Applied Mathematical Sciences, Vol. 10, 2016, no. 7, 345-355 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.510662 A Mathematical Model for Transmission of Dengue Luis Eduardo López Departamento de Ciencias Básicas Institución Universitaria CESMAG, Pasto, Colombia and Universidad Nacional de Colombia, Manizales, Colombia Anibal Muñoz Loaiza Grupo de Modelación Matemática en Epidemiología GMME Facultad de Educación Universidad del Quindío, Armenia, Colombia Gerard Olivar Tost Departamento de Matemáticas y Estadísitica Facultad de Ciencias Exactas y Naturales Universidad Nacional de Colombia, Manizales, Colombia Copyright c 2015 Luis Eduardo López, Anibal Muñoz Loaiza and Gerard Olivar Tost. 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 this paper we propose a mathematical model based on a non-linear system of ordinary differential equations that represent the dynamic of transmission of dengue. It is done a complete stability analysis of the model proposed about the solution of equilibrium, on the base of population growth number of the Aedes aegypti mosquito and the basic reproductive number. Keywords: Mathematical Model, Basic Reproductive Number, Population
346 Luis Eduardo López et al. Growth Number, Dengue, Aedes aegypti 1 Introduction Dengue is a viral disease produced by the virus (DENV) that is transmitted to humans through the bite of the Aedes aegypti mosquito [3]. The transmission cycle starts when an infected Aedes mosquito bites a healthy person, as a result, this human host may infect a susceptible mosquito after it has taken a blood meal from that person. The life cycle of the Aedes aegypti has two phases: the aquatic one (immature phase) and the aerial one (mature phase). In the first one, there are three stages: the egg, the larva (with four evolving phases) and the pupae; while in the second phase the mosquito becomes an adult. The dengue virus is transmitted to humans only by female mosquitoes, since they bite with the target of maturating their eggs and collecting sources of alternate energies. On the other hand, the male mosquitoes feed themselves from plant s nectars [3]. 2 The model From the model proposed, we assumed that human population is constant, that is to say, the growth rate is equal to death rate. The variables and parameters that are used for the model proposed are described in the Table 1 and their transmission dynamics in the Figure 1. The proposed model combines three uncoupled systems as follows. The system (1) represents the logistic growth of the breeding sites. ( ċ i = γ i c i 1 c ) i, i = 1, 2, 3. (1) κ i The system (2) represents the mosquito population growth Aedes aegypti considering its evolving phases. ẋ ẏ ż = δz ɛx = z ( ɛy ) z = φρy 1 ai c i (δ + + ν)z +b (2)
A mathematical model for transmission of dengue 347 Variable/ Parameter Description S(t) Average number of susceptible humans in time t I(t) Average number of infected humans in time t R(t) Average number of recovered humans in time t x(t) Average number of male mosquitoes in time t y(t) Average number of female mosquitoes in time t z(t) Average number of immature mosquitoes in time t c 1 (t) Average number of breeding sites type water tanks in time t c 2 (t) Average number of breeding sites type tire in time t c 3 (t) Average number of breeding sites type waterways in time t N Constant size of human population µ Growth and death rate in human population β Probability of transmission of the virus from mosquito to human ψ Probability of transmission of the virus from human to mosquito θ Rate of recovery ɛ Rate of mosquito s death in mature state ν Rate of mosquito s death in immature state δ Rate of development from immature to male adult Rate of development from immature to female adult φ Probability that a female crosses with a male ρ Average number of fertilized eggs that a female oviposits a 1 Average number of immature mosquitoes in water tanks a 2 Average number of immature mosquitoes in tires a 3 Average number of immature mosquitoes in waterways b Average number of immature mosquitoes formed in other breeding sites γ 1 Rate of growth intrinsic of breeding sites type water tanks γ 2 Rate of growth intrinsic of breeding sites type tires γ 3 Rate of growth intrinsic of breeding sites type waterways κ 1 Maximum number of breeding sites type water tanks in the environment κ 2 Maximum number of breeding sites type tires in the environment Maximum number of breeding sites type waterways in the environment κ 3 Table 1: Variables and Parameters of the model The system (3) represent the virus transmission in human population. Ṡ = µn βψ I S µs N x+y I = βψ I y S (θ + µ)i N x+y Ṙ = θi µr y (3)
348 Luis Eduardo López et al. µn S βψ I N y x+y S I θi R µs µi µr x y δz z ( ) z φρy 1 ai c i +b z ɛx ɛy νz γ i c i (1 c i κ i ) c i 3 Results Figure 1: Transmission of the dengue virus The solution to the system (1) with initial conditional c i (0) = c i0, is of the form κ i c i0 c i (t) =, i = 1, 2, 3. (κ i c i0 )e γ it + c i0 Since γ i > 0, lim t + c i(t) = κ i Therefore, the system (1) has an equilibrium point asymptotically stable with coordinates (κ 1, κ 2, κ 3 ). Thus, ( lim ai c i (t) + b) = a i κ i + b t + doing K = a i κ i + b and replaced in the third equation of the system (2), we obtain the system: ẋ ẏ ż = δz ɛx = z ɛy = φρy ( ) (4) 1 z K (δ + + ν)z
A mathematical model for transmission of dengue 349 Algebraic calculations show that the equilibrium points of the system (4) are: ( ) δk(h 1) K(h 1) K(h 1) Q 1 = (0, 0, 0) and Q 2 =,,, ɛh ɛh h φρ ɛ(δ++ν) where h = aegypti mosquito. Theorem 3.1 (Q 2 local stability) represents the population growth number of the Aedes If h > 1, Q 2 is locally asymptotically stable. Proof 3.1 The Jacobian matrix associated to the linearisation of the system (4) is ɛ 0 δ J(x, y, z) = 0 ɛ 0 φρ ( ) 1 z K φρ y (δ + + ν) K When evaluating the matrix J in the equilibrium point Q 2, we obtain: ɛ 0 δ J(Q 2 ) = 0 ɛ 0 (δ + + ν)h her characteristic polynomial can be written as: φρ h p(λ) = (λ + ɛ) ( λ 2 + (ɛ + (δ + + ν)h)λ + ɛ(δ + + ν)(h 1) ) In this case, if h > 1, the real part of all eigenvalues of the matrix J(Q 2 ) is negative, since they reach the established condition in Routh-Hurwitz in [4]. Therefore, the equilibrium point Q 2 is locally asymptotically stable. This means that if h > 1, the population of the mosquito grows exponentially to equilibrium: x = δk(h 1), y = ɛh K(h 1) ɛh and z = K(h 1) h respectively. Substituting these results in the system (1) we obtain, Ṡ = µn βψ I S µs N δ+ I = βψ I S (θ + µ)i N δ+ Ṙ = θi µr doing, p = S N, q = I N and r = R N,
350 Luis Eduardo López et al. we obtain the system { ṗ = µ(1 p) βψ defined in the set of biological interest q δ+ pq = βψ pq (θ + µ)q (5) δ+ Ω = { (p, q) R 2 : p 0, q 0, p + q 1 } The equilibrium points of the system (5) are ( 1 E 1 = (1, 0) and E 2 =, µ(r ) 0 1) R 0 R 0 (θ + µ) βψ where R 0 = represents the basic reproductive number of the disease. (δ+)(θ+µ) This represents the expected number of secondary cases produced in a completely susceptible population, by a infected mosquito. Theorem 3.2 (E 1 and E 2 local stability) If R 0 < 1, E 1 is locally asymptotically stable in Ω. If R 0 > 1, E 2 is locally asymptotically stable in Ω. Proof 3.2 The Jacobian matrix associated to the linearisation of the system (5) is ( µ βψ J(p, q) = q βψ p ) δ+ δ+ βψ p (θ + µ) δ+ q βψ δ+ When evaluating the matrix J in the equilibrium point E 1, we obtain: ( ) µ βψ J(E 1 ) = δ+ 0 (θ + µ)(r 0 1) her characteristic polynomial can be written as: p(λ) = (λ + µ) (λ + (θ + µ)(1 R 0 )) If R 0 < 1, the real part of all eigenvalues of the matrix J(E 1 ) is negative, therefore, the equilibrium point E 1 is locally asymptotically stable. Now, evaluating the matrix J in the equilibrium point E 2, we obtain: ( ) µr0 (θ + µ) J(E 2 ) = µ(r 0 1) 0 her characteristic polynomial can be written as: p(λ) = λ 2 + µr 0 λ + µ(θ + µ)(r 0 1)
A mathematical model for transmission of dengue 351 In this case, if R 0 > 1, the real part of all eigenvalues of the matrix J(E 2 ) is negative, since they reach the established condition in Routh-Hurwitz in [4]. Therefore, the equilibrium point E 2 is locally asymptotically stable. Proposition 3.1 The system (5) does not have periodic (solutions) orbits in Ω. Proof 3.1 Being f(p, q) = µ(1 p) βψ pq and g(p, q) = βψ pq (θ + µ)q δ + δ + two multi variable functions that represent the right side of the system (5) and l(p, q) = 1 a differentiable continuous function for everything q 0 in Ω. (θ+µ)q Then, applying Dulac s criterion ([1],Theorem 4.8, p. 153) we obtain: p (l(p, q)f(p, q)) + q (l(p, q)g(p, q)) = ( ) µ(1 p) p (θ + µ)q R 0p + q (R 0p 1) µ = (θ + µ)q R 0 this is negative for all (p, q) Ω with q 0. Thus, the system (5) does not have periodic solutions. Now, if q = 0, the system (5) is written as: { ṗ = µ(1 p) q = 0 with (p 0, 0) Ω. Thus, p(t) = 1 (1 p 0 )e µt q(t) = 0 which is not a periodic solution. Because of the system (5) does not have periodic solution in Ω, we make a study of the global stability of the equilibrium points. Theorem 3.3 (E 1 global stability) If R 0 1, the equilibrium point E 1 is globally stable. Proof 3.3 Being V (p, q) = V (E 1 ) = 0 V (p, q) 0 for all (p, q) Ω E 1 q θ+µ a real value function of class C1 (Ω) such that:
352 Luis Eduardo López et al. Deriving V with respect to t, we obtain V = q θ + µ = (R 0p 1)q as R 0 1 and 0 p 1, so R 0 p 1 and therefore, V 0. In this way, according [6] (Theorem 1.1.2, p. 12) V (p, q) represents a Lyapunov s function for the equilibrium E 1 with R 0 1. Thus, E 1 is globally stable under this condition. Now, being S = { (p, q) Ω : V } (p, q) = 0 = {(p, q) Ω : q = 0} the set where the orbital derived from V is zero. So: If R 0 < 1, V = 0 when q = 0. If R 0 = 1, V = 0 when p = 1 or q = 0. Then, in both cases the most invariant set of S is M = {E 1 } In this way, for the LaSalle s theorem ([2], Theorem 4.4, p. 128), the equilibrium E 1 is globally asymptotically stable when R 0 1. Figure 2 shows the vectorial field of (5) for R 0 0.89, where it is observed that all the solutions with initial condition in Ω approximated to the equilibrium point E 1. Figure 2: Vectorial field of (5) with R 0 1
A mathematical model for transmission of dengue 353 Theorem 3.4 (E 2 global stability) If R 0 > 1, the equilibrium point E 2 is globally stable. Proof 3.4 Being E 2 = (p, q ) the equilibrium point of the system (5), with and, p = 1 R 0 and q = µ(r 0 1) R 0 (θ + µ) W (p, q) = (p p ) + (q q ) p ln p p q ln q q a real value function of class C 1 (Ω). So: as so ṗ p = µ(1 p) p Ẇ = ṗ + q p p ṗ q q q = ṗ p (p p ) + q q (q q ) βψ δ + q and q q = βψ p (θ + µ) δ + ( µ Ẇ = p µ βψ ) ( δ + q (p p ) + βψ ) p (θ + µ) (q q ) δ + but therefore, µ = βψ δ + q µ p and (θ + µ) = βψ δ + p Ẇ = µ (p p ) 2 p p 0 Thus, according [5] W (p, q) represents a Lyapunov s function for the equilibrium point E 2. In this way, if R 0 > 1, E 2 is globally stable in Ω. Now, being T = { } (p, q) Ω : Ẇ (p, q) = 0 = {(p, q) Ω : p = p } the set where the W orbital derived is zero. So, at the moment of replacing this condition in the system (5), we obtain, { ṗ = µ (R 0 1) R 0 q = 0 (θ + µ)q
354 Luis Eduardo López et al. Thus, the most invariant set of T is M = {E 2 } In this way, for the LaSalle s theorem ([2], Theorem 4.4, p. 128), the equilibrium E 2 is globally asymptotically stable when R 0 > 1. Figure 3 shows the vectorial field of (5) for R 0 2.75, where it is observed that all the solutions with initial condition in Ω approximated to the equilibrium point E 2 (0.36,0.02) when the time passes. This means that the disease remains in the environment. Figure 3: Campo vectorial de (5) con R 0 > 1 Acknowledgements. The authors thank to their faculties and institutions and to investigation groups: GMME, SIGMA and ABCDynamics. References [1] F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New York, 2012. http://dx.doi.org/10.1007/978-1-4614-1686-9 [2] H. Khalil, Nonlinear Systems, Prentice Hall, New York, 2002. [3] Ministerio de Salud, Presidencia de la Nación, Directrices Para la Prevención y Control de Aedes aegypti, Tercera Edición, Argentina, 2013. [4] J. Murray, Mathematical Biology I: An Introduction, Springer-Verlag, New York, 2001.
A mathematical model for transmission of dengue 355 [5] D. Vargas, Constructions of Lyapunov Functions for Classics SIS, SIR and SIRS, Unidad Académica de Matemáticas, 2009. [6] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990. http://dx.doi.org/10.1007/978-1-4757-4067-7 Received: November 4, 2015; Published: February 4, 2016