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We describe a mathematical model we have recently developed to address some features of the interaction

19 / 01 / 2019 Maths

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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.

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