Section: New Results

Analysis of Dengue Fever SIR Model with time-varying parameters

Participants : Stefanella Boatto [Univ Feder Rio de Janeiro] , Catherine Bonnet, Frédéric Mazenc, Le Ha Vy Nguyen.

Dengue fever is an infectious viral disease occurring in humans that is prevalent in parts of Central and South America, Africa, India and South-east Asia and which causes 390 millions of infections worldwide. We continued this year our study on modeling of dengue epidemics.

We have first considered a SIR model with birth and death terms and time-varying infectivity parameter $\beta \left(t\right)$. In the particular case of a sinusoidal parameter, we showed that the average Basic Reproduction Number $R_o$, introduced in [Bacaër $\&$ Guernaoui, 2006], is not the only relevant parameter and we emphasized the rôle played by the initial phase, the amplitude and the period. For a (general) periodic infectivity parameter $\beta \left(t\right)$ a periodic orbit exists, as already proved in [Katriel, 2014]. In the case of a slowly varying $\beta \left(t\right)$ an approximation of such a solution is given, which is shown to be asymptotically stable under an extra assumption on the slowness of $\beta \left(t\right)$. For a non necessarily periodic $\beta \left(t\right)$, all the trajectories of the system are proved to be attracted into a tubular region around a suitable curve, which is then an approximation of the underlying attractor. Numerical simulations are given [68].

In other to study the effects of urban human mobility on Dengue epidemics, we have considered a SIR-network model (still with birth and death rates). The same model without these rates was introduced in [72].

In the case of constant infection rates, we first examine networks of two nodes. For arbitrary network topologies, some general properties of the equilibrium points are obtained. Then for several specific topologies, we derive explicit expressions of multiple equilibrium points and characterize their stability properties. We extend the study to networks with an arbitrary number of nodes and obtain sufficient conditions for global asymptotic stability of the disease-free equilibrium point.

In the case of time-varying infection rates and networks of arbitrary number of nodes, we introduce a specific topology which leads to a simplification of the network: the dynamics of the total population is described by the classical SIR model. This fact, together with the results of the team on the SIR model, allows a complete characterization of the stability properties of the system, especially the approximation of the epidemic attractor.