to understand the dynamics of COVID-19 anddesign optimal control strategies to control an epidemic. In this work we formulate an optimal control prob-lem with age specific transmission dynamics of COVID-19. The total population in the model is divided intodifferent compartments such as susceptible( S i ), vaccinated but not protected( V i ), ineffectively vaccinated( F i ),Protected( P i ), exposed( E i ), infected( I i ), hospitalized( J i ), recovered( R i ) and deaths( D i ) for i = 1 ,
2. We con-sider two age groups here, the first between 0-40 years and second group with age greater than 40 years. Atany point in time we assume that the individuals will be in one of these compartments. When susceptible indi-viduals in age group i come in close contact with the infected or hospitalized they become exposed to the virusat rates β ij where, β ij is the transmission rate between age groups i and j . Exposed individuals E i progressto the infectious class I i at the rate k (where 1/k is the mean latent period). The term α i e − γτ gives the rateat which infected are hospitalized. Here τ represents the delay in hospitalization and with increasing value ofdelay or γ the rate of movement to J i compartment is less [21], d i and γ are the disease induced death rateand recovery rate of the infected individuals. Hospitalized individuals either recover at the constant rate γ ordie at the age-specific rate d i .We employ time-dependent (age-specific) control functions to measure the effectiveness of age-specific vac-cination and treatment policies aimed at minimizing the number of infected individuals during the pandemic.The control functions µ i ( t ) and µ i ( t ) determine the age-specific vaccination rates of susceptible individuals( S i ) per unit of time for each age group i . We assume that the suceptibles are given both the vaccines togetherat the same time and only those individuals who were vaccinated at time ( t − τ ) will now move to V i , F i or P i compartment. The control variables µ i , µ i represents the age specific treatment rates for infected and hospi-talized population respectively. To make model realistic we assume that there is a time lag between treatmentand recovery represented by τ and τ for infected and hospitalized population respectively. The dynamic modelwith age-specific controls is described by the following system of nonlinear differential equations: dS i dt = ω i − (cid:88) j =1 β ij ( I j + J j ) S i − µ i ( t − τ ) S i ( t − τ ) − µ i ( t − τ ) S i ( t − τ ) − µS i (1) dV i dt = (cid:15) i µ i ( t − τ ) S i ( t − τ ) + γ i µ i ( t − τ ) S i ( t − τ ) − (cid:88) j =1 β ij ( I j + J j ) V i − µV i (2) dF i dt = (cid:15) i µ i ( t − τ ) S i ( t − τ ) + γ i µ i ( t − τ ) S i ( t − τ ) − (cid:88) j =1 β ij ( I j + J j ) F i − µF i (3) dP i dt = (1 − (cid:15) i − (cid:15) i ) µ i ( t − τ ) S i ( t − τ ) + (1 − γ i − γ i ) µ i ( t − τ ) S i ( t − τ ) − µP i (4) dE i dt = (cid:88) j =1 β ij ( I j + J j ) (cid:18) S i + V i + F i (cid:19) − kE i − µE i (5) dI i dt = kE i − d i I i − α i e − γτ I i ( t − τ ) − µ i ( t − τ ) I i ( t − τ ) − γI i (6) dJ i dt = α i e − γτ I i ( t − τ ) − d i J i − µ i ( t − τ ) J i ( t − τ ) (7) dR i dt = γI i + µ i ( t − τ ) J i ( t − τ ) + µ i ( t − τ ) I i ( t − τ ) − µR i (8) dD i dt = d i I i + d i J i − µD ( i ) (9) OBJECTIVES OF THE PROPOSED STUDY
1. To study and compare the dynamics of cumulative infection, hospitalized and mortality with and withoutthe controls.2. To determine which age groups should be prioritized for COVID pandemic vaccination.3. To study and compare the dynamics of infected and hospitalized population with varying efficacies of thevaccine.4. To study and compare the dynamics of infected and death population with varying cost of implementationof vaccination strategy. 3able 1
Symbols Biological Meaning S i Suceptible Population V i effectively Vaccinated but not protected F i ineffectively vaccinated P i Protected Population E i Exposed Population I i infected Population J i hospitalized Population R i recovered Population ω i Rate of entries in each groups β ij transmission rates among different age groups µ i rate of decrease in suceptibles due to first vaccine µ i rate of decrease in suceptibles due to second vaccine µ Natural death rate d disease induced death rates for first infected population d death for second infected group d disease induced death rates for first group hospitailized population d disease induced death rates for second group hospitailized population k infection rates α i rates at which infected are hospitalized µ i recovery rate of infected due to treatment µ i recovery rate of hospitalized due to treatment (cid:15) i , (cid:15) i efficacy of first vaccine γ i , γ i efficacy of second vaccine γ natural recovery rate 4 ositivity and Boundedness For any mathematical model it is fundamental to show that the system of equations considered are positive andhas bounded solutions. We now show that if the initial conditions of the system (3.1)-(3.9) are positive, thenthe solution remain positive for any future time. Using the equations (3.1)-(3.9), we get, dS i dt (cid:12)(cid:12)(cid:12)(cid:12) S i =0 ≥ , dV i dt (cid:12)(cid:12)(cid:12)(cid:12) V i =0 = (cid:15) i µ i S i + γ i µ i S i ≥ ,dF i dt (cid:12)(cid:12)(cid:12)(cid:12) F i =0 = (cid:15) i µ i S i + γ i µ i S i ≥ . dD i dt = d i I i + d i J i (cid:12)(cid:12)(cid:12)(cid:12) J i =0 ≥ dE i dt (cid:12)(cid:12)(cid:12)(cid:12) E i =0 = (cid:88) j =1 β ij ( I j + J j ) (cid:18) S i + V i + F i (cid:19) ≥ . dI i dt (cid:12)(cid:12)(cid:12)(cid:12) I i =0 = kE i ≥ .dJ i dt (cid:12)(cid:12)(cid:12)(cid:12) J i =0 = α i e − γτ I i ≥ . dR i dt (cid:12)(cid:12)(cid:12)(cid:12) R i =0 = γI i + µ i J i + µ i I i ≥ .dP i dt (cid:12)(cid:12)(cid:12)(cid:12) P i =0 = (1 − (cid:15) i − (cid:15) i ) µ i S i + (1 − γ i − γ i ) µ i S i ≥ . Thus all the above rates are non-negative on the bounding planes (given by S i = 0, V i = 0, P i = 0, F i = 0, E i =0, I i = 0, J i = 0, R i = 0, D i = 0) of the non-negative region of the real space. So, if a solution begins in the interiorof this region, it will remain inside it throughout time t . This happens because the direction of the vector fieldis always in the inward direction on the bounding planes as indicated by the above inequalities. Hence, weconclude that all the solutions of the the system (3.1)-(3.9) remain positive for any time t > Boundedness:
Let N i ( t ) = S i ( t ) + V i ( t ) + F i ( t ) + P i + E i + I i + J i + R i + D i Now, dN i dt = dS i dt + dV i dt + dF i dt + dP i dt + dE i dt + dI i dt + dJ i dt + dR i dt + dD i dt = (cid:18) ω i + µ ( I i + J i ) (cid:19) − µN ( t ) ≤ (cid:18) ω i + µ ( I i + J i ) (cid:19) ≤ µN ( t ). This implies that N i ( t ) = C , where C is a constantThus we have shown that the system (2.1)-(2.9) is positive and bounded for each bounded controls considerd.Therefore the biologically feasible region is given by the following set,Ω = (cid:26)(cid:18) S i ( t ) , V i ( t ) , P i ( t ) , F i ( t ) , E i ( t ) , I i ( t ) , J i ( t ) , R i ( t ) , D i ( t ) (cid:19) : N i ( t ) ≤ C, t ≥ (cid:27) R The basic reproduction number which is the average number of secondary cases produced per primary case iscalculated using the next generation matrix method described in [8] at infection free equilibrium. Our system(2.1)-(2.9) has four infected states ( E , E , I , I ). In order to see the behaviour of the optimal strategieswith varying transmissibility we calculate the basic reproduction number. Calculating the jacobian matrix atinfection free equilibrium E (which has only susceptible component) we have,5 ( E ) = − k − µ β S ∗ β S ∗ − k − µ β S ∗ β S ∗ k − d − γ − α e − γτ k − d − γ − α e − γτ or, J ( E ) = F + V where, F describes transmission of new infection and V describes changes in the state including removal bydeath or recovery rate.Matrix F and V are given as, F = β S ∗ β S ∗ β S ∗ β S ∗ V = − k − µ − k − µ k − d − γ − α e − γτ k − d − γ − α e − γτ Calculating the inverse of V we get, V − = − k − µ − k − µ − k ( k + µ )( d + γ + α e − γτ ) − d − γ − α e − γτ − k ( k + µ )( d + γ + α e − γτ ) − d − γ − α e − γτ Now − F V − = β kS ∗ p β kS ∗ q β S ∗ p β S ∗ ( k + µ )( qβ kS ∗ p β kS ∗ q β S ∗ p β S ∗ q where p = ( k + µ )( d + γ + α e − γτ ) q = ( k + µ )( d + γ + α e − γτ )Since the last two rows of matrix − F V − has all zeros as discussed in [8] we define an auxillary matrix andnew matrix K as, E = K = β kS ∗ p β kS ∗ qβ kS ∗ p β kS ∗ q K is given by, R = β kS ∗ ( k + µ )( d + γ + α e − γτ ) + β kS ∗ ( k + µ )( d + γ + α e − γτ ) Now we frame an optimal control problem with vaccination and treatment as controls. Our aim is to studythe role and efficacies of these controls and design an optimal control policy that minimizes that infection anddisease caused mortality. The controls that we consider are as follows:1.
Vaccination:
Vaccination is the most effective method of preventing infectious diseases. The susceptiblesub population are given vaccine to stimulates the body’s immune system to recognize the agent as a threatand destroy it, thereby preventing transmission of the disease among susceptible individual. Vaccination alsofurther helps in recognizing and destroying any of the microorganisms associated with that agent that it mayencounter in the future. The first control that we consider here is vaccination. We assume that combination ofvaccines is given to an infected individual and denote it by variable µ i (first vaccine) and µ i (second vaccine)for two age groups respectively.2. Treatment:
Infected and Hospitalized sub-population are given treatment to reduce the burden ofdisease and control the spread of infection. Studies in [6] suggested the combined use of immunomodulators andantiviral agents as a best treatment strategy to reduce the burden of COVID-19. Therefore the second controlthat we consider here is treatments to infected and hospitalized population. These treatments could be eitherimmunomodelators such as INF, to boost the immune response or anti viral agents like remdesivir, arbidol etc.that inhibits the viral replication. We denote this control variable by µ i and µ i .Let U = ( µ , µ ), U = ( µ , µ ), U = ( µ , µ ) and U = ( µ , µ )The set of all admissible controls is given by U = { ( U , U , U , U ) : U ∈ [0 , U max ] , U ∈ [0 , U max ] , U ∈ [0 , U max ] , U ∈ [0 , U max ] , t ∈ [0 , T ] } In order to reduce the complexity of the problem here we choose to model the control efforts via a linearcombination of the quadratic terms. Also when the objective function is quadratic with respect to the control,differential equations arising from optimization have a known solution. Other functional forms sometimes leadto systems of differential equations that are difficult to solve ([10], [16]). Based on these we now propose anddefine the optimal control problem with the goal to reduce the cost functional defines as follows, J ( U , U , U , U ) = (cid:82) T (cid:18) I ( t ) + I ( t ) + A ( µ ( t ) + µ ( t ) ) + A ( µ ( t ) + µ ( t ) )+ A ( µ ( t ) + µ ( t ) ) + A ( µ ( t ) + µ ( t ) ) (cid:19) dt (3)such that u = (cid:18) µ ( t ) , µ ( t ) , µ ( t ) , µ ( t ) , µ ( t ) , µ ( t ) , µ ( t ) , µ ( t ) (cid:19) ∈ U subject to the system 7 S i dt = ω i − (cid:88) j =1 β ij ( I j + J j ) S i − µ i ( t − τ ) S i ( t − τ ) − µ i ( t − τ ) S i ( t − τ ) − µS i (10) dV i dt = (cid:15) i µ i ( t − τ ) S i ( t − τ ) + γ i µ i ( t − τ ) S i ( t − τ ) − (cid:88) j =1 β ij ( I j + J j ) V i − µV i (11) dF i dt = (cid:15) i µ i ( t − τ ) S i ( t − τ ) + γ i µ i ( t − τ ) S i ( t − τ ) − (cid:88) j =1 β ij ( I j + J j ) F i − µF i (12) dP i dt = (1 − (cid:15) i − (cid:15) i ) µ i ( t − τ ) S i ( t − τ ) + (1 − γ i − γ i ) µ i ( t − τ ) S i ( t − τ ) − µP i (13) dE i dt = (cid:88) j =1 β ij ( I j + J j ) (cid:18) S i + V i + F i (cid:19) − kE i − µE i (14) dI i dt = kE i − d i I i − α i e − γτ I i ( t − τ ) − µ i ( t − τ ) I i ( t − τ ) − γI i (15) dJ i dt = α i e − γτ I i ( t − τ ) − d i J i − µ i ( t − τ ) J i ( t − τ ) (16) dR i dt = γI i + µ i ( t − τ ) J i ( t − τ ) + µ i ( t − τ ) I i ( t − τ ) − µR i (17) dD i dt = d i I i + d i J i − µD i (18)Here, the cost function (3) represents the number of total infected cells, and the overall cost for the im-plementation vaccines and treatments. Effectively, our aim is to minimize the total infected population andthe overall cost. The integrand of the cost function (3), denoted by L ( S, I, V, U , U , U ) = (cid:18) I ( t ) + I ( t ) + A ( µ ( t ) + µ ( t ) ) + A ( µ ( t ) + µ ( t ) ) + A ( µ ( t ) + µ ( t ) ) + A ( µ ( t ) + µ ( t ) ) (cid:19) is called theLagrangian or the running cost.The admissible solution set for the Optimal Control Problem (3)-(3.9) is given byΩ = { ( S i , V i , F i , P i , E i , I i , J i , R i , E i , U , U , U , U ) | S i , V i , F i , P i , E i , I i , J i , R i , D i satisfy(3 . − (3 . } for all u ∈ U EXISTENCE OF OPTIMAL CONTROL
We will show the existence of optimal control functions that minimize the cost functions within a finite timespan [0 , T ] showing that we satisfy the conditions stated in Theorem 4.1 of [12].
Theorem 1.
There exists a 8-tuple of optimal controls (cid:18) µ ∗ ( t ) , µ ∗ ( t ) , µ ∗ ( t ) , µ ∗ ( t ) , µ ∗ ( t ) , µ ∗ ( t ) ,µ ∗ ( t ) , µ ∗ ( t ) (cid:19) in the set of admissible controls U such that the cost functional is minimized i.e., J [ U ∗ , U ∗ , U ∗ , U ∗ ] = min ( U ∗ ,U ∗ ,U ∗ ,U ∗ ) ∈ U (cid:26) J [ U ∗ , U ∗ , U ∗ , U ∗ ] (cid:27) corresponding to the optimal control problem (3)-(3.9).Proof. In order to show the existence of optimal control functions, we will show that the following conditionsare satisfied :1. The solution set for the system (3.1)-(3.9) along with bounded controls must be non-empty, i.e. , Ω (cid:54) = φ .2. U is closed and convex and system should be expressed linearly in terms of the control variables withcoefficients that are functions of time and state variables.3. The Lagrangian L should be convex on U and L ( S i , V i , F i , P i , E i , I i , J i , R i , D i ) ≥ g ( U , U , U , U ), where g ( U , U , U , U ) is a continuous function of control variables such that | ( U , U , U , U ) | − g ( U , U , U , U ) →∞ whenever | ( U , U , U , U ) | → ∞ , where | . | is an l (0 , T ) norm.8ow we will show that each of the conditions are satisfied :1. From Positivity and boundedness of solutions of the system (3.1)-(3.9), all solutions are bounded for eachbounded control variable in U . Also clearly the RHS of the system (3.1)-(3.9) is lipschitz continuous. UsingPicard-Lindelof Theorem[20], we have satisfied condition 1.2. U is closed and convex by definition. Also, the system (3.1)-(3.9) is clearly linear with respect to controlssuch that coefficients are only state variables or functions dependent on time. Hence condition 2 is satisfied.3. Choosing g ( U , U , U , U ) = c ( µ + µ + µ + µ + µ + µ + µ + µ ) such that c = min { A , A , A , A } ,we can satisfy the condition 3.Hence there exists a control 8-tuple ( µ + µ + µ + µ + µ + µ + µ + µ )) ∈ U that minimizes thecost function (3). CHARACTERIZATION OF OPTIMAL CONTROL
We will obtain the necessary conditions for optimal control functions using the Pontryagin’s MaximumPrinciple with delay in state and control variables [13] and also obtain the characteristics of the optimal controls.The Hamiltonian for this problem is given by H = (cid:88) j =1 (cid:18) I i + A ( µ i ( t ) + A µ i ( t ) + A µ i ( t ) + A µ i ( t )) (cid:19) + (cid:88) j =1 λ S i dS i dt + (cid:88) j =1 λ V i dV i dt + (cid:88) j =1 λ F i dF i dt + (cid:88) j =1 λ P i dP i dt + (cid:88) j =1 λ E i dE i dt + (cid:88) j =1 λ I i dI i dt + (cid:88) j =1 λ J i dJ i dt + (cid:88) j =1 λ R i dR i dt Here λ = ( λ S i , λ V i , λ F i , λ P i , λ E i , λ I i , λ J i , λ R i ) is called co-state vector or adjoint vector.Now the Canonical equations that relate the state variables to the co-state variables are given byd λ S i d t = − ∂H∂S i − χ [0 ,T − τ ] ( t ) ∂H ( t + τ ) ∂S i ( t − τ )d λ V i d t = − ∂H∂V i d λ F i d t = − ∂H∂F i d λ P i d t = − ∂H∂P i d λ E i d t = − ∂H∂E i d λ I i d t = − ∂H∂I i − χ [0 ,T − τ ] ( t ) ∂H ( t + τ ) ∂I i ( t − τ ) − χ [0 ,T − τ ] ( t ) ∂H ( t + τ ) ∂I i ( t − τ )d λ J i d t = − ∂H∂J i − χ [0 ,T − τ ] ( t ) ∂H ( t + τ ) ∂J i ( t − τ )d λ R i d t = − ∂H∂R i (19)Substituting the Hamiltonian value gives the canonical system9 λ S i d t = (cid:18) (cid:88) j =1 β ij ( I j + J j ) + µ (cid:19) λ S i − χ [0 ,T − τ ] ( t )( − µ i − µ i ) λ S i ( t + τ ) − χ [0 ,T − τ ] ( t )( (cid:15) i µ i + γ i µ i ) λ V i ( t + τ ) − χ [0 ,T − τ ] ( t )( (cid:15) i µ i + γ i µ i ) λ F i ( t + τ ) − χ [0 ,T − τ ] ( t )(1 − (cid:15) i − (cid:15) i ) µ i + (1 − γ i − γ i ) µ i ) λ P i ( t + τ ) − (cid:88) j =1 β ij ( I j + J j ) λ E i d λ V i d t = (cid:18) (cid:88) j =1 β ij ( I j + J j ) + µ (cid:19) λ V i − (cid:18) (cid:88) j =1 β ij ( I j + J j ) (cid:19) λ E i d λ F i d t = (cid:18) (cid:88) j =1 β ij ( I j + J j ) + µ (cid:19) λ F i − (cid:18) (cid:88) j =1 β ij ( I j + J j ) (cid:19) λ E i d λ P i d t = − µλ P i d λ E i d t = ( k + µ ) λ E i − kλ I i d λ I i d t = − d i + γ ) λ I i − γλ R i + (cid:18) (cid:88) j =1 β ij S j ( λ S j − λ E j ) (cid:19) + (cid:18) (cid:88) j =1 β ij V j ( λ V j − λ E j ) (cid:19) + (cid:18) (cid:88) j =1 β ij F j ( λ F j − λ E j ) (cid:19) + χ [0 ,T − τ ] ( t ) (cid:18) α i e − γτ λ I i ( t + τ ) − α i e − γτ λ J i (cid:19) + χ [0 ,T − τ ] ( t ) (cid:18) µ i ( λ I i ( t + τ ) − λ R i ( t + τ )) (cid:19) d λ J i d t = ( d i ) λ J i + ( (cid:88) j =1 β ij (cid:18) S j ( λ S j − λ E j ) + V j ( λ V j − λ E j ) + F j ( λ F j − λ E j ) (cid:19) + χ [0 ,T − τ ] ( t ) (cid:18) µ i ( λ J i ( t + τ ) − λ R i ( t + τ )) (cid:19) d λ R i d t = − µλ R i along with transversality conditions λ S i ( T ) = 0 , λ V i ( T ) = 0 , λ F i ( T ) = 0 , λ P i ( T ) = 0 , λ E i ( T ) = 0 , λ I i ( T ) =0 , λ J i ( T ) = 0 , λ R i ( T ) = 0 . Now, to obtain the optimal controls, we will use the Hamiltonian minimization condition. Differentiatingthe Hamiltonian with respect to each of the controls and solving the equations, we obtain the optimal controlsin the following. Let x i = (1 − (cid:15) i − (cid:15) i ) S λ P ( t + τ ) , i = 1 , y i = (1 − γ i − γ i ) λ P i ( t + τ ) , i = 1 , µ ∗ = min (cid:26) max (cid:26) χ [0 ,T − τ ] ( t ) (cid:18) λ S ( t + τ ) S − (cid:15) S λ V ( t + τ ) − (cid:15) S λ F ( t + τ ) − x (cid:19) A , (cid:27) , µ max (cid:27) µ ∗ = min (cid:26) max (cid:26) χ [0 ,T − τ ] ( t ) (cid:18) λ S ( t + τ ) S − (cid:15) S λ V ( t + τ ) − (cid:15) S λ F ( t + τ ) − x (cid:19) A , (cid:27) , µ max (cid:27) µ ∗ i = min (cid:26) max (cid:26) χ [0 ,T − τ ] ( t ) (cid:18) λ S i ( t + τ ) S − γ i S λ V i ( t + τ ) − γ i S λ F i ( t + τ ) − y i (cid:19) S i A , (cid:27) , µ i max (cid:27) µ ∗ i = min (cid:26) max (cid:26) χ [0 ,T − τ ] ( t ) (cid:18) λ I i ( t + τ ) − λ R i ( t + τ ) (cid:19) I i A , (cid:27) , µ i max (cid:27) µ ∗ i = min (cid:26) max (cid:26) χ [0 ,T − τ ] ( t ) (cid:18) λ J i ( t + τ ) − λ R i ( t + τ ) (cid:19) I i A , (cid:27) , µ i max (cid:27) Numerical Simulations
In this section, we perform numerical simulations to understand the age specific efficacies of vaccination andthe treatment. This is done by studying the effect of control on the dynamics of the system. Let there exist astep size h > n > T − t = nh . Let m = max ( τ, τ , τ , τ ). For programming point of viewwe consider m knots to left of t and right of T and we obtain the following partition:∆ = (cid:18) t − m = − max ( τ, τ , τ , τ ) .... < t < t = 0 < t ... < t n = t f (= T ) < .... < t n + m ) (cid:19) .Using combination of forward and backward difference approximations,we simulate the results in matlab soft-ware. All the parameter values and the source from which they are taken is given in table 2. Initially, wework with the assumption that the efficacy of both the vaccine is 60 % and later varying the efficacy levelof both the vaccines we plot the the changes in the infection and disease induced mortality. For the initialsimulation we take the values of A i , i = 1 ,
2, the cost associated with vaccination as 10 . We also study theeffects of optimal vaccination strategies on the dynamics of the disease under different vaccination coverages.In this context larger values of of the weights A i mean that the cost associated with vaccination is expensive;hence, the vaccination coverages is less for larger A i . The values for weight constant associated with treatmentfor infected and hospitalized population ( A i , i = 3 ,
4) are taken as 200 and 100. The cost of treatment of thehospitalized population is taken lesser than that of treatment of infected population because it is assumed thatall the facilities are available in the hospital. We have also assumed that the disease induced death rate ofhospitalized is 100 times more than that of infected.In simulation three control strategies are performed A: Implementation of vaccination only strategy to control the spread of COVID-19. B: Implementation of treatment only strategy to control the spread of COVID-19. C: Implementation of both treatment and vaccination strategies to control the spread of COVID-19.Table 2
Parameters Value Source ω i β ij (0.0175,0.0341,0.0319, 0.0339) approximated from[17] µ d .000073 [7] d d .0073 assumed d k α i (0.4, 0.5) [21] (cid:15) i , (cid:15) i γ i , γ i τ τ
12 assumed τ
12 assumed γ τ
10 [11] A i assumed (baseline scenario) A , A .1 Optimal control strategy In this section we evaluate the role of each of the control strategy (vaccination and treatment) in reducing theCOVID-19 burden for two specific age groups considered. Initially, we assume that the efficacy of both thevaccine is 60% and in later sections, we study the effect of increasing the efficacy of vaccine on the infection anddisease induced deaths. In figure 1 we plot the proportion of infected population with time for both the agegroups under different control strategies. In figure 2 and 3 the proportion of hospitalized and disease induceddeath curves are shown. From these figures we observe that the peak in the proportion of infected, hospitalizedand deaths are minimum when treatment and vaccination strategies are followed together compared to theindividual strategies alone. We also observe from figure 1 that with treatment only and combined strategythe peak of infection is reached faster in time compared to no control and vaccination only strategy. Theimplementation of optimal combined therapy leads to the reduction of approximately 50 % in the peak ofinfection for population of age between 0 to 40 followed by a reduction of approximately 53 % for the secondgroup ( >
40) years compared to no control case. The reduction in the peaks of disease induced mortality for firstand second age groups under the combined strategy are approximately 55 % and 62 % respectively comparedto no control case. 12igure 1: (a) Proportion of Infected population for first group(b) Proportion of Infected population for second group13igure 2: (a) Proportion of hospitalized population for first group(b) Proportion of hospitalized population for second groupNow we explore the role of age specific optimal combined strategies on the cumulative infection and diseaseinduced mortality. In figure 4 we plot the cumulative infected and disease induced mortality considering optimalcombined strategy. Comparing the cumulative infected population in absence of controls to the cumulativeinfected population with optimal control strategies on the first age group, we observe from figure 4(a) thatthe reduction in the peaks of cumulative infection is approximately 21 percent. Similarly considering optimalcontrol strategies on second age group, we see that there is approximately 25 percent reduction in the peaks ofcumulative infection. We see that with optimal strategy reduction in the cumulative infection is higher in caseof second group. Therefore, with this observation we claim that in order to reduce the infection to maximumoptimal control strategy should be prioritized to the second age group. The cumulative disease induced mortalityis plotted in figure 4(b) and the cumulative deaths decreased maximum when optimal combined strategy isprioritized to second group. 14igure 3: (a) Proportion of death population for first group(b) Proportion of death population for second group
In the previous sections we had taken the baseline weight constant value related to vaccination ( A i ) as 10 for i=1,2. In this section we study the effects of optimal vaccination strategy on the dynamics of the diseaseunder different vaccination coverage. In the context larger values of of the weights A i means that the costassociated with vaccination is expensive; hence, the vaccination coverages is less for larger A i . We assume thatfor the baseline value of the weight constant the average vaccination coverages is about 60 % and as the cost ofvaccination increases the average vacination coverage reduces.In figure 5 we simulate the effect of varying the cost associated with vaccination. As the value of weightconstant increases, the cost of implementation of vaccination increases resulting in the reduction of vaccinationrates. Due to this there is relatively higher number of infected population compared to the baseline case( A i = 10 ). From figure 5 we see that the infection increase with the increase in the value of the cost forboth the groups. There is almost 20 % and 5% increase in the infected population with the highest cost ofvaccination for second and first age group respectively. The reason for the increase is that large coverages ofoptimal age-specific vaccinations yield increased reductions in the overall number of infected individuals. Here we vary the efficacies of the vaccines and see the effects of varying efficacies in the proportion of infectedand deaths. For the baseline scenario we assume that the efficacy of both the vaccine is 60 % and then we vary15igure 4: (a) Proportion of cumulative infection(b) Proportion of cumulative deathsthe efficacies and see the relative changes in the proportion of infection and death with the baseline case. Fromfigure 6 we see that as efficacy of vaccine increases, infection starts decreasing and it decreases the maximumwith highest efficacy of the vaccine(90%). Figure 7 shows that disease induced mortality also decreases withincreasing efficacy of the vaccine for both the age group considered.16igure 5: Proportion of infection varying weights R ) In this section we study the dynamics of disease and the effect of vaccination strategy with varying transmis-sibility ( R ). Since the severity of the epidemic characterized by the high epidemic peaks which is measuredby the higher values of R , therefore we will observe the prevalence of the cumulative count of the disease byvarying the basic reproduction number. From section 2.1 the basic reproduction number is given by, R = β kS ∗ ( k + µ )( d + γ + α e − γτ ) + β kS ∗ ( k + µ )( d + γ + α e − γτ )With varying values of µ = u = (0 . , . , .
2) the values of R were found to be (7.8, 4.5, 1.9) respectively.From figure 8 it can be observed that epidemic reaches it peak when R0 is around 2.5 with treatment onlystrategy. Whereas with vaccination only strategy and combined optimal strategy the peak is reached muchfaster. In figure 8 (a) we consider the efficacy of vaccine both the vaccine to be 60 % and in figure 8 (b) 90%. varying R in the x-axis in the between 0 to 10, we plot the proportion of cumulative infected populationconsidering different control strategies for two age groups. Our findings suggest that when the epidemic ismild ( R ∈ (1 , . In this work, the total population was divided into 9 different compartments such as suceptibles( S i ), vaccinatedbut not protected( V i ), ineffectively vaccinated( F i ), Protected( P i ), exposed( E i ), infected( I i ), hospitalized( J i ),recovered( R i ) and deaths( D i ) for i=1,2. Firstly, an age specific model representing the dynamics of COVID-19 was formulated and the positivity and boundedness of the model was established. Secondly to study theeffectiveness of the individual vaccine, combination vaccines and treatment an optimal control problem withage specific transmission dynamics of COVID-19 was framed. After which numerical simulation are performed.In simulation three control strategies were performed A: Implementation of vaccination only strategy to control the spread of COVID-19. B: Implementation of treatment only strategy to control the spread of COVID-19. C: Implementation of both treatment and vaccination strategies to control the spread of COVID-19.The implementation of an age specific control strategies lead to the reduction of infection, hospitalized pop-ulation and disease induced deaths (figure 1,2,3). Compared to an individual vaccines strategy, combinationvaccine strategy worked better in minimizing the infection and disease induced deaths. However, the best pos-sible result in minimizing the peaks of infection and disease induced deaths was achieved when both vaccinationand treatment strategies were used. This result is in similar lines to the results obtained in [6, 24].From figures 4, it was observed that in order to reduce the cumulative infection and cumulative diseaseinduced deaths to maximum optimal control strategy must be prioritized to the second age group. Whenthe cost of implementation of vaccination increased there was relatively higher number of infected population18igure 7: Proportion of deaths varying vaccine efficacycompared to the baseline case (figure 5). The reason for these could be that with increasing cost the vaccinationcoverage reduces as a result of which there is increase in the number of infection. Increasing the efficacy of thevaccine also reduces the infection and disease induced deaths (figure 6,7).From figure 8 we observed that larger value of R resulted in the larger pandemic sizes because of the rapidspread of the pandemic. When the epidemic was mild R ∈ (1 , . ACKNOWLEDGEMENTS
The authors from SSSIHL acknowledge the support of SSSIHL administration for this work.
DEDICATION
The authors from SSSIHL and SSSHSS dedicate this paper to the founder chancellor of SSSIHL, BhagawanSri Sathya Sai Baba. The corresponding author also dedicates this paper to his loving elder brother D. A. C.Prakash who still lives in his heart and the first author dedicates this paper to his loving Grandmother.
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