We formulate and analyse a stochastic epidemic model for the transmission dynamics of a tick-borne disease in a single population using a continuous-time Markov chain approach. The stochastic model is based on an existing deterministic metapopulation tick-borne disease model. We compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in tick-borne disease dynamics. The probability of disease extinction and that of a major outbreak are computed and approximated using the multitype Galton-Watson branching process and numerical simulations, respectively. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that a disease outbreak is more likely if the disease is introduced by infected deer as opposed to infected ticks. These insights demonstrate the importance of host movement in the expansion of tick-borne diseases into new geographic areas.
Thresholds for disease extinction provide essential information for the prevention and control of diseases. In this paper, a stochastic epidemic model, a continuous-time Markov chain, for the transmission dynamics of West Nile virus in birds is developed based on the assumptions of its analogous deterministic model. The branching process is applied to derive the extinction threshold for the stochastic model and conditions for disease extinction or persistence. The probability of disease extinction computed from the branching process is shown to be in good agreement with the probability approximated from numerical simulations. The disease dynamics of both models are compared to ascertain the effect of demographic stochasticity on West Nile virus dynamics. Analytical and numerical results show differences in model predictions and asymptotic dynamics between stochastic and deterministic models that are crucial for the prevention of disease outbreaks. It is found that there is a high probability of disease extinction if the disease emerges from exposed mosquitoes unlike if it emerges from infectious mosquitoes and birds. Finitetime to disease extinction is estimated using sample paths and it is shown that the epidemic duration is shortest if the disease is introduced by exposed mosquitoes.
Tuberculosis, an airborne disease affecting almost a third of the world’s population remains one of the major public health burdens globally, and the resurgence of multidrug-resistant tuberculosis in some parts of sub-Saharan Africa calls for concern. To gain insight into its qualitative dynamics at the population level, mathematical modeling which require as inputs key demographic and epidemiological information can fill in gaps where field and lab data are not readily available. A deterministic model for the transmission dynamics of multi-drug resistant tuberculosis to assess the impact of diagnosis, treatment, and health education is formulated. The model assumes that exposed individuals develop active tuberculosis due to endogenous activation and exogenous re-infection. Treatment is offered to all infected individuals except those latently infected with multi-drug resistant tuberculosis. Qualitative analysis using the theory of dynamical systems shows that, in addition to the disease-free equilibrium, there exists a unique dominant locally asymptotically stable equilibrium corresponding to each strain. Numerical simulations suggest that, at the current level of control strategies (with Malawi as a case study), the drug-sensitive tuberculosis can be completely eliminated from the population, thereby reducing multi-drug resistant tuberculosis.
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.
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