Non-seasonal and seasonal relapse model for Q fever disease with comprehensive cost-effectiveness analysis

Q fever (query fever) is classified as one of the likely zoonosis that is transmissible to humans from farm animals. The disease may lead to significant economic losses to livestock owners. The deterministic dynamics of direct transmission, different shedding rates, treatment, the effect of relapse, and seasonality of disease remains an open problem. Therefore, this study introduces direct transmission cycle, different shedding rates, treatment, the effect of relapse, and time-dependent parameters. We obtain control reproduction expressions for the model under different scenarios. Using Lyapunov functions, we establish that the non-seasonal model is globally asymptotically stable. The extension of the model to include seasonal trends shows that there exists a positive periodic solution. The global sensitivity method used shows that imperfect treatment reduces the rate of indirect transmission, but, leads to an increase in direct transmission by symptomatic cattle herds. Optimal control model is also utilized to explore the impact of time-dependent treatment, disinfectant, and separate facilities for animal birthing. We show the control efficacies; and also calculated the average cost-effectiveness ratio, and the incremental cost-effectiveness ratio. We hope that this study further enlightens farm owners on the use of other interventions in controlling Q fever disease in livestock.