Backward Bifurcation as a Desirable Phenomenon: Increased Fecundity Through Infection

Backward or subcritical bifurcation is usually considered an undesirable phenomenon in epidemiology since control measures require a reduction in R₀ not below one but below a much smaller value. However, there are contexts for which a backward or subcritical bifurcation is not a bad thing; it can even be desirable. Such is the case for any characteristic that can be passed to the next generation (genetically fixed or not) and that increases the effective reproductive rate of the host or the total number of individuals. In the present work, we study an epidemiological model consisting of two classes, susceptible and "infected" individuals; the model considers a characteristic that is passed from "infected" to "susceptible" by direct "contact," for instance increased fecundity. We analyze conditions for the appearance of a backward or subcritical bifurcation. We discuss the advantage for the population under infection, since the total number of individuals increases at equilibrium. If one takes that as a proxy for increased fitness, it would increase the species' ecological success. One key element in the model is the fact that "susceptible" individuals have "susceptible" descendants, but "infected" individuals can have "infected" descendants as well as "susceptible" ones. A somehow rare addition for epidemiological models, the fact that "infected" individuals reproduce more rapidly than the susceptible ones, leads to unexpected consequences. Facilitating the "inoculation" increases the total population size, i.e., the backward or subcritical bifurcation appears, with desirable consequences for the population. We show that an increase in the number of susceptible newborns is the main reason for the appearance of a backward or subcritical bifurcation, which induces a bigger population size. We analyze the effect of different combinations of susceptible/infected birth rates. This kind of phenomenon has been observed for bacterial infections in several insects-bacteria and nematodes-bacteria interactions; in particular, it has been intensely studied in interactions of wasps and flies with the genus Wolbachia. It has also been shown in amphibians.