Backwards bifurcations and catastrophe in simple models of fatal diseases

where I is the number of infected people, S is the number of susceptible people, 1 is the total number of people in the population, ! is the transmission rate of the disease, and m is the rate at which individuals leave the infected group. Here IQ means the derivative of I with respect to time, a convention we will use throughout the paper. Equation (1) is applicable to a wide variety of one-group models. Following Castillo-Chavez et al. [3], we allow ! to be a function of 1, allowing a variety of assumptions about mixing. Depending on the type of model, the per-capita removal rate, m, may include the rate of ‘‘background’’ mortality or disease-induced mortality, or transitions to immune, susceptible or quarantined compartments. Note that the number of infectives will increase when S/1'm/! and decrease otherwise. The ratio S/1 gives the proportion of susceptibles in a population, and hence the probability that a given contact of infectious individual is with a susceptible individual, under the assumption of homogeneous mixing. This ratio is at a maximum (generally 1) in a population where the disease is absent, and decreases as the disease begins to invade a population. It is this phenomenon of the disease reducing its own