Modeling cryptosporidiosis in humans and cattle: Deterministic and stochastic approaches

Cryptosporidiosis is a zoonotic disease caused by Cryptosporidium. The disease poses a public and veterinary health problem worldwide. A deterministic model and its corresponding continuous time Markov chain (CTMC) stochastic model are developed and analyzed to investigate cryptosporidiosis transmission dynamics in humans and cattle. The basic reproduction number $R_0$ for the deterministic model and stochastic threshold for the CTMC stochastic model are computed by the next generation matrix method and multitype branching process, respectively. The normalized forward sensitivity index method is used to determine the sensitivity index for each parameter in $R_0$. Per capita birth rate of cattle, the rate of cattle to acquire cryptosporidiosis infection from the environment and the rate at which infected cattle shed Cryptosporidium oocysts in the environment play an important role in the persistence of the disease whereas Cryptosporidium oocysts natural death rate, cattle recovery rate and cattle natural death rate are most negative sensitive parameters in the dynamics of cryptosporidiosis. Numerical results for CTMC stochastic model show that the likelihood of cryptosporidiosis extinction is high when it arises from an infected human. However, there is a major outbreak if cryptosporidiosis emerges either from infected cattle or from Cryptosporidium oocysts in the environment or when it emerges from all three infectious compartments. Therefore to control the disease, control measures should focus on maintaining personal and cattle farm hygiene and decontaminating the environment to destroy Cryptosporidium oocysts.