A mathematical model for mosquito infestation

Journal homepage http://revistas.unitru.edu.pe/index.php/SSMM SELECCIONES MATEMÁTICAS Universidad Nacional de Trujillo ISSN: 2411-1783 (Online) Vol. 06(01): 14 - 18 (2019) A mathematical model for mosquito infestation . Un modelo matemático de la infestación de mosquitos. Roxana López C.∗ Received, May. 01, 2019 Accepted, Jul. 01, 2019 DOI: http://dx.doi.org/10.17268/sel.mat.2019.01.02 Abstract We propose and analyze a mathematical model in ordinary differential equations to describe the dynamics of mosquitoes infested by bacteria. The introduction of some bacteria in mosquitoes population aims to diminish gradually the transmission of vector host-diseases. This is a good strategy of biological control. Keywords. Ordinary differential equation, bacterial infestation , basic reproduction number, local stability, Simulation. Resumen Proponemos y analizamos un modelo matemático que describe la dinámica de los mosquitos infestados por bacterias. La introducción de ciertas bacterias en la población de mosquitos apunta a disminuir gradualmente la transmisión de enfermedades de huesped-vector. Esta es una buena estrategia de control biológico. Palabras clave. Ecuación diferenciales ordinarias, infestación bacterial, número reproductivo básico, Simulación. 1. Introduction. The strategy of introduce bacteria in the mosquitoes population is very important for the biological control of vector in host-vector diseases. The introduction of bacteria in mosquitoes is called bacterial infestation, many researchers propose mathematical predictions for biological control of different host-vector diseases. These kind of differential equation models are described in ([1], [2], [4]) . We refer to the model of infested population only models, type S-I epidemic models. It is important to think that the development of models of intermediate complexity works as a conceptual bridge between simple and complex model as those which implement demographic and epidemiological structure. First, in section 2, we propose and analyze simpler S-I infestation model that differ from the others in the fact that an infested mosquito could produce a susceptible newborn and maintain sufficient simplicity to the model to be used in practical empirical work. we take into account the density dependence due competition for resources. 2. S-I infestation model. The mathematical model is considered through the infestation of a bacterium that inhibits the transmission of the virus that characterizes the diseases by indirect transmission. We divide the population of mosquitoes in aquatic and adult age and with epidemiological subdivision of infested and not infested will be taken into account, those populations are described in Table 1.1. The flow diagram of the infestation dynamics is given in Fig. 2 and the variables are detailed in Table 1.2. ∗ ID ORCID: https://orcid.org/0000-0002-7703-5784, Facultad de Ciencias Matemáticas, Universidad Nacional Mayor de San Marcos, Lima-Perú ([email protected]). . .......................................................................................................................................................................... This work is licensed under the Creative Commons Attribution-NoComercial-ShareAlike 4.0. 14 15 Lopez, R.- Selecciones Matemáticas. 06(01): 14-18 (2019) Variables JU SU JW AW Meaning mosquitoes of aquatic age not infested by the inhibitory bacteria mosquitoes in adulthood not infested by the inhibitory bacteria mosquitoes of aquatic age infested by the inhibitory bacteria mosquitoes in adulthood infested by the inhibitory bacteria TABLE 2.1 Meaning of the variables of the system (3.1) F IGURE 2.1. Flow diagram of the bacterial infestation of mosquitoes The dynamical system is given by (3.1)  dJ U     dt       dJW     dt   dAU     dt         dAW dt = βU AU + (1 − ξ)βW AW − ηU JU − µJU − mJU J , = ξβW AW + σC(A)AU AW − ηW JW − µJW − mJJW , A = ηU JU − αAU − mAAU , = ηW JW − αAW − mAAW . , The parameters of the model (3.1) are constant and are detailed in Table 5. Considerations. It is assumed that aquatic stage mosquitoes reproduce with dinamycs βi Ai where βi (i = U, W ) is the number of newborns produced by an adult mosquito per unit of time, considering that (ξ) is the probability that a infested mosquito in aquatic stage born from an adult infested mosquito . By natural death aquatic mosquitoes and adults leave their aquatic states at a rate µJ (or αA) respectively where µ is the per capita mortality rate of aquatic mosquitoes and α is the per capita death rate of adult mosquitoes. Aquatic mosquitoes migrate to the adult stage at a rate of ηJ where η is the rate of per capita emigration of aquatic mosquitoes to the adult stage. 3. Basic Results. In this section, we study some basic results of the solutions of the system (3.1) which will be very useful to use into the proof of stability and persistence results. 0 Theorem 3.1. For all JU0 , A0U , JW , A0W > 0, there exists JU , AU , JW , AW : (0, ∞) → (0, ∞) which solve 0 , AW = A0W . the system (3.1) with initial conditions JU = JU0 , AU = A0U , JW = JW 16 Lopez, R.- Selecciones Matemáticas. 06(01): 14-18 (2019) Parameters βU βW ηU Meaning adult mosquito maternal transmission rate without bacteria maternal transmission rate of adult mosquito with bacteria average time for a mosquito in aquatic stage without bacteria it becomes an adult mosquito without bacteria average time for a mosquito in aquatic stage with bacteria to become an adult mosquito with bacteria average life time of the mosquito in aquatic stage average life time of the adult mosquito interspecific competition rate among mosquitoes probability that a mosquito in aquatic stage with bacteria born from an adult mosquito with bacteria ηW 1/µ 1/α m ξ TABLE 2.2 Biological meaning of System Parameters (3.1) Theorem 3.2. All solutions of this system (3.1) are bounded. The following reproduction rates are defined   ηU βU , RU = α ηU + µ    ηW βW . RW = α ηW + µ  with biological meaning RU = average number of births of non infested mosquitoes by a typical not infested mosquito during its life , RW = average number of births of mosquitoes per each typical infested mosquito infested during its life Proposition 3.3. If RU > 1, then the system (3.1) has a single equilibrium free of infestation, (JU∗ , A∗U , 0, 0). Proposition 3.4. If RW > 1 and ξ = 1, then the system (3.1) has a single point of total infestation equilibrium ∗ (TI), (0, 0, JW , A∗W ), 4. Global Stability of the Susceptible Extinction Equilibrium. We derive a criteria in the case that the susceptible subpopulation eventually win the competition with the infected subpopulation (the most desirable result). ∗ Theorem 4.1. If RU ≤ 1, RW > 1 and ξ = 1, then the total infestation equilibrium (TI), (0, 0, JW , A∗W ), is 4 global attractor in R+ for the system (3.1). Proof. To show that the TI is a global attractor in R4+ , consider the following Lyapunov function : V (X) = JU + ̺U AU , X = (JU , AU , JW , AW ) , with ̺U > 0 will be chosen later. The derivative of V along a solution of the system (3.1) is   CJU JW CAU AW ηU − 1 JU + (βU − ̺U α)AU − − − mV N . V̇ (X) = (ηU + µ) ̺U ηU + µ J A We choose ̺U = βU . Then α V̇ (X) = (ηU + µ)(RU − 1)JU − CJU JW CAU AW − − mV N . J A 17 Lopez, R.- Selecciones Matemáticas. 06(01): 14-18 (2019) Since RU ≤ 1, then V̇ (X) ≤ 0 for X ∈ R4+ . So as to apply Lyapunov-LaSalle′ s Invariance Principle, we see that : By the boundedness of the solutions (Theorem 3.2) of the system (3.1), we obtain that V̇ (X) ≤ 0 for X ∈ Ωc = R4+ . In the other hand, E = {X ∈ R4+ : V̇ (X) = 0} = {(0, 0, JW , AW )} is an invariant set, as a consequence of the Principle, the trajectory enters M = E and all solutions tend to E as t → ∞. To prove the asymptotic behavior of the TI equilibrium of the system (3.1), we need to consider the convergence theory developed by Thieme for limit equations, described in Appendix B. We have that under the given conditions, any solution of system (3.1) is attracted to the JU AU -plane, then we can decouple the system into the second two equations, it means  σC(J)JU JW  ′  JW = βW AW + − (ηW + µ)JW − mJW J , J (4.1) υC(A)AU AW   A′ = ηW JW + − αAW − mAW A , W A for t → ∞, the corresponding limit system of the above system 4.1 is given by  ′ JW = βW AW − (ηW + µ)JW − mJW (JW + AW ) ≡ F (JW , AW ) , (4.2) A′W = ηW JW − αAW − mAW (JW + AW ) ≡ G(JW , AW ) . We define the function ρ : (0, ∞)2 → R2 by ρ(JW , AW ) = 1 , it follows JW AW  (ηW + µ) m(JW + AW ) βW − − ∂ ∂(ρF ) ∂(ρG) JW A2 AW + = ∂JW ∂AW ∂JW   ηW m(JW + AW ) α + − ∂ − JW AW JW + ∂A   W  βW m m ηW + − 2 − = − 2 − JW AW AW JW <0  for all (JW , AW ) ∈ (0, ∞)2 . Then, by the Bendixson-Dulac criterium , there are no periodic orbits in (0, ∞)2 . The solutions of system ((3.1)) are bounded and there is at least one equilibrium point, so all equilibria is isolated. Since its ω-limit is contained in (0, ∞)2 and (0, ∞)2 is simply connected, we conclude that the ω-limit set is an ∗ ∗ equilibrium. Since (JW , A∗W ) is the only equilibrium in (0, ∞)2 , we have that the ω(JW , AW ) = (JW , A∗W ) of the 2 system ((3.1)). As the ω-limit set of a bounded solution of system ((3.1)) is contained in (0, ∞) , we satisfied all the conditions of Theorem 0.2 (see Appendix B), in consequence, the ω-limit set of system consists of the equilibrium ∗ (JW , A∗W ). ∗ In consequence, the SE = (0, 0, JW , A∗W ) is a global attractor of the system (3.1). 5. Sensitivity Analysis of RW . We used LHS and PRCC methods to estimated the sensitivity of RU .    βW ηW (5.1) RW = α ηW + µW We considered the following values for the parameters βW 1/µW 1/α ηW (0.43, 0.57) (0.01, 0.04) (0.04, 0.07) (0.108, 0.123) • We made 11 simulations to obtain more sensitivity parameter (5.1) • For the graphic, the colors are different tonalities of blue and red. When the tonality of red is more intense the parameter RW will be more sensitive, is the same with the blue tonality color. 18 Lopez, R.- Selecciones Matemáticas. 06(01): 14-18 (2019) F IGURE 5.1. Final Simulation 5.1. Simulations. Interpretation. The more sensitivity parameters are α and β 1) There exists a direct proportional correlation (positive) between RW and β (P RCC ≈ 0.952). It means when we increase the value of β the value of RW will increase. 2) There exists a indirect proportional correlation (negative) between RW and α (P RCC ≈ −0.978). It means when we increase the value of α the value of RW will decrease Parameter β α η µ PRCC 0.952 -0.978 0.228 -0.920 lower bound 0.946 -0.980 0.168 -0.929 upper bound 0.958 -0.975 0.286 -0.909 p-value 0.00 0.00 0.00 0.00 6. Acknowledgements. We gratefully thank MOSTICAW Project and the FONDECYT financial aid. REFERENCES [1] Bliman, P. A. et al , Ensuring successful introduction of Wolbachia in natural populations of Aedes aegypti by means of feedback control, 2015. [2] Bliman, P. A. et al, Global stabilizing feedback law for a problem of biological control of mosquito-borne diseases, Proc. of the 54th IEEE Conference on Decision and Control 2015. [3] Johnson, K. N., Review The Impact of Wolbachia on Virus Infection in Mosquitoes, Viruses 2015. [4] Ndii, M. et al, Modeling the introduction of Wolbachia into Aedes Aegyptus mosquitoe to reduce Dengue transmission, The ANZIAM Journal 2012.