An Optimal Control Problem Applied to Malaria Disease in Colombia

Transcription

1 Applied Mathematical Sciences, Vol. 1, 018, no. 6, 79-9 HIKARI Ltd, An Optimal Control Problem Applied to Malaria Disease in Colombia Jhoana P. Romero-Leiton Escuela de Ciencias Matemáticas y Tecnología Informática Universidad de Investigación de Tecnología Experimental Yachay Tech Hacienda San Miguel de Urcuquí, Ecuador Jessica M. Montoya-Aguilar Departamento de Matemáticas, Universidad del Quindío. Cra 15 Calle 1N, Armenia, Colombia Eduardo Ibargüen-Mondragón Departamento de Matemáticas y Est., Facultad de Ciencias Exactas y Nat. Universidad de Nariño, Pasto, Colombia Copyright c 018 Jhoana P. Romero Leiton et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In Colombia, malaria is a public health problem that affects a large part of its population. Above motivated us to formulate an optimal control problem considering the following control variables: bed nets (BN), intermittent prophylactic treatment in pregnancy (IPTp) and prompt and effective case management (PECM). The results reveal the importance of congenital transmission and death due to disease in the infection outcome. It is imperative that strategies be defined to reduce these risk factors. Mathematics Subject Classification: 34D3, 93D0, 65L05 Keywords: Malaria, infection force, vertical transmission, optimal control problem.

2 80 Jhoana P. Romero-Leiton et al. 1 Introduction Malaria is an infectious disease caused by the parasite of the genus Plasmodium. This infection is usually acquired through the female mosquito bite of the genus Anopheles, although there are other ways of contagion such as blood transfusion or vertical transmission 11]. Tumaco is located on the pacific coast of Colombia, its whether conditions make of this zone a perfect place for malaria transmission 10]. Usually, it is treated with antimalarial drugs, but is important include control strategies with incidence in reduction of the vector population. These strategies consist of fumigations, removal of breeding grounds, use of mosquito nets and use of repellent substances to avoid contact between mosquito and human 16]. There are different factors (biological, physical, environmental, social and economic, among others) that affect the transmission of malaria. The force of infection is another determining factor in the transmission of this disease. Using the mosquito force of infection given in 8] we define β V = β vi H N V, (1) which measures the impact of disease transmission rate when an infected human is introduced into the mosquito population. The qualitative analysis of the mathematical model was made in 9], we formulate an optimal control problem for malaria disease transmission considering control variables associated to bed nets, prompt and effective case management and intermittent prophylactic treatment in pregnancy. The results reveal the importance of congenital transmission, force of infection and death due to disease in the infection outcome. The optimal control problem formulation Following 9], we denote by S H (t), E H (t), I H (t) and R H (t) the populations at time t of susceptible, exposed, infected and recovered humans. On the other hand, S V (t) and I V (t) represent the susceptible and infected mosquito populations at time t. The total human population is given by N H = S H + E H + I H + R H and for mosquitoes its total population is N V = S V + I V. The force of infection for humans is given by β H = β hi V N H, where β h = β HV ɛφ, being β HV the probability of human infection due to the bite of an infected mosquito and ɛφ the per capita mosquito bite rate, and the force of infection for mosquito is defined in the equation (1). With the following control variables u 1 (control variable associated to bed nets (BN)), u (control variable associated to prompt and effective case management (PECM)) and u 3 (control variable associated to intermittent prophy-

3 An optimal control problem applied to malaria 81 lactic treatment in pregnancy (IPTp)) we incorporate in the model defined in 9] next hypothesis: susceptible humans become exposed by contact with infected mosquitoes at a rate (1 u 1 )β H S H, where u 1 0, 1] (u 1 = 0 represents the no efficacy of the bed net while u 1 = 1 means that the use of bed net is completely effective). On the other hand, we assume that infected individuals recover at a rate δ + ξ u, where δ is the spontaneous recovery rate, and ξ 0, 1] is the treatment efficacy. Finally, the birth rate of infected humans is given by (1 u 3 ) λ, where λ is the vertical transmission rate. The control variable u 3 0, 1] (u 3 = 0 represents zero treatment efficacy during pregnancy, while u 3 = 1 means that treatment is completely effective). From above suppositions we obtain the following state equations: ds H de H di H dr H ds V di V = Λ H + ωr H 1 u 1 (t)]β H S H µ H S H = 1 u 1 (t)]β H S H (α + µ H )E H = 1 u 3 (t)] λ I H + αe H δ + ξ u (t) + ρ + µ H ]I H = δ + ξ u (t)]i H (ω + µ H )R H = Λ V 1 u 1 (t)]β V S V µ V S V = 1 u 1 (t)]β V S V µ V I V. () The following performance index or cost function is given by J(x 0, u 1, u, u 3 ) = T c 1 E H + c I H + 1 ( d1 u 1 (t) + d u (t) + d 3 u 3 (t) )], 0 (3) where, c 1 and c represent social costs which depend on the number of malaria infections and the number of mosquito bites; d 1, d y d 3 are relative weights associated with the controls and boundary conditions are x(0) = ( S H, ĒH, ĪH, R H, S V, ĪV ) = x 0 x(t ) = (S Hf, E Hf, I Hf, R Hf, S Vf, I Vf ) = x 1. (4) For the control problem we assume that the initial time is zero, t 0 = 0, the final time t 1 = T is a fixed implementation time of the control strategies, the final state x 1 is variable and the initial state x 0 is an endemic equilibrium. It is important to note that the control functions u i (t) belong to the following set U = {u(t) : u(t) is Lebesgue measurable and 0 u(t) 1, t 0, T ]}, (5)

4 8 Jhoana P. Romero-Leiton et al. Parameter Description (dimension) Value Reference α Rate of progression from exposed to infected (day 1 ) ] µ V Natural death rate of mosquito (day 1 ) ] µ H Natural death rate of human (day 1 ) ] ρ Death rate due to infection (day 1 ) ] δ Recovery rate (day 1 ) ?] β HV Probability of human infection by mosquito (dimensionless) ] β V H Probability of mosquito infection by human (dimensionless) ] ɛφ Mosquito sting rate (day 1 ) ] λ Vertical transmission rate (day 1 ) ?] Λ V Mosquito birth rate (mosq. day 1 ) ] Λ H Birth rate of human (hum. day 1 ) ] ω Rate of immunity loss (day 1 ) ] Table 1: Parameter description and values of the system (). called set of admissible controls. A description of parameters of the model and their values are shown in Table 1..1 Existence of an optimal solution Following the same idea presented in the Theorem.1 from reference 19] (see page 63) we prove the existence of optimal controls. Let U = 0, 1] 3 the control set, v = (u 1, u, u 3 ) U, x = (S H, E H, I H, R H, S V, I V ) and f(t, x, v) the right hand of (), that is f(t, x, v) = Λ H + ωr H (1 u 1 )β H S H µ H S H (1 u 1 )β H S H (α + µ H )E H (1 u 3 )λi H + αe H (δ + ξ u + ρ + µ H )I H (δ + ξ u )I H (ω + µ H )R H Λ V (1 u 1 )β V S V µ V S V (1 u 1 )β V S V µ V I V = (6) It is said that the pair (x 0, v) is feasible if there exists a solution of the control system () in the interval 0, T ] with initial condition x(0) = x 0 and final condition x(t). To prove the existence of the (x, u ) we must verify the following literals: h 1 h h 3 h 4 h 5 h f defined in (6) is of class C 1 on its three components and there is a constant C such that (a) f(t, 0, 0) C (b) f x (t, x, v) C(1 + v ) (c) f u (t, x, v) C.. The set F of feasible pairs is non empty. 3. The set of controls U is convex.

5 An optimal control problem applied to malaria f(t, x, v) = α(t, x) + β(t, x)v. 5. The integrand of the performance index f 0 (t, x, v) defined in (3) is convex in the variable v U. 6. f 0 (t, x, v) c 1 v b c with c 1 > 0 and b > 1. In order to verify the above conditions, we state and prove the following results: Proposition.1. { } (β f v (t, x, v) max h + βv), ξ + (λ/) (Λ µ V µ H + Λ V ), (7) H where f v (t, x, v) = β H S H 0 0 β H S H ξ I H λ/i H 0 ξ I H 0 β V S V 0 0 β V S V 0 0. (8) Proof. Direct calculations show that f v (t, x, v) is given by (8). On the other hand, from the definition of Euclidean norm of a matrix is followed f v (t, x, v) = β h I V ( SH N H ) + β vs V ( SH N H βh I V + β vsv + ξ IH + (λ/) IH (β h + β v) Λ V µ V { (β max h + βv) µ V + (ξ + (λ/) ) Λ H, ξ + (λ/) µ H ) + ξ I H + (λ/) I H µ H } (Λ H + Λ V ). Proposition.. { } (β f(t, 0, 0) max h + βv), ξ + (λ/) (Λ µ V µ H + Λ V ), (9) H where f(t, 0, 0) is the vector defined in (6) evaluated in x = 0 and v = 0.

6 84 Jhoana P. Romero-Leiton et al. Proof. Observe that f(t, 0, 0) = (Λ H, 0, 0, 0, Λ V, 0, ) T. From definition of Euclidean norm we obtain f(t, 0, 0) = { } (β Λ H + Λ V max h + βv), ξ + (λ/) (Λ µ V µ H + Λ V ). H Proposition.3. where f x (t, x, v) C(1 + v ), (10) { } (β C = max h + βv), ξ + (λ/) (Λ µ V µ H + Λ V ). (11) H Proof. The result is obtained by performing a procedure similar to Proposition.1 and.. Proposition.4. The control set U = 0, 1] 3 is convex. Proof. Let X = (x 1, x, x 3 ), Y = (y 1, y, y 3 ) in 0, 1] 3 and γ 0, 1]. Then which implies γx + (1 γ)y 0, 1] 3. Proposition.5. where f is the vector defined in (6). γx i + (1 γ)y i 0, 1], i = 1,..., 3, f(t, x, v) = α(t, x) + β(t, x)v, (1) Proof. The result is obtained directly by taking α(t, x) = Λ H + ωr H β H S H µ H S H β H S H (α + µ H )E H λ/i H + αe H (δ + ρ + µ H )I H δi H (ω + µ H )R H Λ V β V S V µ V S V β V S V µ V I V ; β(t, x) = β H S H 0 0 β H S H ξ I H λ/i H 0 ξ I H 0 β V S V 0 0 β V S V 0 0 (13). Proposition.6. The integrand of the performance index defined in (3) is a convex function (concave upward) on the variable v = (u 1, u, u 3 ).

7 An optimal control problem applied to malaria 85 Proof. Since the integrand of the performance index f 0 is such that f 0 (t, x, v) = c 1 E H + c I H + 1 d1 u 1 + d u + d 3 u 3] = f 1 (t, x) + h(t, v), it is enough to verify that the function 3 h(t, v) = 1 i=1 d i u i is convex on the variable v. Note that h(t, v) is a finite linear combination with positive coefficients of the functions p i (u) = 1/u i, therefore, it is sufficient to verify that the function p(u) defined by p(u) = 1 u is concave upward, which is clear due to d p du = > 0. Proposition.7. The performance index f 0 satisfies Proof. f 0 (t, x, v) 1 min {d 1, d, d 3 }(u 1 + u + u 3). (14) f 0 (t, x, v) = c 1 E H + c I H + c 3 I V + 1 (d 1u 1 + d u + d 3 u 3) 1 (d 1u 1 + d u + d 3 u 3) 1 min {d 1, d, d 3 }(u 1 + u + u 3), Remark.8. Taken b =, c = 0 and c 1 = 1/ min {d 1, d, d 3 } in above proposition the condition 6 is satisfies. From Propositions.1 to.7 and Remark.8 we verify the conditions 1 to 6 of Theorem.1 ( 19], page 63). Therefore, we have the following result which guarantees the existence of optimal controls for the control problem defined by equations (), (3) and (4). Theorem.9. Consider the control problem with state equations (). There exists (x 0, v ) F such that J(x 0, v ) = min {J(x 0, v) : v = (u 1, u, u 3 ) y u i U}, (15) where J(x 0, v) is the performance index defined in (3).

8 86 Jhoana P. Romero-Leiton et al.. Deduction of an optimal solution and adjunct equations In this section we deduce the necessary conditions for the existence of optimal controls by means of the Principle of Pontryagin required to implement the numerical forward-backward sweep method 17]. Let ẋ = f(x, u) the state equations with boundary conditions x(t 0 ) = x 0 and x(t 1 ) = x 1, the index of performance J(x 0, u) and u 1 (t), u (t) and u 3 (t) be continuous piecewise functions in 0, 1]. Let P = (P 1, P, P 3, P 4, P 5, P 6 ) be an extended vector and L be an additional state variable which satisfies L = c 1 E H + c I H + c 3 I V + 1 (d 1u 1 (t) + d u (t) + d 3 u 3 (t) ), (16) with boundary conditions L 0 = 0, L 1 = J(x 0, u). (17) From the above, the Hamiltonian is defined as follows or equivalently H(x, u, P ) = L + P f(x, u). (18) H = c 1 E H + c I H + c 3 I V + 1 (d 1u 1 (t) + d u (t) + d 3 u 3 (t) ) d S H d E H d I H d R H + P 1 + P + P 3 + P 4 = c 1 E H + c I H + c 3 I V + d 1u 1 (t) + d u (t) + P 1 Λ H + ωr H (1 u 1 (t))β H S H µ H S H ] + P (1 u 1 (t))β H S H (α + µ H )E H ] + P 5 d S V + d 3u 3 (t) + P 6 d I V + P 3 (1 u 3 (t))λ/i H + αe H (δ + ξ u (t) + ρ + µ H )I H ] + P 4 (δ + ξ u (t))i H (ω + µ H )R H ] + P 5 Λ V (1 u 1 (t))β V S V µ V S V ] + P 6 (1 u 1 (t))β V S V µ V I V ]. Let P be an extended co-condition vector of dimension n + 1 which defines the following system of adjunct equations P = H x. (19) Since H no depend of L then the system is rewritten as follows P 0 = H L P = H x, (0)

9 An optimal control problem applied to malaria 87 or equivalently P 0 = 0 ] P (N H S H ) 1 = P 1 µ H + (P 1 P ) (1 u 1 (t))β HV I V NH P = (P P 1 ) (1 u 1 (t)) β ] HV I V S H αp NH 3 + P (α + µ H ) c 1 P 3 = (P P 1 ) (1 u 1 (t)) β ] HV I V S H + (P 5 P 6 ) N H (1 u 1 (t)) β V HS V N V + P 3 (δ + ξ u (t) + ρ + µ H ) λ/(1 u 3 (t))] P 4 (δ + ξ u (t)) c P 4 = (P P 1 ) (1 u 1 (t)) β ] HV I V S H + (ω + µ NH H )P 4 P 1 ω ] P (N V S V ) 5 = (P 5 P 6 ) (1 u 1 (t))β V H I H + P NV 5 µ V P 6 = (P 1 P ) (1 u 1 (t)) β ] HV S H + (P 5 P 6 ) (1 u 1 (t)) β ] V HI H S V N H + P 6 µ V c 3. (1) The transversality conditions of are given by N V P i (T ) = 0 i = 1,..., 6. () ] Finally, from the optimality condition H u i following optimal controls = 0 for i = 1,, 3 we obtain the u 1 = u = min{max{0, (P P 1 )β H S H +(P 6 P 5 )β V S V d 1 }, 1} min{max{0, (P 3 P 4 )ξ I H d }, 1} (3) u 3 = min{max{0, P 3λ/I H d 3 }, 1}, where d 1, d y d 3 are diferent of zero..3 Numerical solutions for the control problem In this section we analyze the effects of the control strategies u 1, u y u 3, which are applied on the system () using data of Tumaco (Colombia) which are given on Table 1. Numerical simulations are performed using the forwardbackward sweep method 17], and the implementation time of control strategies is T = 10 days, which is approximately 4 months. The values of the relative weights associated with the controls, the social costs and the effectiveness of antimalarial treatment, are given on Table. The

10 88 Jhoana P. Romero-Leiton et al. Parameters Value References d ] Relative d ] weights d ] Social c Assumed Costs c Assumed treatment ξ ] Table : The values of the parameters associated with the control problem. maximum cost achieved by the implementation of controls is measured in cost units. Here, controls u 1, u and u 3 are implemented simultaneously. In the Figures 1 (a) and (b) it is observed that the exposed human populations E H and infected humans I H decrease rapidly during the first 0 days of strategy implementation, while both populations grow and stabilize in endemic equilibrium without the use of controls. The Figure 1 (c) shows the behavior of the three controls: u 1 and u remain at their upper level of 100 % during the total implementation period of the strategy, while control u 3 reaches and remains at the upper bound of 100% during the first 10 days of strategy implementation, then decreases to its lowest level of 0% after 110 days. The maximum cost achieved by the implementation of the control strategy is 15.4 units of cost. 3 Discussion In Colombia, malaria incidence has declined by more than 40% and mortality by more than 60% in the last 15 years 18]. However, in some regions of the Colombian Pacific, such as Tumaco, there are specific conditions that must be contested to achieve the eradication of malaria 18, 10] In Tumaco, cases of death due to malaria have been substantially reduced. However, cases of congenital malaria have arisen and in addition the incidence of malaria has been concentrated mainly in specific sectors or communes of the municipality. This suggests that the dynamics of transmission is changing, which generates a new public health alert 5]. For this reason in this work, we define some control variables, and we formulate an optimal control problem defined by (3), () and (4). The qualitative analysis of () can be made following a process similar as in, 3, 6, 7, 8], In order to solve the control problem numerically, we demonstrate the existence of optimal solutions and optimal controls, also through Pontryagin Principle we calculated the controls and the adjunct equations required to implement the numerical method. We performed numerical simulations of the control problem with certain

11 An optimal control problem applied to malaria x u 1 =u =u 3 =0 u 1,u,u u 1 =u =u 3 =0 u 1,u,u 3 0 Exposed humans Infected humans H Time (days) (a) Exposed humans Time (days) (b) Infected humans Control u 1 Control u Control u Time Time Time (c) Controls Figure 1: Numerical solutions of the control problem for the exposed and infected humans using the controls u 1 (t),u (t) y u 3 (t) with the values of parameters given on Tables 1 and.

12 90 Jhoana P. Romero-Leiton et al. values for social costs, in all scenarios the transmission of malaria was controlled in a prudent time, which agrees with the theory of Sir Ross MacDonald 15]. The results suggest that in addition to the force of infection, it is important to focus on the vertical transmission rate λ and the rate of spontaneous recovery δ. Acknowledgements. Jhoana P. Romero acknowledge support from Fundación CEIBA. E. Ibarguen-Mondragón acknowledge support from project No /10/017 (VIPRI-UDENAR). References 1] B. T. Grenfell and R.M. Anderson, The estimation of age-related rates of infection from case notifications and serological data, Epidemiology and Infection, 95 (1985), no., ] E. Ibargüen-Mondragón, S. Mosquera, M. Cerón, E.M. Burbano-Rosero, S.P. Hidalgo-Bonilla, L. Esteva, J.P Romero-Leiton, Mathematical modeling on bacterial resistance to multiple antibiotics caused by spontaneous mutations, Biosystems, 117 (014), ] E. Ibargüen-Mondragón, J.P Romero-Leiton, L. Esteva, E. M. Burbano- Rosero, Mathematical modeling of bacterial resistance to antibiotics by mutations and plasmids, Journal of Biological Systems, 4 (016), ] I. Mueller, S. Schoepflin, T.A. Smith, K.L. Benton, M.T. Bretscher E. Lin, B. Kiniboro, P. A. Zimmerman, T. P. Speed, P. Siba and I. Felger, Force of infection is key to understanding the epidemiology of Plasmodium falciparum malaria in Papua New Guinean Children, Proceedings of the National Academy od Sciences, 109 (01), no. 5, ] J. Carmona-Fonseca and A. Maestre, Incidencia de las malarias gestacional, congénita y placentaria en Urabá, Antioquia Colombia, , Revista Colombiana de Obstetricia y Ginecología 60 (009), no. 1, ] J.P. Romero Leiton, E. Ibargüen-Mondragón and L. Esteva, Un modelo matemático sobre bacterias sensibles y resistentes a antibióticos, Matemticas: Ensenanza Universitaria, 19 (011),

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14 9 Jhoana P. Romero-Leiton et al. 18] V. A. Gallo Ortiz and H. B. Valencia, An lisis de la situación de salud del municipio de Tumaco (Perfil epidemiológico), Alcaldía de Tumaco, (01). 19] W.H. Fleming and R.W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, Received: January 6, 018; Published: February 19, 018