Optimal Control of a Delayed SIRS Epidemic Model with Vaccination and Treatment

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1 Acta Biotheor DOI.7/s REGULAR ARTICLE Optimal Control of a Delayed SIRS Epidemic Model with Vaccination and Treatment Hassan Laarabi Abdelhadi Abta Khalid Hattaf Received: 2 February 24 / Accepted: 3 December 24 Ó Springer Science+Business Media Dordrecht 25 Abstract This article deals with optimal control applied to vaccination and treatment strategies for an SIRS epidemic model with logistic growth and delay. The delay is incorporated into the model in order to modeled the latent period or incubation period. The existence for the optimal control pair is also proved. Pontryagin s maximum principle with delay is used to characterize these optimal controls. The optimality system is derived and then solved numerically using an algorithm based on the forward and backward difference approximation. Keywords SIRS epidemic model Incidence rate Delay differential equations Optimal control Vaccination Treatment Introduction Mathematical models provide powerful tools for investigating the dynamics and control of infectious diseases. Previous studies have examined various models for predicting and assessing intervention strategies (see for exemple, Capasso and Serio 978; Driessche and Watmough 2; Kermack and Mckendrick 927; and the references therein). The SIR and SIS assumed implicitly the upper and lower limits of the range of possibilities for host immune response. The SIR epidemic model H. Laarabi (&) K. Hattaf Department of Mathematics and Computer Science, Faculty of Sciences Ben M sik, Hassan II University, P.O Box 7955, Sidi Othman, Casablanca, Morocco abtaabdelhadi@yahoo.fr A. Abta Department of Mathematics and Computer Science, Poly-disciplinary Faculty, Cadi Ayyad University, P.O Box 462, Safi, Morocco K. Hattaf Centre Régional des Métiers de l Education et de la Formation (CRMEF), Derb Ghalef, Casablanca, Morocco

2 H. Laarabi et al. considers recovered individuals to be permanently immune, while the SIS epidemic model considers recovered individuals to be immediately re-susceptible. However, some infections do not fall into either of these extreme categories. This means that it is natural to include the effects of immunity into the mathematical models in order to represent the actual dynamics of epidemic spread and to predict future outbreaks. Such a population dynamics is described by SIRS epidemic models (Mena-Lorca and Hethcote 992). Many SIRS epidemic models in the literature represent dynamics of diseases by systems of ordinary differential equations without delay. However, inclusion of temporal delays in such models makes them more realistic by allowing to describe the effects of disease latency (Hattaf et al. 23). Recently, a number of studies in the literature have been made to study the role of vaccination and treatment on the spread of infectious diseases by using the control theory (see for example Kar and Batabyal 2; Laarabi et al. 22; Lashari and Zaman 22; Lashari et al. 23; Lee and Lashari 24; Lenhart and Workman 27; Zaman et al. 29; and the references therein for detailed description). The optimal control efforts are used to limit the spread of the infectious disease from the population. Our aim is to analyze the consequences of vaccinating the susceptible population and giving treatment to infectious population. These analysis reveal the possibilities to develop strategies that manipulate the level of vaccination and treatment efforts. By treating and vaccinating the population at an appropriate time, it is possible to either reduce or eliminate the disease from the population. In this paper, we consider a general SIRS epidemic model with incubation time and a modified saturated incidence rate, where the susceptibles are assumed to satisfy the logistic equation. The dynamics of this model governed by the following equations (Abta et al. 2, 22): 8 ds ¼ r SðtÞ bsðtþiðtþ SðtÞ >< K þ a SðtÞþa 2 IðtÞ þ drðtþ di ¼ bsðt sþiðt sþ e ls ðaþlþcþiðtþ ðþ þ a Sðt sþþa 2 Iðt sþ dr >: ¼ ciðtþ ðlþdþrðtþ The initial condition for the above system is SðhÞ ¼ ðhþ; IðhÞ ¼u 2 ðhþ and RðhÞ ¼u 3 ðhþ; h 2½ s; Š ð2þ with u ¼ð ; u 2 ; u 3 Þ2ðC þ Þ 3, such that u i ðhþ( s h, i ¼ ; 2; 3Þ. Here C denotes the Banach space Cð½ s; Š; RÞ of continuous functions mapping the interval ½ s; Š into R, equipped with the supremum norm. The nonnegative cone of C is defined as C þ ¼ Cð½ s; Š; R þ Þ. where S is the number of susceptible individuals, I is the number of infectious individuals, R is the number of recovered individuals, l is the natural death of the population, a is the death rate due to the disease, b is the transmission rate, a and a 2 are the parameters that measure the inhibitory effect, c is the recovery rate of the infectious individuals, d denotes the rate at which recovered individuals lose immunity and return to susceptible class, s is the incubation period and e ls is the mortality rate during the incubation period. In the system (), it is assumed that the population growth in susceptible individuals

3 Optimal Control of a Delayed SIRS Epidemic Model is governed by the logistic growth with a carrying capacity K [ as well as intrinsic birth rate constant r [. In this work, we address the question of how to optimally combine the vaccination and the treatment strategies such that the cost of the implementation of the two interventions is minimized while the disease is eradicated within specified period. The organization of this paper is as follows. In Sect. 2, we formulate the optimal control problem and we use the Pontryagin s Maximum principle with delay given in (Göllmann et al. 29) to characterize it. In Sect. 3, we give the numerical method and the simulation results. Finally, we draw conclusions in Sect The Optimal Control Problem We use optimal control strategies in the form of vaccination and treatment to decrease the number of both susceptible and infectious individuals with minimum investment in disease control. The main feature of the present paper is not to consider a special disease but to present a method of how to treat this class of optimization problems. This problem is formulated as an optimal control problem with two control variable (that represent vaccination and treatment strategies). Now we introduce two controls ; u 2 which represents the percentage of susceptible and infected individuals being vaccinated and treated respectively per unit of time. Hence, () becomes 8 ds ¼ r SðtÞ bsðtþiðtþ SðtÞ K þ a SðtÞþa 2 IðtÞ þ drðtþ ðtþsðtþ >< di ¼ bsðt sþiðt sþ e ls þ a Sðt sþþa 2 Iðt sþ ðaþlþcþiðtþ u 2ðtÞIðtÞ ð3þ dr >: ¼ ciðtþ ðlþdþrðtþþðtþsðtþþu 2 ðtþiðtþ SðÞ ¼S ; IðÞ ¼I ; RðÞ ¼R ð4þ It is easy to show that there exists a unique solution ðsðtþ; IðtÞ; RðtÞÞ of system (3) with initial data ðs ; I ; R Þ2ðC þ Þ 3. In addition, for biological reasons, we assume that the initial data for system (3) satisfy S ðtþ; I ðtþ; R ðtþ; t 2½ s; Š: The problem is to minimize the objective (cost) functional given by Z tf Jð ; u 2 Þ¼ A SðtÞþA 2 IðtÞþ 2 B u 2 ðtþþ 2 B u 2 2 ðtþ Subject to the differential equations (3), where the first two terms in the objective functional represent benefit of susceptible and infected populations that we wish to reduce and the parameters A and A 2 are positive constants to keep a balance in the size of SðtÞ and IðtÞ, respectively. We use in the second term in the objective functional, (as it is customary), the quadratic term 2 B iu 2 i ; i ¼ ; 2, where B i is a positive weight ð5þ

4 H. Laarabi et al. parameter which is associated with the control u i ðtþ and the square of the control variable reflects the severity of the side effects of the vaccination and treatment. Our target is to minimize the objective functional defined in (5) by decreasing the number of infected and susceptible individuals, by using possible minimal control variables ð ðtþ; u 2 ðtþþ. In other words, the control variable ð ðtþ; u 2 ðtþþ 2 U ad represent the percentage of susceptible and infected individuals being vaccinated and treated respectively per unit of time and U ad is the control set defined by U ad ¼fu ¼ð ; u 2 Þju i ðtþ measurable; u i ðtþu max i ; t 2½; t f Š; i ¼ ; 2g; where u max is maximum attainable value for and u max 2 is maximum attainable value for u Existence of an Optimal Control The existence of the optimal control pair can be obtained using a result by Fleming and Rishel (975) and by Lukes (982). Theorem 2. There exists control functions u ðtþ; u 2ðtÞ so that Jðu ðtþ; u 2 ðtþþ ¼ min Jð ðtþ; u 2 ðtþþ ð ;u 2 Þ2U ad Proof To prove the existence of an optimal control pair it is easy to verify that:. The set of controls and corresponding state variables is nonempty. 2. The admissible set U ad is convex and closed. 3. The right hand side of the state system (3) is bounded by a linear function in the state and control variables. 4. The integrand of the objective functional is convex on U ad. 5. There exists constants x [, x 2 [ and q [ such that the integrand LðS; I; ; u 2 Þ of the objective functional satisfies: LðS; I; ; u 2 Þx 2 þ x ðj j 2 þju 2 j 2 Þ q 2 : The result follows directly from (Fleming and Rishel 975). h 2.2 Characterization of the Optimal Control Before characterizing the optimal control pair, we first define the Lagrangian for the optimal control problem (3 5) by: LðS; I; ; u 2 Þ¼A SðtÞþA 2 IðtÞþ 2 B u 2 ðtþþ 2 u 2 2 ðtþ and the Hamiltonian H for the control problem by: HðS; I; R; ; u 2 ; k i ; tþ ¼LðS; I; ; u 2 Þþ Xi¼3 k i f i i¼ ð6þ where k i, i ¼ ; 2; 3 are the adjoint functions to be determined suitably.

5 Optimal Control of a Delayed SIRS Epidemic Model Next, By applying Pontryagin s Maximum principle with delay given in (Göllmann et al. 29) to the Hamiltonian H, we obtain the following theorem Theorem 2.2 Given optimal controls u ðtþ, u 2 ðtþ and solutions S ðtþ, I ðtþ and R ðtþ of the corresponding state system (5) and (3), there exists adjoint variables k, k 2 and k 3 that satisfy dk ðtþ ¼ A þ k ðtþ r 2S K u k 3 ðtþu K ðtþ v ½;tf sšk 2 ðt þ sþðe ls K Þ dk 2 ðtþ ¼ A 2 þ k ðtþk 2 þ k 2 ðtþðl þ a þ c þ u 2 Þ k 3ðtÞðc þ u 2 Þ v ½;tf sšk 2 ðt þ sþk 2 e ls dk 3 ðtþ ¼ k ðtþd þ k 3 ðtþðl þ dþ where K ¼ and K 2 ¼ bi ðþa 2 I Þ ðþa S þa 2 I Þ 2 bs ðþa S Þ ðþa S þa 2 I Þ 2 with transversality conditions k i ðt f Þ¼; i ¼ ; 2; 3: ð8þ ð7þ Furthermore, the optimal control pair u ðtþ is given by u ðk ðtþ k 3 ðtþþs ðtþ ðtþ ¼max min B u ðk 2 ðtþ k 3 ðtþþi ðtþ 2ðtÞ ¼max min ; u max ; u max 2 ; ; ð9þ ðþ Proof Using the Pontryagin s Maximum principle with delay in state, we obtain the adjoint equations and transversality conditions such that: dk ðtþ ¼ oh os v oh ½;t f sš ðt þ sþ; os s dk 2 ðtþ ¼ oh oi v oh ½;t f sš ðt þ sþ; oi s dk 3 ðtþ ¼ oh or v oh ½;t f sš ðt þ sþ; or s and by using the optimality conditions we find oh ¼ B u ðtþ k ðtþs þ k 3 ðtþs ¼ ; o oh ¼ u ðtþ k 2 ðtþi þ k 3 ðtþs ¼ ; ou 2 which gives k ðt f Þ¼ k 2 ðt f Þ¼ k 3 ðt f Þ¼ at ¼ u ðtþ at u 2 ¼ u 2 ðtþ ðþ

6 H. Laarabi et al. u ðtþ ¼ ð k ðtþ k 3 ðtþþs ðtþ and u 2 B ðtþ ¼ ð k 2ðtÞ k 3 ðtþþi ðtþ Using the property of the control space, we obtain 8 u ðtþ ¼ if ðk ðtþ k 3 ðtþþs ðtþ >< B u ðtþ ¼ ð k ðtþ k 3 ðtþþs ðtþ ð if \ k ðtþ k 3 ðtþþs ðtþ >: B u ðtþ ¼umax if B ðk ðtþ k 3 ðtþþs ðtþ B u max 8 u 2 ðtþ ¼ if ðk 2 ðtþ k 3 ðtþþi ðtþ >< u 2 ðtþ ¼ ð k 2ðtÞ k 3 ðtþþi ðtþ ð if \ k 2ðtÞ k 3 ðtþþi ðtþ >: u 2ðtÞ ¼umax 2 if ðk 2 ðtþ k 3 ðtþþi ðtþ u max 2 \u max \u max 2 So the optimal control pair is characterized as (9) and (). h The optimal control pair and the state are found by solving the following optimality system, which consists of the state system (3), the adjoint system (7), boundary conditions (4) and (8), and the characterization of the optimal control pair ðu ; u 2Þ (9) and (): ds ðtþ ¼ r S S bs I K þ a S þ a 2 I þ dr ðk ðtþ k 3 ðtþþs max min ; S B ;u max di ðtþ ¼ e ls bs ðt sþi ðt sþ þ a S ðt sþþa 2 I ðaþlþcþi ðt sþ ðk 2 ðtþ k 3 ðtþþi max min ;u max 2 ; I dr ðtþ ¼ ci ðlþdþr ðk ðtþ k 3 ðtþþs þ max min ;u max ; S B ðk 2 ðtþ k 3 ðtþþi ðtþ þ max min ;u max 2 ; I dk ðtþ ¼ A þ k ðtþ r 2S k ðtþ u k 3 u K v ½;t f sšk 2 ðt þ sþðe ls K Þ dk 2 ðtþ ¼ A 2 þ k ðtþk 2 þ k 2 ðl þ a þ c þ u 2 Þ k 3ðc þ u 2 Þ v ½;t f sšk 2 ðt þ sþk 2 e ls dk 3 ðtþ ¼ k ðtþd þ k 3 ðl þ dþ ð2þ with k ðt f Þ¼;k 2 ðt f Þ¼;k 3 ðt f Þ¼;SðÞ¼S ;IðÞ¼I ;RðÞ¼R ; and K 2 ¼. bi ðþa 2 I Þ ðþa S þa 2 I Þ 2 bs ðþa S Þ ðþa S þa 2 I Þ 2 where K ¼

7 Optimal Control of a Delayed SIRS Epidemic Model 3 Numerical Results and Discussions In this section, we solve numerically the optimality system (2) and we present the results found. We note that the optimality system is a two-point boundary value problem, with separated boundary conditions at times t ¼ and t ¼ t f. Solving the system (2) requires an iterative scheme developed by Hattaf and Yousfi (22). This involves use of an appropriate algorithm. Let there exists a step size h [ and integers ðn; mþ 2N 2 with s ¼ mh and t f ¼ nh. For reasons of programming, we consider m knots to left of and right of t f, and we obtain the following partition: D ¼ t m ¼ s\ \t \\t \ \t n ¼ t f \ \t nþm Then, we have t i ¼ ihð m i n þ mþ. Next we define the state and adjoint variables SðtÞ; IðtÞ; RðtÞ; k ðtþ; k 2 ðtþ; k 3 ðtþ and ðtþ; u 2 ðtþ in terms of nodal points Algorithm Step for i = m,...,, do S i = S, I i = I, R i = R, u i = andui 2 =,, end for for i = n,...,n + m, do λ i =,λi 2 =,λi 3 =, end for Step2 S i+ = S i + h r Si K S βs i i I i +α S i+α 2 I i + dr i u i S i, I i+ = I i + h e μτ βsi m Ii m +α S i m+α 2I i m (α + μ + γ u i 2 )I i, R i+ = R i + h γ I i (μ + d)r i + u i S i + u i 2 )I i, λ n i = λ n i + h A + λ n i r( 2Si K ) i ui λ n i 3 u i χ [,t f τ](t n i )λ n i+m 2 e μτ i+, λ n i 2 = λ n i 2 + h A 2 + λ n i u i ) χ [,t f τ](t n i )λ n i+m 2 e μτ i+ 2, λ n i 3 = λ n i 3 + h λ n i d + λ n i 3 (μ + d), i+ = λn i λ n i 3 B S i+ i+ 2 = λn i 2 λ n i 3 I i+ u i = max min i+, u max, i+ 2 + λ n i 2 (α + μ + γ + u i 2 ) λn i 3 (γ + u i 2 = max min i+ 2, u max 2, Step3 for i =,..., n, do write S (t i ) = S i, I (t i ) = I i, R (t i ) = R i, u (t i) = u i and u 2 (t i) = u i 2, end for

8 H. Laarabi et al. S i, I i, R i, k i, ki 2, ki 3, ui and ui 2. Now a combination of forward and backward difference approximation, we obtain the Algorithm. For the simulations, we use the parameter values given in Table. These realistic hypothetical parameter values are chosen in the case when the population is not vaccinated and not treated ð ¼ u 2 ¼ Þ, and when the disease persists in the population. In addition, we choose the following initial values: S ¼ 2 peoples, I ¼ 5 peoples and R ¼ peoples. The weight constant values in the objective functional are A ¼, A 2 ¼ B ¼ 5 and ¼. We investigate and compare numerical results in the following three possible strategies for the control of the disease. 3. Only Vaccination Control (u 2 = ) With this strategy, only the vaccination control is used to optimize the objective function JðuÞ while the treatment control u 2 is set to zero. In Fig., we observe that there is a significant decrease in the number of susceptible individuals vaccinated compared with those not vaccinated, with a slight decrease in the number of infected individuals and an increase the number of recovered individuals. 3.2 Only Treatment Control ( = ) With this strategy, only the treatment control u 2 is used to optimize the objective function JðuÞ while the vaccination control is set to zero. In Fig. 2, we observe that there is a significant decrease in the number of infected individuals treated compared with those not treated, we also see that there is no change between treated and untreated susceptible individuals. Table Parameters, their symbols and values used in the model (3) Parameters Descriptions Values l Natural death of the population. UT a Death rate due to disease. UT a Parameter that measure the inhibitory effect. people a 2 Parameter that measure the inhibitory effect. people b Transmission rate. people UT c Recovery rate.4 UT r Intrinsic birth rate.5 UT K Carrying capacity 3 people d Rate at which recovered individuals lose immunity. UT s Time incubation UT Efficacy of vaccination u 2 Efficacy of treatment UT denotes the unit of time which can be expressed in day or year

9 Optimal Control of a Delayed SIRS Epidemic Model S(t) 3 25 = u 2 =, u 2 = I(t) 7 6 u = u = 2, u 2 = R(t) = u 2 =, u 2 = Control u Fig. Evolution of different classes of untreated individuals with vaccination (marked by the solid lines) and without vaccination (marked by the dashed lines) for time delay s ¼ S(t) u = u = 2 u =, u I(t) = u 2 = =, u 2 5 = u 2 = =, u 2.8 R(t) Control u Fig. 2 Evolution of different classes of unvaccinated individuals with treatment (marked by the solid lines) and without treatment (marked by the dashed lines) for time delay s ¼ We see that the control in Fig. always needs to be the maximal while the control in Fig. 2 can be reduced after the first days. Hence, we can firstly apply more of the vaccination control in order to reduce the the number of susceptible individuals to below certain threshold. After, we start to apply more of the treatment control to decrease the number of infected individuals, and we begin to decrease this control after days of treatment.

10 H. Laarabi et al. S(t) 3 u =, u = 2 25, u I(t) =, u 2 =, u 2 R(t) =, u 2 =, u 2 Control u Control u Fig. 3 Evolution of different classes of individuals with both controls (marked by the solid lines) and without controls (marked by the dashed lines) for time delay s ¼ 3.3 Combined Vaccination and Treatment Strategy With this strategy, we use both the vaccination control and the treatment control u 2 to optimize the objective function JðuÞ. In Fig. 3, we observe that there is a significant decrease in the number of infected individuals and susceptible individuals controlled compared with those not controlled. 4 Conclusion In this paper, we studied optimal combination of vaccination and treatment strategies for driving infectious diseases with cure and vaccine towards eradication within a specified period for delayed SIRS model with latent period and a modified saturated incidence rate with varying size population. Existence for the optimal control pair is established, Pontryagin s maximum principle with delay is used to characterize these optimal controls, and the optimality system is derived. For the numerical simulation, we propose an algorithm based on the forward and backward difference approximation. Our numerical results show that the optimal strategy

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