Optimal Control of an SIR Model with Delay in State and Control Variables

Abstract

We will investigate the optimal control strategy of an SIR epidemic model with time delay in state and control variables. We use a vaccination program to minimize the number of susceptible and infected individuals and to maximize the number of recovered individuals. Existence for the optimal control is established; Pontryagin’s maximum principle is used to characterize this optimal control, and the optimality system is solved by a discretization method based on the forward and backward difference approximations. The numerical simulation is carried out using data regarding the course of influenza A (H1N1) in Morocco. The obtained results confirm the performance of the optimization strategy.

1. Introduction

For a long time, infectious diseases have caused several epidemics, leaving behind them not only millions of dead and infected individuals but also severe socioeconomic consequences. Nowadays, mathematical modeling of infectious diseases is one of the most important research areas. Indeed, mathematical epidemiology has contributed to a better understanding of the dynamical behavior of infectious diseases, its impacts, and possible future predictions about its spreading. Mathematical models are used in comparing, planning, implementing, evaluating, and optimizing various detection, prevention, therapy, and control programs. Many influential results related to the development and analysis of epidemiological models have been established and can be found in many articles and books (see, e.g., [1–3]).

Epidemiological models often take the form of a system of nonlinear, ordinary, and differential equations without time delay. However, for various biological reasons, the real dynamic behavior of an epidemic depends not only on its current state but also on its past history. Thus, to reflect the real behavior of some diseases, many researchers have proposed and analyzed more realistic models including delays to model different mechanisms in the dynamics of epidemics like latent period, temporary immunity and length of infection (see, e.g., [4–8] and the references therein).

To the best of our knowledge, including time delay in both state and control variables in the context of an epidemic controlled model has not been studied. There have been some works (like [9, 10]) which study an optimal control problem with time delay but only in the state variable. In this paper, we will investigate the effect of a vaccination program in the case of an SIR (susceptible-infected-recovered) epidemic model with time delay in the control and the state variables. To do this, we will consider an optimally controlled SIR epidemic model with time delay, where the control means the percentage of susceptible individuals being vaccinated per time unit, and the time delay represents the required time so that a vaccinated susceptible person moves from the susceptibles class to the recovered class. We use optimal control approach to minimize the number of susceptible and infected individuals and to maximize the number of recovered individuals during the course of an epidemic.

This paper is organized as follows. In Section 2, we will present a mathematical model with time delay and a control term. The analysis of optimization problem is presented in Section 3. In Section 4, we will give a numerical appropriate method and the corresponding simulation results. Finally, the conclusions are summarized in Section 5.

2. Mathematical Model

We consider the SIR epidemic model with constant total population size. The population is divided into three disease-state compartments: susceptible individuals (𝑆), people who can catch the disease; infectious (infective) individuals (𝐼), people who have the disease and can transmit the disease; recovered individuals (𝑅), people who have recovered from the disease. We assume that an individual can be infected only through contacts with infectious individuals and that immunity is permanent. The transitions between different states are described by the following parameters:(i)Λ is the recruitment rate of susceptibles; (ii)𝛽 is the effective contact rate; (iii)𝑑 is the natural mortality rate; (iv)𝛾 is the recovery rate; (v)𝜖 is the disease induced death rate. The population dynamics is given by the following system of ordinary differential equations subject to nonnegative initial conditions: 𝑑𝑆(𝑡)(𝑡)𝐼(𝑡)𝑑𝑡=Λ−𝛽𝑆𝑁(𝑡)(𝑡),(𝑡)−𝑑𝑆𝑑𝐼(𝑡)𝐼(𝑡)𝑑𝑡=𝛽𝑆𝑁(𝑡)−𝐼(𝑡),(𝑡)𝛾+𝑑+𝜖𝑑𝑅(𝑡)(𝑡),𝑑𝑡=𝛾𝐼−𝑑𝑅 (1) where 𝑆(0)=𝑆₀, 𝐼(0)=𝐼₀, 𝑅(0)=𝑅₀, and 𝑁(𝑡)=𝑆(𝑡)+𝐼(𝑡)+𝑅(𝑡) is the total population number at time 𝑡.

The strategy of the control we adopt consists of a vaccination program; our goal is to minimize the level of susceptible and infected individuals and to maximize the recovered individuals. Into the model (1) we include a control 𝑢 that represents the percentage of susceptible individuals being vaccinated per time unit. In order to have a realistic model, we need to take into account that the movement of the vaccinated susceptible individuals from the class of susceptibles into the recovered class is subject to delay. Thus, the time delay is introduced in the system as follows: at time 𝑡 only a percentage of susceptible individuals that have been vaccinated 𝜏 time unit ago, that is, at time 𝑡−𝜏, are removed from the susceptible class and added to the recovered class. So the mathematical system with time delay in state and control variables is given by the nonlinear retarded differential equations: 𝑑𝑆(𝑡)(𝑡)𝐼(𝑡)𝑑𝑡=Λ−𝛽𝑆𝑁(𝑡)(𝑡)()𝑆(),(𝑡)−𝑑𝑆−𝑢𝑡−𝜏𝑡−𝜏𝑑𝐼(𝑡)𝐼(𝑡)𝑑𝑡=𝛽𝑆𝑁(𝑡)−𝐼(𝑡),(𝑡)𝛾+𝑑+𝜖𝑑𝑅(𝑡)(𝑡)()𝑆().𝑑𝑡=𝛾𝐼−𝑑𝑅+𝑢𝑡−𝜏𝑡−𝜏 (2) In addition, for biological reasons, we assume, for 𝜃∈[−𝜏,0], that 𝑆(𝜃), 𝐼(𝜃), and 𝑅(𝜃) are nonnegative continuous functions and 𝑢(𝜃)=0. Note that the control 𝑢 is assumed to be integrable in the sense of Lebesgue, bounded with 0≤𝑢≤𝑏<1, and 𝑏 is a given constant.

To show the existence of solutions for the control system (2), we first prove that the system (2) is dissipative; that is, all solutions are uniformly bounded in a proper subset Ω∈ℝ³₊.

Let (𝑆,𝐼,𝑅)∈ℝ³₊ be any solution with nonnegative initial conditions. Adding equations of (2) we get 𝑑𝑁𝑑𝑡=Λ−𝑑𝑁−𝜖𝐼<Λ−𝑑𝑁. (3) After integration, using the constant variation formula, we have 𝑁(𝑡)≤Λ𝑑+𝑁(0)𝑒^−𝑑𝑡. (4) It then follows that (𝑡)≤Λ0≤𝑁𝑑as𝑡⟶∞. (5) Therefore all feasible solutions of the system (2) enter into the region Ω=(𝑆,𝐼,𝑅)∈ℝ³₊Λ∶𝑁≤𝑑. (6) Then we can rewrite (2) in the following form: 𝑑𝑋𝑑𝑡=𝐴𝑋+𝐹𝑋,𝑋_𝜏=𝐺𝑋,𝑋_𝜏, (7) where 𝑋(𝑡)=𝑆(𝑡)𝐼(𝑡)𝑅(𝑡),0,𝐹𝑋(𝑡)𝐴=−𝑑000−𝑑+𝛾+𝜀0𝛾−𝑑,𝑋_𝜏(𝑡)=(𝑡)𝐼(𝑡)Λ−𝛽𝑆𝑁(𝑡)−𝑢_𝜏(𝑡)𝑆_𝜏(𝑡)𝛽𝑆(𝑡)𝐼(𝑡)𝑁(𝑡)𝑢_𝜏(𝑡)𝑆_𝜏(𝑡),𝑢_𝜏(𝑡)=𝑢(𝑡−𝜏),𝑋_𝜏(𝑡)=𝑋(𝑡−𝜏). (8)

The second term on the right-hand side of (7) satisfies 𝐹𝑋₁(𝑡),𝑋_1𝜏(𝑡)−𝐹𝑋₂(𝑡),𝑋_2𝜏(𝑡)≤𝑀₁𝑋₁(𝑡)−𝑋₂(𝑡)+𝑀₂𝑋_1𝜏(𝑡)−𝑋_2𝜏(𝑡), (9) where 𝑀₁ and 𝑀₂ are some positive constants, independent of the state variables 𝑆(𝑡), 𝐼(𝑡), and 𝑅(𝑡), and 𝑋₁(𝑡)−𝑋₂(𝑡)=𝑆₁(𝑡)−𝑆₂(𝑡+𝐼₁)(𝑡)−𝐼₂(𝑡)+𝑅₁(𝑡)−𝑅₂(𝑡,𝑋_1𝜏)(𝑡)−𝑋_2𝜏(𝑡)=𝑆_1𝜏(𝑡)−𝑆_2𝜏(𝑡+𝐼_1𝜏)(𝑡)−𝐼_2𝜏(𝑡)+𝑅_1𝜏(𝑡)−𝑅_2𝜏(𝑡). (10) Here 𝑆_𝑖𝜏(𝑡)=𝑆_𝑖(𝑡−𝜏),𝐼_𝑖𝜏(𝑡)=𝐼_𝑖(),𝑅𝑡−𝜏_𝑖𝜏(𝑡)=𝑅_𝑖(𝑡−𝜏)for𝑖=1,2. (11) Moreover, we get 𝐺𝑋₁,𝑋_1𝜏−𝐺𝑋₂,𝑋_2𝜏≤𝑀𝑋₁−𝑋₂+𝑋_1𝜏−𝑋_2𝜏, (12) where 𝑀=max𝑀₁+‖𝐴‖;𝑀₂<∞. (13)

Thus, it follows that the function 𝐺 is uniformly Lipschitz continuous. From the definition of the control 𝑢(𝑡) and the restriction on 𝑆(𝑡), 𝐼(𝑡), and 𝑅(𝑡)≥0, we see that a solution of the system (7) exists (see [11]).

3. The Optimal Control Problem

Our goal is to minimize the number of susceptible and infected individuals and to maximize the number of recovered individuals during the course of an epidemic. Mathematically, for a fixed terminal time 𝑡_𝑓, the problem is to minimize the objective functional 𝐽(𝑢)=^𝑡𝑓₀𝐴₁𝑆(𝑡)+𝐴₂𝐼(𝑡)−𝐴₃𝑅(𝑡)+𝐴₄2𝑢²(𝑡)𝑑𝑡, (14) where 𝐴_𝑖≥0 (for 𝑖=1,…,4) denote weights that balance the size of the terms. In other words, we seek the optimal control 𝑢^∗ such that 𝐽𝑢^∗=min{𝐽(𝑢)∶𝑢∈𝒰}, (15) where 𝒰 is the set of admissible controls defined by 𝒰=𝑢∶0≤𝑢≤𝑏<1,𝑡∈0,𝑡_𝑓,𝑢isLebesguemeasurable. (16)

3.1. Existence of an Optimal Control

The existence of the optimal control can be obtained using a result by Fleming and Rishel in [12].

Theorem 1. Consider the control problem with system (2). There exists an optimal control 𝑢^∗∈𝒰 such that 𝐽𝑢^∗=min_𝑢∈𝒰𝐽(𝑢). (17)

Proof. To use an existence result in [12], we must check the following properties. (1)The set of controls and corresponding state variables is nonempty. (2)The control set 𝒰 is convex and closed. (3)The right-hand side of the state system is bounded by a linear function in the state and control variables. (4)The integrand of the objective functional is convex on 𝒰. (5)There exist constants 𝑐₁,𝑐₂>0 and 𝜌>1 such that the integrand 𝐿(𝑆,𝐼,𝑅,𝑢) of the objective functional satisfies 𝐿(𝑆,𝐼,𝑅,𝑢)≥𝑐₂+𝑐₁|𝑢|^2𝜌/2. (18)An existence result by Lukes [13] was used to give the existence of solution of system (2) with bounded coefficients, which gives condition 1. The control set is convex and closed by definition. Since the state system is linear in 𝑢, the right side of (2) satisfies condition 3, using the boundedness of the solution. The integrand in the objective functional (14) is convex on 𝒰. In addition, we can easily see that there exist a constant 𝜌>1 and positive numbers 𝑐₁ and 𝑐₂ satisfying 𝐿(𝑆,𝐼,𝑅,𝑢)≥𝑐₂+𝑐₁|𝑢|^2𝜌/2. (19)

3.2. Characterization of the Optimal Control

In order to derive the necessary condition for the optimal control, Pontryagin’s maximum principal with delay given in [14] was used. This principal converts (2), (14), and (15) into a problem of minimizing a Hamiltonian, 𝐻, defined by 𝐻=𝐴₁𝑆(𝑡)+𝐴₂𝐼(𝑡)−𝐴₃𝑅(𝑡)+𝐴₄2𝑢²(𝑡)+³_𝑖=1𝜆_𝑖𝑓_𝑖, (20) where 𝑓_𝑖 is the right side of the differential equation of the 𝑖th state variable.

Theorem 2. Given an optimal control 𝑢^∗∈𝒰 and solutions 𝑆^∗, 𝐼^∗, and 𝑅^∗ of the corresponding state system (2), there exist adjoint functions 𝜆₁, 𝜆₂, and 𝜆₃ satisfying 𝜆̇₁=−𝐴₁+𝜆₁𝑑+𝜆₁−𝜆₂𝛽𝐼^∗𝑁^∗+𝜒_{[0,𝑡𝑓−𝜏]}(𝑡)𝜆⁺₁−𝜆⁺₃𝑢^∗,𝜆̇₂=−𝐴₂+𝜆₁−𝜆₂𝛽𝑆^∗𝑁^∗+𝑑+𝛾+𝜖𝜆₂−𝛾𝜆₃,𝜆̇₃=𝐴₃+𝜆₃𝑑 (21) with the transversality conditions 𝜆₁𝑡_𝑓=𝜆₂𝑡_𝑓=𝜆₃𝑡_𝑓=0. (22) Furthermore, the optimal control 𝑢^∗ is given by 𝑢^∗(𝑡)𝜆=min𝑏,max0,⁺¹−𝜆⁺₃𝐴₄𝜒_{[0,𝑡𝑓−𝜏]}(𝑡)𝑆^∗, (23) where 𝜆⁺_𝑖(𝑡)=𝜆_𝑖(𝑡+𝜏) for 𝑖=1,…,3.

Proof. The adjoint equations and transversality conditions can be obtained by using Pontryagin’s maximum principle with delay in the state and control variables [14] such that 𝜆̇₁=−𝜕𝐻(𝑡)𝜕𝑆(𝑡)−𝜒_{[0,𝑡𝑓−𝜏]}(𝑡)𝜕𝐻(𝑡+𝜏)𝜕𝑆(𝑡−𝜏),𝜆₁𝑡_𝑓𝜆̇=0,₂=−𝜕𝐻𝜕𝐼,𝜆₂𝑡_𝑓𝜆̇=0,₃=−𝜕𝐻𝜕𝑅,𝜆₃𝑡_𝑓=0. (24) The optimal control 𝑢^∗ can be solved from the optimality condition: 𝜕𝐻𝜕𝑢+𝜒_{[0,𝑡𝑓−𝜏]}(𝑡)𝜕𝐻(𝑡+𝜏)𝜕𝑢_𝜏=0. (25) That is, 𝐴₄𝑢+𝜒_{[0,𝑡𝑓−𝜏]}(𝑡)𝜆⁺₃−𝜆⁺₁𝑆=0. (26) By the bounds in 𝒰 of the control, it is easy to rewrite 𝑢^∗ in the form (23).

4. Numerical Simulation

In this section, we give a numerical method to solve the optimality system which is a two-point boundary value problem, with separated boundary conditions at times 𝑡₀=0 and 𝑡_𝑓.

Let there exist a step size ℎ>0 and (𝑛,𝑚)∈ℕ² with 𝜏=𝑚ℎ and 𝑡_𝑓−𝑡₀=𝑛ℎ. For reasons of programming, we consider 𝑚 knots to left of 𝑡₀ and right of 𝑡_𝑓, and we obtain the following partition: Δ=𝑡_−𝑚=−𝜏<⋯<𝑡₋₁<𝑡₀=0<𝑡₁<⋯<𝑡_𝑛=𝑡_𝑓<𝑡_𝑛+1<⋯<𝑡_𝑛+𝑚. (27)

Then, we have 𝑡_𝑖=𝑡₀+𝑖ℎ(−𝑚≤𝑖≤𝑛+𝑚). Next, we define the state and adjoint variables 𝑆(𝑡), 𝐼(𝑡), 𝑅(𝑡), 𝜆₁, 𝜆₂, 𝜆₃, and the control 𝑢 in terms of nodal points 𝑆_𝑖, 𝐼_𝑖, 𝑅_𝑖, 𝜆^𝑖₁, 𝜆^𝑖₂, 𝜆^𝑖₃, and 𝑢_𝑖. Now using combination of forward and backward difference approximations, we obtain the Algorithm 1.

Step 1: For 𝑖=−𝑚,…,0  do 𝑆𝑖=𝑆0, 𝐼𝑖=𝐼0, 𝑅𝑖=𝑅0, 𝑢𝑖=0 End for For 𝑖=𝑛,…,𝑛+𝑚  do 𝜆𝑖1=0, 𝜆𝑖2=0, 𝜆𝑖3=0 End for Step 2: For 𝑖=0,…,𝑛−1, do 𝑆𝑖+1=𝑆𝑖+ℎ(Λ−𝑢𝑖−𝑚𝑆𝑖−𝑚)1+ℎ(𝛽(𝐼𝑖/𝑁)+𝑑) 𝐼𝑖+1=𝐼𝑖1+ℎ(𝛾+𝑑+𝜀−𝛽(𝑆𝑖+1/𝑁)) 𝑅𝑖+1=𝑅𝑖+ℎ(𝛾𝐼𝑖+1+𝑢𝑖−𝑚𝑆𝑖−𝑚)1+ℎ𝑑 𝜆1𝑛−𝑖−1=𝜆1𝑛−𝑖+ℎ(𝐴1+𝛽(𝐼𝑖+1/𝑁)𝜆2𝑛−𝑖+(𝜆3𝑛−𝑖+𝑚−𝜆1𝑛−𝑖−1+𝑚)𝜒[0,𝑡𝑓−𝜏](𝑡𝑛−𝑖)𝑢𝑖)1+ℎ(𝑑+𝛽(𝐼𝑖+1/𝑁)) 𝜆2𝑛−𝑖−1=𝜆2𝑛−𝑖+ℎ(𝐴2+𝜆3𝑛−𝑖𝛾−𝜆1𝑛−𝑖−1𝛽(𝑆𝑖+1/𝑁))1+ℎ(𝛾+𝑑+𝜀−𝛽(𝑆𝑖+1/𝑁)) 𝜆3𝑛−𝑖−1=𝜆3𝑛−𝑖−𝐴3ℎ1+𝑑ℎ 𝑇𝑖+1=(𝜆1𝑛−𝑖−1+𝑚−𝜆3𝑛−𝑖−1+𝑚)𝐴4𝜒[0,𝑡𝑓−𝜏](𝑡𝑖+1)𝑆𝑖+1 𝑢𝑖+1=min(𝑏,max(0,𝑇𝑖+1)) Step 3: For 𝑖= 0,…, n, write 𝑆∗(𝑡𝑖)=𝑆𝑖, 𝐼∗(𝑡𝑖)=𝐼𝑖, 𝑅∗(𝑡𝑖)=𝑅𝑖, 𝑢∗(𝑡𝑖)=𝑢𝑖 End for

Algorithm 1 

The numerical simulations were carried out using data regarding the course of the influenza A (H1N1) in Morocco. The initial conditions and parameters of the system (2) are taken from [15, 16], while the time delay value is taken from [17]: 𝛽=0.3095,Λ=1174.17,𝑑=3.9139×10⁻⁵,𝛾=0.2,𝜀=0.0063,𝜏=10, (28) the initial conditions for the ordinary differential system were 𝑆(0)=30×10⁶,𝐼(0)=30,𝑅(0)=28, (29) and the transversality conditions for the ordinary differential system were 𝜆_𝑖𝑡_𝑓=0(𝑖=1,…,3). (30)

Figure 1 indicates that the number of susceptible individuals (𝑆) decreases more rapidly in the case with control. It reaches 7.399×10⁵ at the end of the vaccination period against 6.732×10⁶ in case without control, that is, a reduction of 5.992×10⁶.

Figure 1 

The function 𝑆 with and without control.

Figures 2, 3, and 4 represent the number of infected individuals (𝐼) with control (solid curve) and without control (dashed curve). It shows that in the presence of a control, the number of infected individuals (𝐼) decreases greatly. The maximum number of infected individuals in the case with control is 1.787×10⁴ and is 4.404×10⁶ in the case without control; then the efficiency of our strategy in reducing the spread of infection is nearly 99.52%.

Figure 2 

The function 𝐼 with and without control.

Figure 3 

The function 𝐼 with control.

Figure 4 

The function 𝐼 without control.

Figure 5 shows that the number of people removed begins to grow notably from 24th day instead of 48th day in the absence of control. Moreover the number of recovered individuals at the end of the vaccination period is 2.926×10⁷ instead of 2.342×10⁷, which represents an increase of 5.840×10⁶ cases.

Figure 5 

The function 𝑅 with and without control.

Figures 6 and 7 represent, respectively, the optimal control and optimal value of the cost. The curves start to increase during the first month because of the high infection level; then they drop off steadily which is because of the constant and steady eradication of the infection.

Figure 6 

The optimal control 𝑢∗.

Figure 7 

The optimal value of the cost.

5. Conclusion

The work in this paper contributes to a growing literature on applying optimal control techniques to epidemiology. We proposed a more realistic controlled model by including time delay which represents the needed time for the migration from the susceptible class to the recovered class after vaccination. The optimal control theory has been applied in the context of an SIR model with time delay in state and control variables, and that includes a control 𝑢 that represents the percentage of susceptible individuals being vaccinated per time unit. By using Pontryagin’s maximum principle, the explicit expression of the optimal controls was obtained. Simulation results indicate that the proposed control strategy is effective in reducing the number of susceptible and infected individuals and maximizing the recovered individuals.

Acknowledgment

The authors would like to thank the anonymous referee for his/her valuable comments on the first version of the paper, which have led to an improvement in this paper. Research reported in this paper was supported by the Moroccan Systems Theory Network.