A Model for the Population Dynamics of the Vector Aedes aegypti (Diptera: Culicidae) with Control

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1 Applied Mathematical Sciences, Vol., 8, no. 7, 3-35 HIKARI Ltd, A Model for the Population Dynamics of the Vector Aedes aegypti Diptera: Culicidae) with Control Julián Alejandro Olarte García, Aníbal Muñoz Loaiza and Carlos Alberto Abello Muñoz Grupo de Modelación Matemática en Epidemiología GMME) Facultad de Educación Universidad del Quindío Armenia-Quindío, Colombia Copyright c 8 Julián Alejandro Olarte García 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 The epidemics of dengue, zika, chikunguya and yellow fever can be prevented by fighting against their main vector Aedes aegypti. This paper presents a mathematical model described by a set of ordinary nonlinear differential equations, which depend on the dynamics of the vector and uses the theory of optimal control considering methods and associated costs, application of insecticides, environmental sanitation and use of vector traps, to deduce the most sustainable strategy that eliminates the population of female mosquitoes. Optimal control is achieved by applying the Pontryagin Maximum Principle. Keywords: Aedes aegypti, optimal control, Pontryagin Maximum Principle, simulation Introduction Historically, World Health Organization efforts to control vectors in the Americas region resulted in the elimination of Aedes aegypti populations in many of the tropical and subtropical countries by the 97s, however, as is often the case in the field of public health, when a health threat disappears, the control program ceases to exist. Resources dwindled, control programmes collapsed,

2 3 Julián Alejandro Olarte García et al. infrastructures dismantled, and fewer specialists were trained and deployed. The mosquitoes and the diseases that they transmit returned with force to an environment in which they remained few intact defenses [6]. The consequences of this dramatic reappearance are best illustrated by the recent history of dengue, zika, chikungunya, and yellow fever and recognizing the limited existence of effective drugs and vaccines for the treatment of almost all these diseases, prevention of infections and control of vectors become a component essential for reducing the burden of vector-borne diseases [7]. Therefore, the main purpose of most programs contemporary is to reduce the densities of vector populations as much as possible and keep them at low levels, and where feasible, efforts should also be made to reduce the longevity of adult female mosquitoes []. There are several control procedures used to inhibit the proliferation of the vector, all of which have their pros and cons and they can be classified as mechanical, chemical and biological control [6]. In mechanical control, public health officials are responsible to visit households and destroy, alter, remove or recycle non-essential containers that can harbor eggs, larvae or pupae of the vector. The chemical control uses pesticide on surface treatments with residual effect or as spatial treatments and aims to eliminate the adult and inmature mosquitoes. Although very effective, it can kill different mosquitoes and cause health problems in population; however, choosing those products that are safer, highly efficient, with a very low degree of toxicity and with minimal or no possibility of contamination of the environment usually generate acceptance of communities [9]. Biological control consists of the introduction of living organisms that feed, compete, eliminate and parasitize the immature vector states in the water deposits. Another method of control is vectorial traps, which although originated for entomological surveillance, in several countries they have been modified to eliminate immature or adult populations of A. aegypti [6, 6]. In all these vector control efforts, minimizing labor and costs while maximizing reduction of mosquitoes is a priority and control mathematical models are a very valuable tool for identifying the optimum solution. Optimal control theory has been used successfully to make decisions involving biological or medical models, whose desired outcome, goal and performance of control actions depend on the particular situation []. For example, Rodrigues, Monteiro and Torres ) present an application of the optimal control theory to dengue epidemics, where the cost functional depends not only on the costs of medical treatment of the infected people but also on the costs related to educational and sanitation campaigns [4]. Thomé, Yang and Esteva ) analyze the minimal effort to reduce the fertile female mosquitoes, searching for the optimal control by means of the Pontryagin s Maximum Principle, when sterile male mosquitoes produced by irradiation) are introduced as biological

3 A model for the population dynamics of the vector Aedes aegypti control, besides the application of insecticide []. Dias, Wanner and Cardoso 5) analyze the dengue vector control problem in a multiobjective optimization approach, in which the intention is to minimize both social and economic costs, using a dynamic mathematical model representing the mosquitoes population [8]. Rafikov, Rafikova and Yang 5) formulate an infinite-time quadratic functional minimization problem of A. aegypti mosquito population and they analyze different scenarios in which the three efforts of management of the population chemical insecticide control, sterile insect technique control, and environmental carrying capacity reduction) are combined in order to assess the most sustainable policy to reduce the mosquito population [7]. Masud, Kim, B. and Kim, Y. 7) analyzed optimal implementations of control with specific prevention measures for dengue transmission in the standard hosthumans)-vectormosquito) SIR-SI model [8]. In this article, we consider the model proposed in [5] and extend it to a mathematical control model that describes the dynamics of the population of two states aquatic and aerial of A. aegypti, covering only female mosquitoes since they represent the threat in the transmission of diseases), whose forms of intervention to reduce the infestation consist of environmental sanitation and the application of insecticides, combined with the use of vector traps. The model The mathematical model describes the dynamics of an Aedes aegypti population female only) when a certain control program implements integrated economically sustainable methods that have an impact on the reproduction and ecological plasticity of mosquitoes. Here, we consider the life cycle of the female mosquito population divided into two compartments: the aquatic phase or immature stages egg, larva and pupa) and the aerial phase or adult stages. We denote as x t): average number of adult mosquitoes at time t, x t): average number of immature mosquitoes at time t, and C i t) for i =,, 3: average number of epidemiologically important breeding places at time t. The life cycle of the mosquito and their breeding sites involve the following constant parameters: ω, rate of development of the immature stage to the adult stage; ɛ, mortality of the mature mosquito; φ, rate of oviposition of female mosquitoes; f, fraction of eggs that give birth to female mosquitoes; π, mortality rate of immature stages; r i, i =,, 3, intrinsic growth rate of the i-th breeding site; K i, carrying capacity of the breeding site in category i in the environment. The two management efforts of the mosquito population are insecticides and lethal traps, which we will call control perifocal; or insecticides and environmental sanitation, which we will call control focal. The application of the focal control, u t), implies to proportionally decrease the variable x t) per unit time; and the variable x t) decreases proportionally per unit time, due

4 34 Julián Alejandro Olarte García et al. to the application of the perifocal control, u t). The system of differential equations that describes the growth dynamics of the A. aegypti population using these controls is: x t) = ωx t) ɛx t) u t)x t) h ) dt x t) = fφx t) x ) a) t) π + ω)x t) u t)x h ) dt Kt) { Ci t) = r i C ) it) C i t) h 3 ) i =,, 3) b) dt K i with {ω, ɛ, φ, π, θ i, r i, K i } R +, f, ), u t) [, ], u t) [, ] and initial conditions: x ) = x, x ) = x, C i ) = C i i =,, 3) The trajectories of the solutions of the system exist in the compact and positively invariant set: x { x } x = C ω ) jk, C R5 + : x j j =, ; < Ci K i, i =,, 3, ɛ C 3 in which K = i= 3 θ i K i > denotes the fixed carrying capacity for the population of immature mosquitoes. This expression represents the maximum amount of immature forms that can lodge the three types of more productive breeding places in a locality, taking into account for each category of container its carrying capacity K i ) and its average pupal productivity θ i ). The model a)-b) can be simplified in the following way. First, the compact equation for the three types of breeding sites can be decoupled from the system. Secondly, the size of Kt) is controlled by human intervention and independent of the mosquito population. In fact, dependent on the analytical solution of the nonlinear equation b), Kt) is given by: Kt) = 3 i= θ i C i K i C i + K i C i )e r it i =,, 3) Thus, assuming that r i >, Kt) will converge rapidly or not, depending on r i ) to the maximum capacity of eggs-larvae-pupae in the middle, K, then

5 A model for the population dynamics of the vector Aedes aegypti system a)-b) become the following autonomous system: x dt = ωx t) ɛx t) u t)x t) h ), x ) = x x dt = fφx t) x ) t) ) π + ω + u t))x t) h ), x ) = x K The simplest model will be analyzed in the compact subset: { [ ] } z x = R +: x ω ɛ K, x K x 3). Controllability Some systems can be brought into a position of equilibrium, by applying an input and after a finite period of time. In order to know if the solution to a problem of design of a control system exists, one must arrive at specific conclusions regarding its controllability. Since there is no theory of controllability applicable to non-linear systems, the system ) is linearized in the neighborhood of the stable stationary solution ẑz = [ˆx, ˆx, ˆx 3 ], considering deviations of said stationary solution and the deviation of the control u = [ u, u ] : z = z ẑẑẑ, u = u ûûû to have the system in deviations of the form: z) = A z z) + B u u) 4) equivalently, d dt [ ] x = y [ ] [ ] [ ] [ ] a a x b u + a a y b u where a = h ẑẑẑ,ûûû) x a = h ẑẑẑ,ûûû) x b = h ẑẑẑ,ûûû) u = ɛ + û ), a = h ẑẑẑ,ûûû) x = fφ ˆx ), K = ω, a = h ẑẑẑ,ûûû) x = fφ ˆx K π + ω + û ), = ˆx, b = h 3ẑẑẑ,ûûû) u = ˆx. The system 4) is said to be completely controllable if the state of the system can be transfered from the zero state z at any initial time t to any terminal state z f within a finite time t f t. Here, when we say that the system can he

6 36 Julián Alejandro Olarte García et al. transferred from one state to another, we mean that there exists a piecewise continuous input u L [, t f ]; R ), t t f, which brings the system from one state to the other []. A necessary and sufficient condition, due to R. E. Kalman 956), for complete controllability of the state of a finite-dimensional linear system with invariant matrices A and B, similar to 4), is that rg [B AB... A n B] = n 5) where [B AB... A n B] is the Kalman controllability matrix []. As an alternative to this condition, one may determine the number of columns of C or rows) that are linearly independent [3]. For the system ), the controllability matrix is [ ] b a C = b a a b b a b a This matrix has a non-zero diagonal because b ii = ˆx i. Then the two rows of C are non-zero and form a linearly independent set, or equivalently, the first two columns of C contain pivot and are then linearly independent. Therefore, rg C = rg C = 3. As the dimension of the linear system of state 4) is 3 = rg C, then said system is completely controllable. 3 The optimal control problem In this section, we use the optimal control theory to analyze the population growth of Aedes aegypti. Our objective is to solve the following problem: given the initial population size in the two compartments, x, x and x 3, find the best strategy in terms of combined efforts, insecticides, lethal traps and suppression of mosquito breeding foci that would minimize the number of mosquitoes in an infested area, while also keeping the execution of the control program at low cost. Mathematically, the problem is to minimize the following functional objective: Jut)) = tf x t) + x t) + κ u t) + κ ) u t) dt 6) where the coefficients κ, κ > are constant weights on the controls, and u i L [, t f ]; [, ] ), subject to the initial value problem ). In other words, it is a question of determining a vectorial function u = u t) in the set of permissible functions Γ such that Ju ) Ju), u Γ, or equivalently: Ju, u ) = min u Γ Ju, u ) 7)

7 A model for the population dynamics of the vector Aedes aegypti donde Γ = { u L [, t f ]; R ) : u = [u, u ], u i }. This functional objective includes the costs that health providers have budgeted for the application of controls) during a specified period of time; κ u is the necessary cost to carry out focal control activities and κ u is the necessary cost to carry out perifocal control activities. The basic framework of an optimal control problem is to prove the existence of the optimal control and then characterize the optimal control through the optimality system. Proposition 3. Given the control problem with system ), there exists u = [u, u ] Γ such that min u Γ Ju) = Ju ). Proof. The existence of an optimal control pair can be tested using the results of Fleming and Rishel [5], Theorem 4..), which asks to check the following hypothesis. i) The set of controls and corresponding state variables is not empty: a result of existence by Lukes [], Theorem 9..) gives the existence of the system solution ) with bounded coefficients. ii) The control set is convex and closed by definition. iii) The right side of the system ) is bounded by a linear function in the state and the control: since the state solutions are a priori bounded. iv) The integrand in the functional target 6), L, is clearly convex in Γ and, moreover, there are constants δ, δ > and σ > such that L satisfies Lt, zt), ut)) δ ut) σ δ, just choose δ = min{κ, κ }, σ = and δ > is arbitrary. 3. Maximum Principle In 956 S. L. Pontryagin established conditions necessary to find optimal controls, what is known as the Maximum Principle [4]. This principle converts ),6) and 7) into the problem of minimizing pointwise a Hamiltonian, H, with respect to u and u : H zt), ut), λt) ) = L zt), ut) ) + λ i h i where zt) = [ x t), x t) ] is the vector of state variables, ut) = [ u t), u t) ] is the vector of controls, λt) = [ λ t), λ t) ] is the vector of adjoint or conjugated variables, h i is the right side of the i th differential equation of the system ) and L zt), ut) ) is the integrand defined in 6). Namely, ) ) + λ ωx ɛ + u )x H z, u, λ ) = x + x + = + λ fφx x K i= κ u + κ u ) ) π + ω + u )x

8 38 Julián Alejandro Olarte García et al. We obtain the optimality condition with the help of the Lagrangian, which is formed by adding a penalized term to the criterion. So, the Lagrangian L z, u, λ ) is L z, u, λ ) = H z, u, λ ) + ρ i u i + i= ρ i+ u i ), i= where ρ j j =,, 3, 4) are multipliers of penalty and satisfy ρ i u i =, ρ i+ u i ) = i =, ) 8) By applying the Pontryagin s maximum principle [4] and the existence of an optimal control, we obtain the following proposition: Proposition 3. If u and u is an optimal control couple that minimizes Ju, u ) over Γ with corresponding states x and x, then there exists adjoint functions λ and λ verifying ) dλ dt = ɛ + u )λ fφ dλ dt = ωλ + x K fφ x K + π + ω + u λ ) λ with the transversality conditions: λ t f ) = λ t f ) =. Futhermore, the optimal control u = [u, u ] is given by [ { { u = min max, λ } } { { x,, min max, λ } x, } ] 9) κ κ Proof. The adjoint equations and the transversality conditions can be obtained using dλ dt = H ) z z, u, λ, λt f ) = From the first order condition, L u =, the optimal control is solved. This is, or equivalently, κ i u i λ i x i + ρ i ρ i+ = i =, ) u i = λ ix i ρ i + ρ i+ κ i i =, ) ) According to 8), we distinguish three cases:

9 A model for the population dynamics of the vector Aedes aegypti i) On the set {t : < u i t) < }; ρ i =, ρ i+ =. Replacing in ), u i = λ i x i. κ i ii) On the set {t : u i t) = }; ρ i, ρ i+ =. Replacing in ), ρ i = λ i x i. κ i κ i iii) On the set {t : u i t) = }; ρ i =, ρ i+. Replacing in ), ρ i+ = λ i x i. κ i κ i Clearly by case i), the inequalities of cases ii) and iii) must satisfy < x i { κ i } <, for which it is necessary, with respect to case ii) that u λi i = min x i,, { } κ i and with respect to case iii) that u λ i i = max, x i. The joint analysis of κ i the three cases allows us to characterize the control u i = u i t) for i =, of the form 9). 3. Optimality System The optimality system arises by incorporating the state system with given initial conditions and the adjoint system with the transversality conditions, coupled with the characterization of the control vector. Thus, we have ẋ = ωx ɛ + u )x ẋ = fφx x ) π + ω + u )x K λ = ɛ + u )λ fφ x ) λ K ) ) λ = ωλ + fφ x K + π + ω + u λ { { u = min max, λ } } { { x,, u = min max κ Initial conditions: x = x ), x = x ). Terminal conditions: λ t f ) =, λ t f ) =. 4 Numerical illustration, λ κ x } }, Numerical solutions to the optimality system ) are executed using MA- TLAB with the parameter values and initial conditions reported in la Table. It is important note that the parameters values were chosen such that the total λ i

10 3 Julián Alejandro Olarte García et al. fφω ɛπ+ω) population never goes into extinction and it yields > in the absence of control strategies i.e. when u = u = ). The entomological parameters of the vector were determined by evaluating the polynomial functions of Yang et al. 9) at C derived from empirical data [3], while the remaining parameters are hypothetical. An iterative method is used for solving the optimality system. We start to solve the state equations with a initial guess for the controls over the simulated time using a forward fourth order Runge-Kutta scheme. Because of the transversality conditions, the adjoint equations are solved by a backward fourth order Runge-Kutta scheme using the current iteration solution of the state equations. Then, the controls are updated by using a convex combination of the previous controls and the value from the characterization 9). This is done iteratively until we obtain convergence. For more detail see []. Table : Parameters for the life cycle model. modelmmodlmldl. of Parameters and initial conditions for the life cycle model and their values.o.f Parameter φ ɛ π ω Value Parameter f x x K Value Figure -I) represents the population of immature mosquitoes in the system without control dashed line) and ) continuous line) using perifocal control as the only elimination method. We appreciate in the scenario with control, that the aquatic phase increases until about the 3th hour of development and survival of the offspring and reaches its maximum number of x t) = 75, then decreases severely, reaching x t) = 555 on the 3th day when the control disappears. In omission of any of the controls, the aquatic phase grows prolifically and begins to stabilize after the 3th hour. Figure -II) represents the population of mature mosquitoes in the system without control dashed line) and ) continuous line) using perifocal control as the only elimination method. We appreciate in the scenario with control, that the maximum density of the aerial phase is x ) = 4 during the whole vector control campaign, then it drops sharply in the period -5 days and, in the course of the next 5 days, flocks of mosquitoes are slowly disappearing. In omission of any of the controls, the aerial phase grows prolifically from time zero. In Figure -III) we see that before the marginal application of focal control, the number of immature mosquitoes increases rapidly at the beginning, until day.5, but its growth becomes quite slow later; after the application of

11 Aquatic phase Aerial phase Aquatic phase Aerial phase A model for the population dynamics of the vector Aedes aegypti... 3 the focal control, the size of the aquatic phase increases during the entire implementation period, more quickly until before the first day and at the end of the control campaign. The difference detected between the two scenarios is that the focal control manages to reduce the density of the aquatic phase in a non-controlled model by up to 6%. In Figure -IV) we see that with the application of focal control as the only control measure against any control effort, the number of mature mosquitoes increases monotonously within 3 days of the eradication campaign. The notable difference between the two scenarios is that the focal control manages to reduce the density of the aerial phase in the non-controlled model by up to %. I) II) # # Without control With control 4 Without control With control Figure : The graphs show densities of mature x t) and immature x t) mosquito populations with perifocal control and without control. III) VI) # 6 3. # Without control With control 5.5 With control Without control Figure : The graphs show densities of mature mosquitoes x t) and immature x t) with focal control and without control.

12 Control Control 3 Julián Alejandro Olarte García et al. The controls u t) and u t) plotted in Figure 3 are functions of time with κ = and κ =, respectively. The control u t) depends largely on the adjoint variable λ t) and the values of κ ; the control u t) depends to a large extent on the adjoint variable λ t) and the values of κ. V) VI) Control u t) Control u t).5 Control u t) Control u t) Figure 3: Graphs of control functions applying V): focal control without perifocal control; VI): perifocal control without focal control. Figure 4 illustrates the responses of the evolutionary phases to the same control method; in subfigure VII), x t) dashed line) and x t) solid line) were mapped under the action of the perifocal control, and in subfigure VIII), x t) dashed line) and x t) solid line) were mapped under the action of the focal control. We observe that in situation VII) both curves decline from day.5, but x t) becomes slowly smaller and reduces its distance of x t), which exceeds it by 75 at the end of the campaign. Very different is the situation VIII) where both curves rise from the first day, showing a future tendency to recover its steady state beyond the thirtieth day. Figure 5 shows how the subpopulations decrease due to integrated control focal-perifocal) or perifocal control only. The continuous curve reveals that both subpopulations fall to zero level in an approximate time of days. With respect to the segmented curve, it can be seen that the densities of the immature phase and the adult phase are greater by implementing a single measurement than integrating both controls. The abysmal dissimilarity occurs between the curves of subfigure IX), while the curves of subfigure X) remain almost the same for the first 3 hours and separate almost parallel. The controls u t) and u t) plotted in Figure 6 are functions of time with κ = κ =, respectively. The control u t) depends largely on the adjoint variable λ t) and the values of κ ; the control u t) depends to a large extent on the adjoint variable λ t) and the values of κ.

13 Aquatic phase Aerial phase Subpopulation Subpopulation A model for the population dynamics of the vector Aedes aegypti VII) VIII) # 6 4 # Aquatic phase Aerial phase 4 Aquatic phase Aerial phase Figure 4: The graphs represent the biological phases of A. aegypti in the same cartesian plane using a single control in each situation: VII) perifocal control, VIII) focal control. IX) X) # 6 3 # Figure 5: Each graph represents a single biological phase of A. aegypti in the same cartesian plane with combined controls continuous line) and perifocal control dotted line). 5 Conclusion In this document, we study the optimal combination of control strategies to lead to the reduction of Aedes aegypti infestation within a specific period and area. We consider a stratified model in the aquatic and aerial phases of the life cycle and its densities, including focal and perifocal controls as measures against infestation by public health agents. We apply the Pontryagin Maximum Principle to characterize the controls and derive the optimization system. The numerical simulations of the resulting optimality system showed that the implementation of combined controls focal-perifocal) is the best strategy to choose, since it reduces the number of female mosquitoes in both phases fast enough and in greater abundance, in addition, the application must remain

14 Control u t) Control u t) 34 Julián Alejandro Olarte García et al. XI) XII) Figure 6: Graphs of the control functions by applying the focal control with the perifocal control. highly effective for 5 days for certain appropriate parameter values such that equilibrium with infestation is asymptotically stable. Acknowledgements. The authors are very grateful with Grupo de Modelación Matemática en Epidemiología GMME) and its institution, Universidad del Quindío. References [] D.L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering, Academic Press, New York, 98. [] H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Vol., Wiley-Interscience, New York, 97. [3] H.M. Yang, M.L.G. Macoris, K.C. Galvani, M.T.M. Andrighetti and D.M.V. Wanderley, Assessing the effects of temperature on the population of Aedes aegypti, the vector of dengue, Epidemiology & Infection, 37 9), [4] H.S. Rodrigues, M.T.T. Monteiro and D.F. Torres, Dynamics of dengue epidemics when using optimal control, Mathematical and Computer Modelling, 5 ), [5] J.A.O García and A.M. Loaiza, Un modelo de crecimiento poblacional de Aedes aegypti con capacidad de carga logística, Revista de Matemtica: Teoría y Aplicaciones, 8), 79-3.

15 A model for the population dynamics of the vector Aedes aegypti [6] M.J. Nelson, Aedes Aegypti: Biología y Ecología, Washington, D.C: OPS, 986. [7] M. Rafikov, E. Rafikova and H.M. Yang, Optimization of the Aedes aegypti control strategies for integrated vector management, Journal of Applied Mathematics, 5 5), [8] M.A. Masud, B.N. Kim and Y. Kim, Optimal control problems of mosquito-borne disease subject to changes in feeding behavior of Aedes mosquitoes, Biosystems, 56 7), [9] R. Rodríguez, Estrategias para el control del dengue y del Aedes aegypti en las Américas, Revista Cubana de Medicina Tropical, 54 ), 89-. [] R.C. Thomé, H.M. Yang and L. Esteva, Optimal control of Aedes aegypti mosquitoes by the sterile insect technique and insecticide, Mathematical Biosciences, 3 ), [] R. Barrera, Recomendaciones para la vigilancia de Aedes aegypti, Biomédica, 36 6), [] S. Lenhart and J.T. Workman, Optimal Control Applied to Biological models, Crc Press, 7. [3] S.I. Grossman and J.J. Flores, Álgebra Lineal, McGraw-Hill,. [4] T.L. Vincent and W.J. Grantham, Nonlinear and Optimal Control Systems, John Wiley & Sons, 997. [5] W.H. Fleming and R.W. Rishel, Deterministic and stochastic Optimal Control, Springer-Verlag, New York, [6] World Health Organization, Dengue: Guidelines for Diagnosis Treatment Prevention and Control, New Edition 9, World Health Organization. [7] World Health Organization, A Global Brief on Vector-Borne Diseases, Geneva: WHO, 4. [8] W.O. Dias, E.F. Wanner and R.T. Cardoso, A multiobjective optimization approach for combating Aedes aegypti using chemical and biological alternated step-size control, Mathematical Biosciences, 69 5), Received: February, 8; Published: March, 8