A hybrid model for the effects of treatment and demography on malaria superinfection

As standard mathematical models for the transmission of vector-borne pathogens with weak or no apparent sterilizing immunity, Susceptible-Infected-Susceptible (SIS) systems such as the Ross-Macdonald equations are a useful starting point for modeling the impacts of interventions on prevalence for diseases that cannot superinfect their hosts. In particular, they are parameterizable from quantities we can estimate such as the force of infection (FOI), the rate of natural recovery from a single infection, the treatment rate, and the rate of demographic turnover. However, malaria parasites can superinfect their host which has the effect of increasing the duration of infection before total recovery. Queueing theory has been applied to capture this behavior, but a problem with current queueing models is the exclusion of factors such as demographic turnover and treatment. These factors in particular can affect the entire shape of the distribution of the multiplicity of infection (MOI) generated by the superinfection process, its transient dynamics, and the population mean recovery rate. Here we show the distribution of MOI can be described by an alternative hyper-Poisson distribution. We then couple our resulting equations to a simple vector transmission model, extending previous Ross-Macdonald theory.