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On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrome (AIDS). I: Single population models. (English) Zbl 0715.92029
Summary: We investigate systematically the role played by the reproductive number (the number of secondary infections generated by an infectious individual in a population of susceptibles) on single group populations models of the spread of HIV/AIDS. Our results for a single group model show that if \(R\leq 1\), the disease will die out, and strongly suggest that if \(R>1\) the disease will persist regardless of initial conditions. Our extensive (but incomplete) mathematical analysis and the numerical simulations of various research groups support the conclusion that the reproductive number R is a global bifurcation parameter. The bifurcation that takes place as R is varied is a transcritical bifurcation; in other words, when R crosses 1 there is a global transfer of stability from the infection- free state to the endemic equilibrium, and vice versa. These results do not depend on the distributon of times spent in the infectious categories (the survivorship functions).
Furthermore, by keeping all the key statistics fixed, we can compare two extremes: exponential survivorship versus piecewise constant survivorship (individuals remain infectious for a fixed length of time). By choosing some realistic parameters we can see (at least in these cases) that the reproductive numbers corresponding to these two extreme cases do not differ significantly whenever the two distributions have the same mean. At any rate a formula is provided that allows us to estimate the role played by the survivorship function (and hence the incubation period) in the global dynamics of HIV.
These results support the conclusion that single population models of this type are robust and hence are good building blocks for the construction of multiple group models. Our understanding of the dynamics of HIV in the context of mathematical models for multiple groups is critical to our understanding of the dynamics of HIV in a highly heterogeneous population.

MSC:
92D30 Epidemiology
34D05 Asymptotic properties of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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