A Simulation Model for the Chikungunya with Vectorial Capacity

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1 Applied Mathematical Sciences, Vol. 9, 2015, no. 140, HIKARI Ltd, A Simulation Model for the Chikungunya with Vectorial Capacity Steven Raigosa Osorio, Eliécer Aldana Bermúdez and Anibal Muñoz Loaiza Grupo de Modelación Matemática en Epidemiología (GMME) Facultad de Educación Universidad del Quindío Armenia, Quindío-Colombia Copyright c Steven Raigosa Osorio, Eliécer Aldana Bermúdez, Anibal Muñoz Loaiza. 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 We have formulated a simulation model with base in non-linear ordinary differential equations following the formalism stated by Sir Ronald Ross for the Malaria, using the therm of incidence, the vectorial capacity of the Aedes aegypti and the sinusoidal-like temperature effect in the probabilities of virus transmission to the susceptible people and to the non-infected mosquitoes. We determine and simulate the vectorial capacity C v (T ) and the epidemic threshold, Basic Reproduction Number, in terms of temperature R 0 (T ) and time R 0 (t). Moreover, we have simulated the infected population and female mosquito population carrying virus using Maple. The phase plane is obtained using previously reported data. Keywords: Vectorial capacity, Basic Reproduction Number, Sir Ronald Ross model, Simulation model, Transmission probabilities time and temperature dependent, Chikungunya 1 Introduction Chicungunya is an alpha-virus transmitted mainly through the Aedes aegypti mosquito bite. Its propagation is dengue-like, it cause intense fever that could

2 6954 Steven Raigosa Osorio et al. convert it in hemorrhagic fever, that is potentially mortal [1]. Other symptoms are headache, muscle pain, and joint pain, that could persist several months [2]. This virus it was detected for the first time in Tanzania (1952), until 2004 it was reported the infection in Africa, Asia, Europa, Indian and Pacific oceans. In 2007 the virus spread reach Italy, that infection was generated by Aedes albopictus in the Emilia-Romagna region. At the end of 2013 it has reported outbreaks in America, with cases expanding through the Caribbean region [2, 3]. It is known that A. aegypti is one of the most efficient vectors for arbovirus due to its anthropophilic origin, frequently it bites several times before completing the Oogenesis and proliferates in a narrow proximity to the humans. Some factors could influence the transmission dynamics of the virus, including environmental and climatic factors, interactions between guests, pathogens, and immune factors of the population [4]. The modeling of diseases transmitted by vectors has increased in the past years, displaying a public health problematic around the world. Actually, there exist several mathematical models applied to the transmission dynamics of the Chikungunya by Aedes aegypti [5, 6, 7, 8, 9, 10], as well as some papers treating the dynamics of the population growth of the mosquitoes, including different factors. The epidemiological models are generally described with dynamic systems, that help us to describe the connection between different epidemiological variables. The main goal of these models is trying to describe a system as real as possible. One of the most important aspects to take into account in mathematical modeling, infection process simulations, and vector transmitted pathologies is the weather [11, 12]. Particularly, variations of the temperature [11, 12, 13, 14, 15] influence the development of the life cycle of the vectors and it could imply epidemic outbreaks with high incidence rate. The entomological parameters related to these models and with the temperature-dependent vectorial capacity C v (T ) [13, 14, 16], that describes the ability to propagate the disease between the host, viruses and vector [13, 14, 16, 17]. On the other hand, the basic reproduction number R 0 determines the average number of secondary cases generated by an infected people in a susceptible population [18].

3 A simulation model for the Chikungunya with vectorial capacity The Model We have formulated an epidemiological model for the transmission dynamics of the Chikungunya by Aedes aegypti mosquitoes, using as base ordinary nonlinear differential equations that read the dynamics following the work by S.R. Ross [17]. We have introduced the time-dependent vectorial capacity in the incidence of vector and people. The dynamic system equations are: with φ(t ) = v(0) = v 0. du dt dv dt ɛ C ασ(t ) v(t ), ψ = = φ(t )(1 u)v θu (1) = ψ(t )(1 v)u ɛv (2) ɛ C αβ(t )m v(t ) and initial conditions u(0) = u 0, The model have the following variables and parameters: u = u(t): is the fraction of infected people by Chikungunya, v = v(t): is the fraction of Chikungunya carriers, (1 u): the fraction of susceptible people, (1 v): fraction of non-infected mosquitoes, M: total population of female mosquitoes, N: total people population at time t, α: it is the bites number by person each day, β(t ): is the probability of dengue transmission to a susceptible person as function of temperature [11, 13], σ(t ): is the probability that a mosquito acquires dengue virus, while viremic bites a person, as function of temperature [11, 13], m: is the mosquito number by person, θ: is the infected people recovery rate, ɛ: is the death rate of the mosquitoes carrying the infection due to environmental factors, and C v (T ) is the vectorial capacity. where, C v (T ) = α2 β(t )σ(t )m ɛ β(t ) = T (T ) T (4) this virus transmission probability depends on the temperature that increases when 12.4 o C < T < 28 o C, decreases for T > 28 o C, and it is zero if T > 32.5 o C [11, 13]. And the virus transmission probability from people to mosquito is lineal for the interval 12.4 o C < T < 32.5 o C [11, 13]. (3) σ(t ) = T (5) We consider a temperature function depending on the sinusoidal-like time, T (t) = ϱ + ξ sin 2π t, with ϱ the annual average temperature and ξ the temperature variation amplitude [15]. R 0 (T ) is defined as function of T using 365 equations (1-5). R 0 (T ) = α2 β(t )σ(t )m (6) ɛθ

4 6956 Steven Raigosa Osorio et al. or it is expressed in terms of vectorial capacity R 0 (T ) = C v(t ) θ Moreover, the vectorial capacity can be expressed in terms of the extrinsic incubation period n and the survival rate of mosquitoes p [8]. C v (T ) = mα2 β(t )σ(t )p n ln p, C vr (T ) = α2 β(t )σ(t )e ɛn The relative vectorial capacity is suitable to compare the epidemic potential in space and time [13]. A high C vr (T ) value indicates high risk of epidemic. Results show that temperature is an environmental variable that promotes the presence of vectors [13]. n = 4 + e T provides an estimation for the extrinsic incubation period using some experimental findings in a range of temperature 12 o C < T < 35 o C [16]. 3 Results and conclusiones The graphics of the functions (3)-(8) are obtained using the Maple software, with previously reported data from [8, 11, 16]. ɛ (7) (8) Figura 1.Transmission probability with parameters α =1.45, ɛ = and m =0.1. The first graphic shows the behavior of the virus transmission probabilities to the people and mosquitoes in a range of temperature (12.4 C, 32.5 C), according with the fitted functions for the Aedes aegypti mosquito [11, 16]. We have observed for 28 C that the maximum transmission probability is The virus transmission probability from people to mosquitoes increases in a linear way with a maximum value of 1 for 26 C. Here, the probability reaches his higher value, indicating that this function fits to a real scenario.

5 A simulation model for the Chikungunya with vectorial capacity 6957 Figura 2. Vectorial capacity C v (T ) and C v (t), α =1.45, ɛ = and m =0.1. Considering a sinusoidal-like function (6) for annual average temperature C and a variation amplitude 10 C, the transmission probability to mosquitoes shows two stationary picks in an approximate period of 270 days. While, the transmission probability to humans shows only one peak in the same period of time (270 days). Figure 2 shows that the vectorial capacity increases in a range of temperature from (12.4 C to 28 C). On the other hand, it decreases from 28 C and higher temperatures. The vectorial capacity evolves in time as the virus transmission to the mosquitoes does it. Figura 3.Epidemic threshold R 0 (T ) and R 0 (t) with θ = , α =1.45, ɛ = and m =0.1. According with (8), the epidemic threshold R 0 is inversely proportional to the infected-people recovery rate with respect to the vectorial capacity. With variations in the speed of growth, as seen in Figure 3.

6 6958 Steven Raigosa Osorio et al. Figura 4.Phase plane simulation of the population with θ = , α =1.45, ɛ =0.0595, m =0.1, ϱ =22.45 and ξ =10. Figure 4 shows the time evolution of the infected-people with the Chikungunya virus. This one stabilizes quickly in a high value, while the carrier mosquitoes stabilizes periodically with tiny picks around 25%. The phase plane shows that the initial trajectories for different initial populations tend to achieve an equilibrium point. We have observed a great epidemiological impact of Chikungunya in the proposed model with the described functions solved using entomological and geographical parameters. Acknowledgements. AML thanks Grupo de Modelación Matemática en Epidemiología (GMME), Facultad de Educación, Universidad del Quindío, Colombia; and also thanks M. E. Dalia Marcela Muñoz Pizza and M. C. J. Guerrero-Sánchez (IFUAP-BUAP). References [1] Laith Yakob, Archie C. A. Clements, A Mathematical Model of Chikungunya Dynamics and Control: The Major Epidemic on Reunion Island, PLoS ONE, 8 (2013), e [2] Miguel Lugones Botell and Marieta Ramirez Bermudez, Virus Chicungunya, Revista Cubana De Medicina General Integral, 30 (2014). [3] CDC, Chikungunya Virus, Department of Health and Human Services, [citado Jun 2014], Atlanta, Georgia, US, Available at

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