Abstract
In this paper we build and analyze two stochastic epidemic models with death. The model assumes that only susceptible individuals (S) can get infected (I) and may die from this disease or a recovered individual becomes susceptible again (SIS model) or completely immune (SIR Model) for the remainder of the study period. Moreover, it is assumed there are no births, deaths, immigration or emigration during the study period; the community is said to be closed. In these infection disease models, there are two central questions: first it is the disease extinction or not and the second studies the time elapsed for such extinction, this paper will deal with this second question because the first answer corresponds to the basic reproduction number defined in the bibliography. More concretely, we study the mean-extinction of the diseases and the technique used here first builds the backward Kolmogorov differential equation and then solves it numerically using finite element method with FreeFem++. Our contribution and novelty are the following: however the reproduction number effectively concludes the extinction or not of the disease, it does not help to know its extinction times because example with the same reproduction numbers has very different time. Moreover, the SIS model is slower, a result that is not surprising, but this difference seems to increase in the stochastic models with respect to the deterministic ones, it is reasonable to assume some uncertainly.
Highlights
The ordinary differential equations in epidemic models has been a well-known topic for some time, there exist classic book like ([2], Chap. 10) or [3] and more recent monographs: [4] and [5]
It is assumed there are no births, deaths, immigration or emigration during the study period; the community is said to be closed. In these infection disease models, there are two central questions: first it is the disease extinction or not and the second studies the time elapsed for such extinction, this paper will deal with this second question because the first answer corresponds to the basic reproduction number defined in the bibliography
All these deterministic models serve as a framework for formulating analogous stochastic models and as a source of comparison with the stochastic models
Summary
The ordinary differential equations in epidemic models has been a well-known topic for some time, there exist classic book like ([2], Chap. 10) or [3] and more recent monographs: [4] and [5]. The other one is explained in [11] [12] and it is used in this paper This technique begins by assuming different probabilities of the changes and calculating means and covariance matrix to obtain a stochastic differential system. These comments would explain the results in [13]. As already commented in the abstract, we will solved numerically the backward Kolmogorov equation using finite element method with FreeFem++. These authors et al have used this technique en several paper: [13] [15] [16] [17] or [18] with very hopeful results to spread more complex problems. The codes for the numerical tests are available on request
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
