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Models for an arenavirus infection in a rodent population: consequences of horizontal, vertical and sexual transmission
1.  Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 794091042, United States, United States 
2.  Department of Biological Sciences, Texas Tech University, Lubbock, TX 794091031, United States 
Machupo virus is transmitted among rodents through horizontal (direct contact), vertical (infected mother to offspring) and sexual transmission. The immune response differs among rodents infected with Machupo virus. Either rodents develop immunity and recover (immunocompetent) or they do not develop immunity and remain infected (immunotolerant). We 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 represent susceptible, immunocompetent, immunotolerant and recovered rodents, denoted as $S$, $I^t$, $I^c$ and $R$, respectively. A unique diseasefree equilibrium (DFE) is shown to exist and a basic reproduction number $\mathcal R_0 $ is computed using the next generation matrix approach. The DFE is shown to be locally asymptotically stable if $\mathcal R_0<1$ and unstable if $\mathcal R_0>1$.
Special cases of the general model are studied, where there is only one immune stage, either $I^t$ or $I^c$. In the first model, $SI^cR^c$, it is assumed that all infected rodents are immunocompetent and recover. In the second model, $SI^t$, it is assumed that all infected rodents are immunotolerant. For each of these models, the basic reproduction numbers are computed and their relationship to the basic reproduction number of the general model determined. For the $SI^t$ model, it is shown that bistability may occur, the DFE and an enzootic equilibrium, with all rodents infectious, are locally asymptotically stable for the same set of parameter values. A simplification of the $SI^t $ model yields a third model, where the sexes are not differentiated, and therefore, there is no sexual transmission. For this third simplified model, the dynamics are completely analyzed. It is shown that there exists a DFE and possibly two additional equilibria, one of which is globally asymptotically stable for any given set of parameter values; bistability does not occur. Numerical examples illustrate the dynamics of the models. The biological implications of the results and future research goals are discussed in the conclusion.
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