Mathematical models for the transmission dynamics of infectious diseases have aided our understanding of the important factors that drive epidemic patterns. In this paper, we formulate and analyze a stochastic epidemic model, a continuous-time Markov chain, in order to understand rotavirus dynamics with a contaminated environment. The assumptions of the deterministic model are utilized in the formulation of the corresponding stochastic model. We perform both local and global stability analyses of the equilibria of the deterministic model with respect to the basic reproduction number. The extinction threshold for the stochastic model and conditions for either disease extinction or persistence are derived by employing the branching process to the infectious classes only. It is shown that the probability of rotavirus extinction obtained from the branching process is in excellent agreement with the numerically approximated probability. Numerical results indicate that the probability of rotavirus extinction is the highest if the contaminated environment introduces the virus into a totally susceptible population at the beginning of the epidemic process. Thus, a major rotavirus outbreak is likely if the virus emanates from infectious children at the onset of the epidemic. Results of sensitivity analysis showed that shedding of the virus into the environment by infectious children is the most sensitive parameter of the model. Further, it is shown that decreasing the shedding rate leads to an increase in the probability of disease extinction and vice versa. This, therefore, implies that disposal of stool of infectious children should be well managed if efforts to curb further spread of the disease or even eliminating it are to bear desirable fruits.
We formulate and analyze a deterministic mathematical model for the transmission dynamics of schistosomiasis with treatment of both humans and bovines and mollu-sciciding of the contaminated environment. The model effective reproduction number is derived and analytical results show that the disease-free and endemic equilibria are both locally and globally asymptotically stable. Pontryagin’s Maximum Principle which uses both Lagrangian and Hamiltonian principles with respect to a time-dependent constant is used to establish the existence of the optimal control problem and to derive the necessary conditions for optimal control of the disease. Mollusciciding of the contaminated environment has a major impact on disease control. However, combining it with treatment could help mitigate the spread of the disease compared to applying each control measure individually. Numerical simulations are performed to support theoretical results.
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