The dynamics of mathematical models for machupo viral infection in a rodent population



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Machupo virus is a zoonotic disease that is spread by wild rodents. In humans this disease is known as Bolivian hemorrhagic fever as it was first identified in an outbreak in Bolivia. Humans are exposed to this disease through urine or feces or saliva of infected animals. The mortality rate in humans is approximately 30%. Thus it is very important to study the spread of the disease in rodent populations so that it's spread to the human population can be prevented.

We begin by giving some background on Machupo viral infection in rodents. Machupo virus is transmitted horizontally, vertically, and sexually. In addition, rodents respond differently to infection depending on various conditions. Either rodents develop immunity and recover (referred to as immunocompetent) or they do not develop immunity and remain infected (referred to as immunotolerant). We use this information to formulate a general deterministic model for male and female rodents consisting of eight differential equations, four for females and four for males. The four states in the differential equation are susceptible, immunocompetent, immunotolerant and recovered, denoted as S, It, Ic and R, respectively. We compute the disease-free equilibrium (DFE) and study the dynamics for this model near the DFE. A basic reproduction number ${\mathcal R}_0 $ is computed and it is shown that the DFE is locally asymptotically stable if R0<1. The basic reproduction number shows important relationships among the various model parameters that determine whether an outbreak will occur (R0>1).

Since the general model with eight differential equations is difficult to analyze, we consider special cases of this model. We study SIt and SIcRc models. In the first model SIt all infected individuals are assumed immunotolerant and in the second model SIcRc it is assumed that all infected individuals are immunocompetent. For these models, we compute a basic reproduction number R0 and show the DFE is locally asymptotically stable if R0<1. For a simple SI model similar to the SIt, model but which is not differentiated by the sex of the rodent, the dynamics are completely understood. There exists a DFE and possibly two endemic equilibria.

Finally, we formulate stochastic differential equations for the general model and all the other special cases. For the stochastic models, the extinction state is an absorbing state. Therefore, if population sizes become very small, it is possible for complete population extinction, even though this may not be the case for the deterministic models. We illustrate some of the analytical results with numerical examples. Numerical approximations for the solution of the differential equations are graphed and compared to a sample path of the stochastic differential equations. Our future goals are to obtain better estimates for the model parameters and to use these models to predict when outbreaks in rodent populations are likely to occur.



Mathematical modeling, Machupo virus, Deterministic