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We describe a mathematical model we have recently developed to address some features of the interaction of deer
mice infected with Hantavirus in terms of simple differential equations involving the mice population. As one of the
consequences of our calculations we provide a possible mathematical answer to the question of what may be
responsible for the observed “refugia” in the North American Southwest. Specifically, we show how environmental
factors could lead to the extinction of the infection in localized areas and its persistence in other localized areas from
which, under favorable conditions it can spread again.
To describe population growth, bM represents births of mice. Given that infected mice are made, not
born, and that all mice are born susceptible, bM mice are born at a rate proportional to the total density:
all mice contribute equally to the procreation [3]. To describe deaths, MS I
c − ,
represent deaths for
natural reasons, at a rate proportional to the corresponding density. To include effects of competition for
shared resources, the terms −MS,IM / K represent a limitation process in the population growth. Each
one of these competition terms is proportional to the probability of an encounter of a pair formed by one
mouse of the corresponding class, susceptible or infected, and one mouse of any class. The reason for this
is that every mouse, either susceptible or infected, has to compete with the whole population. The
“carrying capacity” K , in this process, characterizes in a simplified way the capacity of the medium to
maintain a population of mice. Higher values of carrying capacity represent higher availability of water,
food, shelter and other resources that mice can use to thrive [9]. Finally, to model the primary process,
infection, we take aMSMI
to represent the number of susceptible mice that get infected, due to an
encounter with an infected (and consequently infectious) mouse. This rate a could generally depend on
the density of mice, for example due to an increased frequency of fight when the density is too high and
the population feels overcrowded [5]. However, we will assume it to be constant for the sake of
simplicity. The fact that the infection is chronic, infected mice do not die of it, and infected mice do not
lose their infectiousness probably for their whole life [3, 5-7], supports this description.
