Optimal Control Applied to the Spread of Influenza A(H1N1)

Transcription

1 Applied Matematical Sciences, Vol. 6, 2012, no. 82, Optimal Control Applied to te Spread of Influenza AH11 M. El ia 1, O. Balatif 2, J. Bouyagroumni, E. Labriji, M. Racik Laboratoire d Analyse Modélisation et Simulation, Département de Matématiques et d Informatique, Faculté des Sciences Ben M Sik, Université Hassan II Moammedia, B.P 7955, Sidi Otman, Casablanca, Maroc Abstract Te aim of tis work is te application of optimal control to te study of te impact of vaccination on te spread of influenza A H11. Seeking to reduce te infected group, we use a control simulating a vaccination program. Te Pontryagin s maximum principle is used to caracterize te optimal control. Te optimality system is derived and solved numerically. Keywords: Optimal control, influenza A H11, Pontryagin s maximum principle 1 Introduction Influenza A H is an acute respiratory disease of umans appeared in Te contamination is mainly by air, tat is to say, couging and sneezing. Te virus can survive 8 to 48 ours in te open air, depending on te nature of te surface on wic it rests. It caused an influenza epidemic in te monts tat followed its appearance. Given te scale of te epidemic, te WHO described te pandemic in June In August 2010, te world came into a post-pandemic stage according to WHO. However, it is only a warning, te virus is still circulating widely around te globe. To deal wit tis epidemic, many governments ave decided to implement toug measures; for example, in several countries, most governments ave 1 Corresponding autor. elia moamed@yaoo.fr 2 Corresponding autor. balatif.mats@gmail.com

2 4058 M. El ia et al opted for mass vaccination plans as a precaution of te epidemic. In tis context and given tat te optimal control teory as been used successfully to make decisions involving biological or medical models, we propose to study, in tis paper, te impact of a vaccination campaign on te spread of te influenza AH11. Te paper is organized as follows. In section 2, we present a matematical model describing te spread of te influenza AH11 wit a control term. Te analysis of optimization problem is presented in section 3. In section 4, we give a numerical appropriate metod and te simulation corresponding results. Finally, te conclusion are summarized in section 5. 2 Matematical model In tis article we consider te simple model SIR used in [1]. Te ost population is divided into tree epidemiological classes: namely susceptible S, infective I and removed R. denotes te total population. We assume tat an individual can be infected only troug contacts wit infectious individuals. Tis model is governed by te following system of ordinary differential equations ds dt = Λ µs βs I di dt = βs I µ + d + r I dr = ri µr dt were all parameters are non negative and defined as follows Parameter Definition β Effective contact rate Λ Recruitment rate µ atural mortality rate d H11 induced mortality rate r Recovery rate Table 1: Parameter definitions We introduce into te above mentioned model a control ut representing te vaccination rate at time t. Te control ut is te fraction of susceptibles individuals being vaccinated per unit time. We assume tat all susceptibles vaccinees are transferred directly to te removed class. Te matematical

3 Control of AH system wit control is given by te nonlinear differential equations ds dt = Λ µs βs I us di dt = βs I µ + d + r I dr = ri µr + us dt 1 wit S0 0, I0 0 and R0 0 are given and t = St + It + Rt for all t. 3 Te optimal control problem In tis section we use te optimal control teory to analyze te beavior of te model 1. Our goal is to reduce te infected individuals and te cost of vaccination. Matematically, te problem is to minimize te objective functional Ju = t 0 fit + A 2 u2 tdt 2 were te parameter A 0 denotes te weigt on cost. And t f represents te duration of te vaccination program. In oter words, we seek te optimal control u suc tat J u = min {Ju : u U} 3 were U is te set of admissible controls defined by U = {ut : 0 u 0.9, t [0, t f ], u is Lebesgue mesurable} Te Pontryagin s maximum principle [2] converted 1, 2, 3 into problem of minimizing an Hamiltonian, H, defined by H = It + A 2 u2 t + 3 λ i f i 4 i=1 were f i is te rigt side of te differential equation of te i t state variable. By applying te Pontryagin s maximum principle [2] and te existence result of optimal control from [3], we obtain te following teorem:

4 4060 M. El ia et al Teorem 1 Tere exists an optimal control u, and corresponding solution S, I and R, tat minimizes Ju over U. Moreover, tere exists adjoint functions, λ 1, λ 2 and λ 3 verifying λ 1 = λ 1 µ + λ 1 λ 3 u + λ 1 λ 2 β I λ 2 = 1 + λ 1 λ 2 β S + λ 2µ + r + d λ 3 r λ 3 = λ 3 µ wit te transversality conditions λ 1 t f = λ 2 t f = λ 3 t f = 0 Futermore, te optimal control u is given by Proof. u = min0.9, max0, λ 1 λ 3 S 5 A Te existence of optimal control can be proved by using te results from [3] see Teorem 2.1. Te adjoint equations and transversality conditions can be obtained by using Pontryagin s Maximum Principle suc tat λ 1 = H, λ S 1t f = 0 λ 2 = H I, λ 2t f = 0 λ 3 = H, λ R 3t f = 0 Te optimal control u can be solve from te optimality condition, H u = 0 tat is H u = Au λ 1S + λ 3 S = 0 By te bounds in U of te controls, it is easy to obtain u in te form of 5 4 umerical simulations In tis section we first present an iterative metod for te numerical solution of te optimality system. ext, we present numerical results obtained using Matlab.

5 Control of AH Algoritm Te numerical algoritm presented below is a semi-implicit finite difference metod. We discretize te interval [t 0, t f ] at te points t i = t 0 + i i = 0, 1,..., n, were is te time step suc tat t n = t f, [4]. ext, we define te state and adjoint variables St, It, Rt, λ 1, λ 2, λ 3 and te control u in terms of nodal points S i, I i, R i, λ i 1, λ i 2, λ i 3 and u i. ow a combination of forward and backward difference approximation is used as follows: Te Metod, developed by [5] and presented in [6] and [7], is ten read as: S i+1 S i I i = Λ µs i+1 βs i+1 ui S i+1 I i+1 I i I i+1 = βs i+1 µ + d + r I i+1 R i+1 R i = ri i+1 µr i+1 + u i S i+1 By using a similar tecnique, we approximate te time derivative of te adjoint variables by teir first-order backward-difference and we use te appropriate sceme as follows = 1 1 µ u β I = 1 + λ1 n i 1 λ2 n i 1 β S i+1 + λn i 1 2 µ + r + d 3 r = 1 3 µ Te algoritm describing te approximation metod to obtain te optimal control is te following Algoritm 2 Step1: S0 = S 0, I0 = I 0, R0 = R 0, λ i t f = 0i = 1,..., 3, u0 = 0. Step 2: for i = 0,..., n 1, do: S i+1 = S i + Λ 1 + µ + β I i + ui I i+1 = 1 + µ + r + d β S i+1 R i+1 = R i + ri i+1 + u i S i µ I i

6 4062 M. El ia et al 1 1 = 1 2 = β I i µ + β I i = λn i µ λ n i 1 1 λ3 n i 1 + λn i + ui 1 λ1 n i 1 β S i+1 3 u i 3 r + λn i β S i+1 µ + r + d Si+1 T i+1 = A u i+1 = min0.9, max0, T i+1 end for Step 3: for i = 0,..., n, write: S t i = S i, I t i = I i, R t i = R i, u t i = u i. end for 4.2 umerical results Te numerical simulations are carried out using Matlab and using te following parameter values and initial conditions: Initial conditions S I0 30 R0 28 Parameter Values β Λ µ d 0.63% r 0.2 ote tat te initial conditions and te parameter values Λ, µ, d and r are taken from [1] and te parameter β is calculated from β = R 0 µ + d + r wit R 0 = 1.9 Te graps below, allow us to compare canges in te number of infected, susceptible and removed individuals before and after te introduction of control. In figure 1 we notice tat in presence of control te number I decreases greatly. Tus te maximum number of I would be in te presence of control against oterwise. It s nearly 87% of te effectiveness of te vaccination campaign.

7 Control of AH Te figure 2 indicates tat te number S decreases more rapidly during te vaccination campaign. It reaces at te end of tis campaign against in te absence of control, i.e a reduction of cases. Te figure 3 sows tat te number of people removed begins to grow notably from 22 nd day instead of 41 st days in te absence of control. Moreover te number R at te end of te vaccination period is instead of , wic represents an increase of cases. Finally, figure 4 gives a representation of te optimal control u. Figure 1: te function I wit and witout control

8 4064 M. El ia et al Figure 2: te function S wit and witout control Figure 3: te function R wit and witout control Figure 4: te optimal control u

9 Control of AH Conclusion In tis work we discussed an efficient numerical metod based on optimal control to study te impact of a vaccincation strategy on te spread of te influenza AH11. Te numerical simulation of bot te systems i.e wit control and witout control, sows tat tis strategy elps to reduce te number of infected and susceptible individuals and increase te number of te removed individuals greatly. Te results obtained sows also tat te effectiveness of vaccination campaign can reac 87%. References [1] K. Hattaf,. Yousfi- Matematical Model of te Influenza AH11 Infection- Advanced Studies in Biology, vol. 1, 2009 [2] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Miscenko, Te Matematical Teory of Optimal Processes, Wiley, ew York, [3] H.Raj josi, S. Lenart, M Y Li, L. Wang, Optimal control metods applied to disease models-contemporary Matematics, Volume 410, 2006 [4] A. B. Gumel, K. C. Patidar, and R. J. Spiteri, editors, Asymptotically Consistent on-standard Finite-Difference Metods for Solving Matematical Models Arising in Population Biology, R. E. Mickens and Worl Scientific, Singapore, [5] A. B. Gumel, P.. Sivakumar, and B. M. Saai, A matematical model for te dynamics of HIV-1 during te typical course of infection, Tird world congress of nonlinear analysts, 2001, 47: [6] K. Hattaf, M. Racik, S. Saadi, Y. Tabit et. Yousfi- Optimal Control of Tuberculosis wit Exogenous Reinfection- Applied Matematical Sciences, Vol. 3, 2009, no. 5, [7] J. Karrakcou, M. Racik, and S. Gourari, Optimal control and Infectiology: Application to an HIV/AIDS Model, Applied Matematics and Computation, 2006, 177: Received: Marc, 2012