Mathematical Analysis of Nipah Virus Infections Using Optimal Control Theory

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1 Jornal of Applied Mathematics and Physics, 06, 4, 099- Pblished Online Jne 06 in SciRes. Mathematical Analysis of Nipah Virs nfections Using Optimal Control Theory Jakia Sltana, Chandra N. Podder Department of Compter Science and Engineering, Green University of Bangladesh, Dhaka, Bangladesh Department of Mathematics, University of Dhaka, Dhaka, Bangladesh Received March 06; accepted 3 Jne 06; pblished 6 Jne 06 Copyright 06 by athors and Scientific Research Pblishing nc. This work is licensed nder the Creative Commons Attribtion nternational License (CC BY). Abstract The optimal se of intervention strategies to mitigate the spread of Nipah Virs (NiV) sing optimal control techniqe is stdied in this paper. First of all we formlate a dynamic model of NiV infections with variable size poplation and two control strategies where creating awareness and treatment are considered as controls. We intend to find the optimal combination of these two control strategies that will minimize the cost of the two control measres and as a reslt the nmber of infectios individals will decrease. We establish the existence for the optimal controls and Pontryagin s maximm principle is sed to characterize the optimal controls. The nmerical simlation sggests that optimal control techniqe is mch more effective to minimize the infected individals and the corresponding cost of the two controls. t is also monitored that in the case of high contact rate, controls have to work for longer period of time to get the desired reslt. Nmerical simlation reveals that the spread of Nipah virs can be controlled effectively if we apply control strategy at early stage. Keywords Nipah Virs (NiV), Optimal Control, Existence of the State, Existence of the Objective Fnctional, Pontryagin s Maximm Principle, Transversality Condition, Optimality Condition, Hamiltonian (H). ntrodction Mathematical modeling has become an important tool for analyzing the spread as well as control of infectios diseases. t is also a sefl tool for the measrement of the effect of different strategies for controlling the spread of infectios diseases within a poplation. n recent years epidemiological modeling of infectios disease transmission has had an increasing inflence on the theory and practice of disease management and control []. There How to cite this paper: Sltana, J. and Podder, C.N. (06) Mathematical Analysis of Nipah Virs nfections Using Optimal Control Theory. Jornal of Applied Mathematics and Physics, 4,

2 are a nmber of different methods for calclating the optimal control for a specific mathematical model. For example, Pontryagin s maximm principle [] allows the calclation of the optimal control for a system of ordinary differential eqation with a given constraint. Here the optimal control strategy is sed to minimize the infected individals and to maximize the total nmber of recovered individals. Nipah virs, a member of the gens Henipavirs, a new class of virs in the Paramyxoviridae family, has drawn attention as an emerging zoonotic virs in soth-east and soth-asian region [3]. This emerging infectios disease has become one of the most alarming threats of the pblic health mainly de to its periodic otbreaks and the high mortality rate [4]. Epidemiology is the stdy of the distribtion and determinants of health related states or events in specified poplations and the application of epidemiology is to control of health problems. The crcial point is that epidemiology concerns itself with poplations or grops of poplation in contrast to clinical medicine, which deals with individals (patients). Therefore, epidemiology describes health and disease in terms of freqencies and distribtions of determinants and conditions in a poplation or in a specific grop of a poplation. Althogh Nipah virs has cased only a few otbreaks, it infects a wide range of animals and cases severe disease and death in people, making it a pblic health concern [5]. Treatment is mostly symptomatic and spportive as the effect of antiviral drgs is not satisfactory. So the very high case fatality addresses the need for adeqate and strict control and preventive measres. This paper deals with application of optimal control to a dynamic model of Nipah Virs (NiV) infections and its possible control and preventive strategy with the help of optimal control techniqe. Or aim is to minimize the total nmber of infectios individals and the cost which is related for creating awareness and treatment.. Formlation of Model Nipah virs infection is a zoonotic virs and transmitted first from animal to hman. Once it has been transmitted to hman, then it contines to be transmitted throgh hman to hman (HH) by the close contact of infected individals de to its highly infectivity [6]. Let s sppose that S( t ), ( t ) and R( t ) denote the nmber of individals in the ssceptible, infectios and recovered classes at time t respectively. The total poplation at time t is represented by N( t) = S( t) + ( t) + R( t). We consider the following system of non-linear differential eqation, is a type of standard SR disease model, to describe the dynamics of Nipah Virs (NiV) infections in the commnity. = ν β µ = β ( γ + µ + α) = γ µ = ν α µ, S t N t S t t S t t S t t t R t t R t N t N t t N t () with initial conditions, S 0 = S 0, 0 = 0, R 0 = R 0, N 0 = N, () and where, the parameter β represents the effective contact rate, ν is the natral birth rate, µ is the natral mortality rate, γ is the recovery rate and α represents the disease indced death rate. Since there is no proper vaccination program or appropriate drgs for NiV infections, so in the model we introdce two control strategies, namely, creating awareness ( ) among the commnity abot the risky areas be- t measres the effort to be needed to fore otbreak of the disease and the treatment ( ). Here the control increase awareness which reslts in the redction of the transmission rate ( β ) and the control the effort reqired for giving health cares for the infected people to redce the infected individals. Now the NiV model with two control strategies is given below: = ν β ( µ + ) = β ( γ + µ + α) = γ µ () + + = ν α µ, S t N t S t t S t t S t t t t R t t R t S t t N t N t t N t t measres (3) 00

3 with initial conditions, S 0 = S 0, 0 = 0, R 0 = R 0, N 0 = N. (4) Here or main objective is to minimize the total nmber of infected individals and to redce the cost which is needed for creating awareness and treatment on a specified time interval. For the flfillment of or prpose, we work with the following objective fnction which is similar as [7]. T Minimize J( ( t), ( t) ) = A ( t) + ( B + B ) dt 0 where, B and B are weight parameters that help to balance the corresponding costs. We define the control set as follows: {( ) [ ]} U = t, t : 0 t, 0 t, t 0, T. n the objective fnction, A represents the total nmber of infected individals, B represents the cost for creating awareness and B represents the cost for treatment. 3. Existence of the Optimal Control for NiV Model 3.. Existence of the State Adding first three eqations of the system (3) we get, On integrating we get, So we have From the forth eqation of (3) we have and then S + + R = ν N t µ S t α t µ t µ R t S + + R = ν N t µ S + + R t α t S + + R ν N t S + + R ν S + + R t d ( S + + R) S + + R ν d. t [ ] [ ] T S + + R e ν S + + R = M, t 0, T S t M, t M and R t M. So, finally we get N ( t ) M. Since S( t ), ( t ), R( t ) and 3.. Existence of the Objective Fnctional ( ν µ ), N t N t ( ν µ ) T [ ] N t N e = M, t 0, T N t are bonded above, so there exists soltion for the system (3). By proving the following theorem we can establish the existence of the objective fnctional: Theorem. Consider the control problem with system (3). Then there exists optimal controls (, ) minimize (, ) J over the control set U. i.e., that 0

4 ( ) J( ) J, = min,. (5), U Proof: To se an existence reslt in [8], we mst check the following properties [9]. ) The set of controls and corresponding state variables is non-empty. ) The control set U is convex and closed. 3) The right-hand side of the state system is bonded by a linear fnction in the state and control variables. 4) The integrand of the objective fnctional is convex on U and is bonded below by k + k (, ) η with k > 0, k > 0 and η >. To prove the above theorem we need to se the following theorem and 3. Theorem. Let each of the fnctions,, F F Fn Fn F Fn and the partial derivatives,,,,,, x x x x n be continos in a region of tx,, x,, xn space defined by α< t < βα, < x < β,, αn < x < βn, and let the point ( t , x, x,, xn ) be in. Then there is an interval [ t t0 ] < h in which there exists a niqe soltion ( x = φ ( t),, xn = φn( t) ) of the system of differential eqations that also satisfies the initial conditions (,,, n ), ( ) x = F tx x x = F tx,,, x, ( ) x = F tx,,, x, n n n be a system of n differential eqations with initial condi- i 0 i. f each of the fnctions,, n and the partial derivatives F F Fn Fn n+,,,,,, are continos in space, then there exists a niqe soltion x xn x xn x = σ ( t),, xn = σn( t) that satisfies the initial conditions. Now with the help of the above two theorems we prove the for conditions of theorem (). Proof of theorem : To se an existence reslt in [8], we mst check the following properties [9]. ) The set of controls and corresponding state variables is non-empty. ) The control set U is convex and closed. 3) The right-hand side of the state system is bonded by a linear fnction in the state and control variables. ( 0) ( 0) n( 0) Theorem 3. Let xi = Fi( tx,,, xn) for i [, n] 0 tions x ( t ) = x for i [, n] F F x t = x, x t = x,, x t = x n 4) The integrand of the objective fnctional is convex on U and is bonded below by (, ) k > 0, k > 0 and η >. Proof of ): Let s consider, ds F tsrn dt = d F dt = n (,,,, ) ( tsrn,,,, ) d R F tsrn 3 dt = (,,,, ) d N = F 4 ( tsrn,,,, ), dt where, F, F, F 3 and F 4 bildp the right hand side of the system (3). Let ( t) = C and k + k η with t = C for some constants C, C. F, F, F 3 and F 4 mst be linear and they are also continos everywhere. Moreover, the partial derivatives of F, F, F 3 and F 4 with respect to all state are constants and they are continos everywhere. So by following the theorem 3, we can say that there exists an niqe soltion S = σ ( t), = σ ( t), R σ t N = σ t which satisfies the initial conditions. Therefore, the set of controls and corresponding =, 3 4 n 0

5 state variables is non-empty. Hence the condition ) is satisfied. Proof of ): By definition, U is closed. Take any controls, Additionally, observe that θ Hence, θ ( θ) satisfied. Proof of 3): We consider, 0 + for all, The state system is given below: Now we rewrite the system in matrix form: where, and ( θ) 0 θ +. θ and ( θ) ( θ). Then θ + θ θ + θ =. U and θ [ 0, ]. Then U and θ [ 0,]. Therefore, U is convex, and condition ) is F ν N S F K F S+ + γ 3 F ν 4 N. ds F tsrn dt = d F dt = (,,,, ) ( tsrn,,,, ) d R F tsrn 3 dt = (,,,, ) d N = F 4 ( tsrn,,,, ). dt S S ( t) F( t, X, ) m t, X ( t) + n t, R R ( t) N N S ν 0 k 0 0 m t, = R 0 γ 0 0 N ν S S n t, = R S + N 0 which gives a linear fnction of the controls and defined by time and state variables. Then we can find ot the bond of the right hand side. t is noted that all parameters are constant and greater than or eqal to zero. Therefore we can write, (6) (7) (8) 03

6 (,,, ) + + (, ) p X + ( ( t), ( t) ) FtX m X S t t where S, are bonded and p incldes the pper bond of the constant matrix. Hence we see that the right hand side is bonded by a sm of the state and the control. Therefore, condition 3) is satisfied. Proof of 4): For the proof of the condition 4) we se the reslt in [0] and [Fleming and Rishel (975)]. The control and the state variables are non-negative vales and are non-empty. n the minimization problem, the necessary convexity of the objective fnctional in is satisfied. The control variables, U is also convex and closed by definition. Frthermore, from [0] we see that the integrand in the objective fnctional which is A ( t) + A + A3 is convex on the control set U. Now we have to prove that J(, ) k + k (, ) η with k > 0, k > 0 and η >. Here, J(, ) A( t) B B = + + J(, ) A( t) B B + + J A t B B B B (, ) + ( + ) [ let, + = ] = k + k(, ) k > that depends on the pper bonds of where, 0 dition 4) is satisfied. 4. Characterization of the Optimal Control t. We can also see that η = >, k > 0. Hence, con- n order to derive the necessary condition for the optimal control, we se pontryagin s maximm principle []. This principle converts the system and the objective fnctional into a problem minimizing pointwise a Hamiltonian H with respect to and. n the objective fnction, the vale A is the balancing parameter, B and B are the weight parameters balancing the cost. Here we can see from the system (3) that R appears only in the recovered class. So, when we bild p the optimality system, we will ignore the recovered class. By sing pantraygin s Maximm principle we first derive the Hamiltonian which is given below H ( S,, N ) = A ( t) + B + B + λs ( νn β S µ S S ) + λ β γ + µ + α + λ ν µ α ( S ) N ( N ) where, λ S, λ, λ N are the associated adjoints for the state S,, N respectively. By differentiating the Hamiltonian (H) with respect to each state variable, we find the differential eqation for the associated adjoint. Hence, the adjoint system is, with the final conditions, ( ) λ = λ µ + + β λβ S S λ = λ βs + λ γ + µ + α λ βs + λ + λ α A S N λ = λν+ λ ν µ N S N ( T) = ( T) = ( T) = 0. λ λ λ S N So by differentiating the Hamiltonian with respect to two controls and we obtain: (9) (0) 04

7 H = B Sλ S = 0 = 0 SλS = B and H = = 0 5. Optimality System State eqations: with initial conditions, Adjoint eqations: Transversality eqations: B λ = 0 λ =. B = ν β ( µ + ), = β ( γ + µ + α) = γ µ + +, = ν α µ, S t N t S t t S t t S t t t t R t t R t S t t N t N t t N t S 0 = S 0, 0 = 0, R 0 = R 0, N 0 = N. () ( ). λ = λ µ + + β λβ. S S λ = λ βs + λ γ + µ + α λ βs + λ + λ α A. S N λ = λν+ λ ν µ N S N Characterization of the optimal control and ( T) ( T) ( T) 0. S N, () (3) λ = λ = λ = (4) and : SλS 0 if < 0, B SλS SλS = if 0, B B SλS if >. B λ 0 if < 0, B λ λ = if 0, B B λ if >. B (5) (6) 05

8 n compact notion we can write, and SλS = min, max 0, B λ = min, max 0,. B (7) (8) 6. Nmerical Reslts and Discssions Nmerical soltions to the optimal system are exected sing MATLAB. The considered two controls (, ) depend on the adjoints λ S, λ and λ N of the state variables S, and N respectively. We simlate the model withot control and with control and then we compare the reslts. We considered the nmerical vale of the controls and in between zero(0) and one() as they are not 00 percent effective. We also monitored the effectiveness of the weight parameter to see how the control is related to weight fnction. n this simlation we assmed the initial vales of S, and N as proportions instead of whole nmbers. The parameter vales sed in the simlations are presented in the following Table. Figre depicts the importance of the controls to the disease dynamics. From the graphs, we see that the control has a positive impact to redce infection ntil the controls are effective enogh. t is also clear from the figres that the disease can be controlled over finite period of time after imposing control strategies. Figre and Figre 3 monitored the impact of the parameter (α) of disease indced death rate. Here we see if α is at a low rate (α = 0.3) then the controls work effectively and as a reslt there is a significant redction of the infections. When controls do not work it reslted the increase of infected individals. On the other hand for the higher rate of α (where awareness does not work, = 0 and the treatment works for a short period of time) there is a sharp decrease of infection de to death reslting the existence of fewer recovered people. Figre 4 and Figre 5 show the comparative sitation of the disease dynamics for low and high contact rates. n the case of low contact rate (β = 0.), the infectios individals decrease ntil the controls work effectively and as a reslt there is a notable increment of recovered individal. On the other hand, for the very high contact rate (β = ), which reslted a severe disease brden, the controls work for a longer period of time to redce the disease brden. Figre 6 and Figre 7 show the inflence of the varios weight parameters. Here we notice that for low Table. Description and parameter vales of the NiV model. Variable Description nitial vales S 0 nitial ssceptible individals 0.90 [assmed] 0 nitial infected individals 0.05 [assmed] R 0 nitial recovered individals 0.05 [assmed] Parameters Description nitial vales ν Birth rate 0.03 [assmed] µ Mortality rate 0.00 [assmed] β Contact rate 0.75 [7] γ Recovery rate [assmed] α Disease indced death rate 0.0 [assmed] A Weight parameter 0 [7] B Weight parameter [7] B Weight parameter [7] T Nmber of years 6 [assmed] 06

9 Figre. NV model with control and withot control, parameter vales are taken from Table. Figre. NiV model with low disease indced death rate, α = 0.3 and other parameter vales are taken from Table. 07

10 Figre 3. NiV model with high disease indced death rate, α = 3 and other parameter vales are taken from Table. Figre 4. NiV model with low contact rate, β = 0. and other parameter vales are taken from Table. 08

11 Figre 5. NiV model with high contact rate, β = and other parameter vales are taken from Table. Figre 6. NiV model with low weight parameters, B = 0., B = 0.3 and other parameter vales are taken from Table. 09

12 Figre 7. NiV model with high weight parameters, B =, B = 3 and other parameter vales are taken from Table. weight parameters (B = 0., B = 0.3) the infectios individals decrease sharply for first few years (as the controls work at maximm level). t is also noticed that the infected individals start to increase when the effectiveness of the controls start to decrease. n the case of high weight parameter vales (B =, B = 3) the high effectiveness of the controls are monitored and as a reslt there is a sharp redction of infection dring that effective level. 7. Conclsions The important findings are given below: A comparison between with and withot control strategy is monitored. The effect of control parameters is very mch notable for redcing the infected individals to control the disease dynamics. The controls need to be effective for longer period of time in case of high incidence. The optimal control is mch more effective to minimize the infected individals (as a reslt recovered individals will be maximized) and also to minimize the cost of the two control measres. For low weight parameter vales, the controls show their effectiveness at a maximm level. From the simlations it is monitored that the optimal combination of treatment and creating awareness is very prominent for disease elimination. Acknowledgements The athor, JS acknowledge, with thanks, the spport in part of the National Science and Technology (NST), Dhaka. The athors are gratefl to the reviewers for their constrctive comments. References [] Chong, H.T., Hossain, M.J. and Tan, C.T. (008) Differences in Epidemiologic and Clinical Featres of Nipah Virs Encephalitis between the Malaysian and Bangladesh Otbreaks. Nerology Asia, 3, 3-6. [] Pontryagin, L.S., Boltyanskii, V.G., Gamkrelize, R.V. and Mishchenko, E.F. (96) The Mathematical Theory of Op- 0

13 timal Processes. New York, Wiley. [3] (008) Nipah Virs nfections. WHO Report, Asia-Pacific Region, World Health Organization. [4] (0) National Gideline for Management, Prevention and Control of Nipah Virs nfection inclding Encephalitis, Directorate General of Health Services. Ministry of Health and Family Welfare, Government of the People s Repblic of Bangladesh. [5] (004) Nipah Virs: Vaccination and Passive Protection Stdies in a Hamster Model. Jornal of Virology, 78, [6] Biswas, M.H.A. (04) Optimal Control of Nipah Virs (NV) nfections: A Bangladesh Scenario. Jornal of Pre and Applied Mathematics: Advances and Applications,, [7] Bakare, E.A., Nwagwo, A. and Danso-Addo, E. (04) Optimal Control Analysis of an SR Epidemic Model with Constant Recritment. nternational Jornal of Applied Mathematical Research, 3, [8] Fleming, W.H. and Rishel, R.W. (975) Deterministic and Stochastic Optimal Control. Springer Verlag, New York. [9] Ysf, T.T. and Benyah, F. (0) Optimal Control of Vaccination and Treatment for an SR Epidemiological Model. World Jornal of Modelling and Simlation, 8, [0] Hsieh, Y. and She, S. (00) The Effect of Density-Dependent Treatment and Behavior Change on the Dynamics of HV Transmission. Jornal of Mathematical Biology, 43,