A stochastic epidemic model for the dynamics of two pathogens in a single tick population

Understanding tick-transmitted pathogens in tick infested areas is crucial for the development of preventive and control measures in response to the increasing cases of tick-borne diseases. A stochastic model for the dynamics of two pathogens, Rickettsia parkeri and Rickettsia amblyommii, in a single tick, Amblyomma americanum, is developed and analysed. The model, a continuous-time Markov chain, is based on a deterministic tick-borne disease model. The extinction threshold for the stochastic model is computed using the multitype branching process and conditions for pathogen extinction or persistence are presented. The probability of pathogen extinction is computed using numerical simulations and is shown to be a good estimate of the probability of extinction calculated from the branching process. A sensitivity analysis is undertaken to illustrate the relationship between co-feeding and transovarial transmission rates and the probability of pathogen extinction. Expected epidemic duration is estimated using sample paths and we show that R. amblyommii is likely to persist slightly longer than R. parkeri. Further, we estimate the duration of possible coexistence of the two pathogens.