Analysis of a Mathematical Model for Dengue - Chikungunya

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1 Applied Mathematical Sciences, Vol. 11, 217, no. 59, HIKARI Ltd, Analysis of a Mathematical Model for Dengue - Chikungunya Oscar A. Manrique A., Dalia M. Muñoz P., Anibal Muñoz L., Mauricio Ropero P., Steven Raigosa O., Juan C. Jamboos T. and Francisco Betancourt B. Grupo de Modelación Matemática en Epidemiología (GMME) Maestría en Biomatemáticas, Facultad de Educación Facultad de Ciencias Básicas y Tecnologías Universidad del Quindío, Quindío, Colombia Copyright c 217 Óscar A. Manrique A. 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 A dynamical system of non-linear ordinary differential equations which describes the Dengue-Chikungunya infectious process is reported. In this model it is considered the presence of two viruses transmitted by the same vector. Taking into account this fact, we have determined the epidemic threshold, basic reproduction number, using the next generation matrix. The simulations of the differential equations system are carried out with the MATLAB software. Keywords: Mathematical model, Dengue-Chikungunya, Basic Reproduction Number, Next generation matrix, Infectious process 1 Introduction Chikungunya virus is an arbovirus which belongs to the Alfavirus gender of the Togaviridae family. A vector responsible of its transmission is Aedes aegypti, the same species which transmits Dengue fever [7, 2]. Dengue is one of the most important pathogens which affects tropical and substropical regions. It is estimated that annually occur more than 5 million

2 2934 Oscar A. Manrique A. et al. infections, 5, hospitalizations, and 2, deaths because of it. than 1 countries have reported Dengue fever outbreaks [6]. More In Colombia restlessness have appeared due to a possible outbreak, which could be dangerous because till now do not exist any vaccine with approved commercialization. In this way, a safe, effective, and affordable vaccine would represent an advance in its control [1]. Also, it may be a tool to achieve the World Health Organization (WHO) objective which is the reduction of the Dengue morbidity at less than 25% and its mortality at least in a 5% for 22 [6]. Moreover, a vaccine against Chikungunya symptoms does not exist. The National Health Institute in its epidemiological bulletin of the 39 week, 216 published the following charts showing the number of Dengue and Chikungunya positive cases in the country and making a comparison with data of 215 [4]. Figure 1: Dengue and Chykungunya cases ( )[4]. In this paper, it is set a differential equations model to describe the dynamics of the infectious process of the mosquito with the human population. Coexistence between both pathologies (Dengue and Chikungunya) is considered. The epidemic threshold is described using the next generation method. Finally, the sensitivity analysis of the basic reproduction number (R ) is carried out. 2 Model It is set and studied a model for the transmission dynamics of Dengue - Chikungunya viruses which are caused by Aedes aegypti vector, considering susceptible populations, humans infected, and carrier and non-carrier mosquitoes. This model is governed by the following assumptions: the susceptible human population and non-carrier mosquitoes with a constant growth rate. It is considered only one Dengue serotype and vertical transmission is not considered. Also, co - infection is not considered in the mosquito and the total population of mosquitoes is variable. The total people population is variable as well.

3 Analysis of a mathematical model for Dengue-Chikungunya 2935 This model contemplates the following variables: x 1 : average number of susceptible persons to both kinds of virus, x 2 : average number of Chikungunya infected persons, x 3 : average number of persons infected by Dengue, x 4 : average number of persons recovered from chikungunya virus and susceptible to contract Dengue, x 5 : average number of recovered persons from dengue virus and susceptible to acquire chikungunya virus, x 6 : average number of recovered persons from chikungunya virus which acquire dengue, x 7 : average number of recovered persons from dengue virus which contract chikungunya, x 8 : average number of persons which are free of both infections, y 1 : average number of non-carrier mosquitoes, y 2 : average number of carrier mosquitoes, and N : total human population. Also, the parameters of the model are: : a constant flux of susceptible population, ρ : constant flux of non-carrier mosquitoes, µ : rate of natural death of human population, ɛ : mortality rate of mosquitoes, σ : transmission probability of Dengue or Chikungunya virus from infected people to non - carrier mosquitoes, β 1 : transmission probability of Dengue from carrier mosquitoes to susceptible persons, β 2 : transmission probability of Chikungunya from carrier mosquitoes to susceptible persons, α 1 : recovery rate of infected persons by Dengue, α 2 : recovery rate of infected persons by Chikungunya, γ recovery rate of persons infected by Chikungunya which previously recovered of Dengue and later get infected by Chikungunya, θ : recovery rate of persons infected by Dengue which previously recovered from Chikungunya and later get infected by Dengue, f : fraction of persons which get infected by Dengue, g : fraction of persons which get infected by Chikungunya after recovered by Dengue, h : fraction of persons which get infected by Dengue after recovered by Chikungunya, 1 (f + g + h) : fraction of persons infected by Chikungunya virus, and Γ = σ x 3 y N 1 + σ x 2 y N 1 + σ x 7 y N 1 + σ x 6 y N 1. So, the dynamic system which plays the infectious process is, dx 1 dx 2 dx 3 dx 4 dx 5 y 2 = β 1 f y 1 + y 2 x y 2 1 β 2 a y 1 + y 2 x 1 µx 1 y 2 = β 2 (1 (f + g + h)) y 1 + y 2 x 1 (α 2 + µ)x 2 y 2 = β 1 f y 1 + y 2 x 1 (α 1 + µ)x 3 y 2 = α 2 x 2 β 1 h y 1 + y 2 x 4 µx 4 y 2 = α 1 x 3 β 2 g y 1 + y 2 x 5 µx 5

4 2936 Oscar A. Manrique A. et al. dx 6 dx 7 dx 8 dy 1 dy 2 y 2 = β 1 h y 1 + y 2 x 4 (θ + µ)x 6 y 2 = β 2 g y 1 + y 2 x 5 (γ + µ)x 7 = γx 7 + θx 6 µx 8 = ρ Γ ɛy 1 = Γ ɛy 2 where,, µ, ɛ, ρ, γ, α 1, α 2, θ >, < β 1, β 2, h, g, f < 1, a = 1 (f + g + h) and their initial conditions a x 1 () = x 1, x 2 () = x 2, x 3 () = x 3, x 4 () = x 4, x 5 () = x 5, x 6 () = x 6, x 7 () = x 7, x 8 () = x 8, y 1 () = y 1, y 2 () = y 2. The epidemiological sense region is defined as: Ω = { (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, y 1, y 2 ) R 1 + : < N µ, M ρ } ɛ The flux diagram including human populations, infected and free of infection, and carrier and non-carrier mosquitoes is depicted in Figure 2. y β 1f 1 x y 1 +y 1 2 x 3 α 1x 3 x 5 y β 2g 1 x y 1 +y 5 2 x 7 γx 7 x 1 y β 2(1 f g h) 1 x y 1 +y 1 2 µx 1 µx 3 µx 5 µx 7 x 8 µx 8 x 2 α 2x 2 x4 β 2h y 2 y 1 +y 2 x 4 x 6 µx 2 µx 4 µx 6 θx 6 ρ y 1 Γ y 2 ɛy 1 ɛy 2 Figure 2: Flux diagram of the dynamics.

5 Analysis of a mathematical model for Dengue-Chikungunya Simulations The differential equations system were carried out using the MATLAB software. The values of the parameters are taken from the literatura [8, 9, 3], and including the following initial conditions x 1 () = 1, x 2 () = 3, x 3 () = 2, x 4 () =, x 5 () =, x 6 () =, x 8 () =, x 7 () =, x 8 () =, y 1 () = 1 y y 2 () = 1. The following table depicts the values of each parameter included in the model. Parameter ρ µ ɛ σ β 1 β 2 Value Parameter α 1 α 2 γ θ f g h Value Table 1: Parameters of the model x x x x x 5 6 x x x 1 5 Figure 3: Susceptible populations behavior: x 1, x 4, x 5 and x 8. In Figure 3 is depicted the behavior of x 4, x 5, and x 1. This figure shows that the susceptible populations have a similar behavior, such populations tend to stabilize around 1 5 but with different maximum values. Figure 4 shows the behavior of the infected people by Chikungunya, in this figure x 2 achieves a maximum value of 365 persons in approximately 18 days, and descend in a precipitate way between 2 and 4 days. Then, the persons recovered by Dengue infection which get infected by Chikungunya present a

6 2938 Oscar A. Manrique A. et al. similar behavior, but those ones achieving a maximum value of 16 and stabilizing at arount 16 days. From Figure 5 we are able to see the behavior x x Figure 4: Behavior of populations infected by Chikungunya. of the infected populations by Dengue, having those ones a critical number of infected persons at around 3 days. Finally, Figure 6 depicts the behavior of the mosquitoes at time, two different behaviors are observed, one increases in time (y 1 ) and the other decreases (y 2 ) y 1 5 y Figure 6: Mosquitoes behavior.

7 Analysis of a mathematical model for Dengue-Chikungunya x x Figure 5: Behavior of the populations infected by Dengue virus. 4 Results To obtain the basic reproduction number we have considered the following infectious stages, namely, x 3, x 7, x 2, x 6, and y 2. Following the next generation matrix approach we have that the R is defined as: σβ 1 f R = ɛ(α 1 + µ) + β 2σ(1 (f + g + h)) ɛ(α 2 + µ) (1) The additive and multiplicative effects of R indicate that the vector could transmit Dengue or Chikungunya to the susceptible population. The terms β 1f ɛ and β 2(1 (f+g+h)) indicate incidence of new cases of Dengue or Chikungunya ɛ to the susceptible populations, respectively, during the lifetime of the vector. σ The α 1 expression corresponds to the incidence of Dengue to the non-carrier +µ σ mosquitoes. Moreover, α 2 represents the incidence of Chikungunya to the +µ non-carrier mosquitoes.

8 294 Oscar A. Manrique A. et al. References [1] Carmen Acosta-Bas, I. Gomez-Cordero, Biología y métodos diagnósticos del dengue, Rev. Biomed., 16 (25), [2] T. Carrada, L. Vázquez, I. López, La ecología del dengue y el Aedes aegypti. Tercera parte, Salud Pública México, 26 (1984), no. 3, [3] Departamento Administrativo Nacional de Estadística, Colombia, Proyecciones anuales de población por sexo y edad Estudios Censales No. 4. [4] Instituto Nacional de Salud, Boletín epidemiológico, semana 39 de [5] O.A. Montesinos-Lopez, C.M. Hernández-Suarez, Modelos matemáticos para enfermedades infecciosas, Salud Pública México, 49 (27), no. 3, [6] Organizacion Mundial de la Salud, Preguntas y respuestas sobre las vacunas contra el dengue:estudio de fase III de la vacuna CYD-TDV en América Latina, (214). [7] B.N. Restrepo Jaramillo, Chikungunya virus infection, CES Med., 28 (214), no. 2, [8] L.D. Restrepo-Alape, H.D. Toro-Zapata, & A. Muñoz-Loaiza, Modelo matemático para el control químico con resistencia del Aedes aegypti (Diptera: Culicidae), Revista de Salud Pública, 12 (21), no. 6, [9] N. Surapol, T. Walaipun, T. Ming, Control of the Transmission of Chikungunya Fever Epidemic Through the use of Adulticide, American Journal of Applied Sciences, 8 (211), no. 6, Received: February 9, 217; Published: December 4, 217