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Backwards bifurcations and catastrophe in simple models of fatal diseases. (English) Zbl 0917.92022
Our purpose in writing this paper was to provide an intuitive explanation of the mechanisms that drive backwards bifurcations in some simple disease models, and thus to make it possible to explore in what circumstances such bifurcations might occur in more realistic disease models and hence in what circumstances we should look for the dynamical signature of the backwards bifurcations – breakpoints above which the disease can persist for certain values of \(R_0\), and the possibility of ‘catastrophic’ fast dynamics as underlying parameters change slowly – in real-world disease systems.
The original models on which our analysis is based are AIDS models. It is natural to consider the possibility that backwards bifurcations of the type discussed here may occur in AIDS, with men making up the core groups and women the victim groups, since male homosexual transmission is thought to be very important to the spread of AIDS in many places. For this to cause backwards bifurcations, however, AIDS would need to change the population structure in such a way as to increase the proportion of men’s sexual contacts that were with other men. This is certainly possible, but does not intuitively seem likely. In general, models of sexual mixing involve complicating social factors beyond the scope of this discussion. An exploration of the possibility of core/victim backwards bifurcations in AIDS would certainly be of interest, however. Another difficulty with AIDS modeling is that, directly contrary to our discussion of slow-changing parameters, it seems likely that behavioral ‘parameters’ in a disease like AIDS change quickly compared to the time scale of disease spread itself.
For a wide variety of other diseases, it is at least plausible that a ‘core’ group of healthier, more active people might be both more important than other groups at spreading the disease, and less affected by it. For backwards bifurcations to be caused by the mechanism discussed here, however, it would be necessary for such a disease to cause enough mortality or morbidity to substantially change the structure of the mixing population, which sharply reduces the range of possibilities. It is possible that this mechanism for backwards bifurcations would be more relevant in studies of animal diseases. Studies have demonstrated that diseases like anthrax and myxomatosis have had profound effects on animal populations.

MSC:
92D30 Epidemiology
34C23 Bifurcation theory for ordinary differential equations
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