This paper considers the Susceptible-Infected-Vaccinated-Recovered (SIRV) deterministic model with a non linear force of infection and treatment, where individual humans that are vaccinated losses their vaccination after some time and become vulnerable to infections. The basic reproduction number R 0 obtained from the model system is an epidemic threshold that determines if a disease will continue to ravage the human population or not. The model state equations considered in this paper possess two steady-state solutions such that if R 0 < 1, the infection-absent steady-state solutions are locally and globally asymptotically stable. Also, if R 0 > 1, a unique infection-persistent steady-state solutions are established, which is also locally and globally asymptotically stable. Thus, it leads to the persistence of infections in the human host population. Finally, numerical simulations were carried out to validate our theoretical results.
Coinfection by Plasmodium species and Toxoplasma gondii in humans is widespread, with its endemic impact mostly felt in the tropics. A mathematical model is formulated as a first-order nonlinear system of ordinary differential equations to describe the coinfection dynamics of malariatoxoplasmosis in the mainly human and feline susceptible host population in tropical regions. Comprehensive mathematical techniques are applied to show that the model system is bounded, positive and realistic in an epidemiological sense. Also, the basic reproduction number (R omt ) of the coinfection model is obtained. It is shown that if R omt < 1, the model system at its malaria-toxoplasmosis absent equilibrium is both locally and globally asymptotically stable. The impact of toxoplasmosis and its treatment on malaria, and vice versa, is studied and analyzed. Sensitivity analysis was performed to investigate the impact of the model system parameters on the reproduction number of the transmission of malaria-toxoplasmosis coinfection. Simulations and graphical illustrations were made to validate the results obtained from the theoretical model.
In this paper, a mathematical model describing the transmission dynamics of onchocerciasis with distributed delays in infection incubation and recovery in humans and blackfly host population is formulated. We showed that the delayed model is positively invariant and bounded. Also, we obtain the onchocerciasis-free and endemic steady-state solutions as well as the basic reproduction number [Formula: see text] of the delayed onchocerciasis model. We found that the delayed onchocerciasis model is locally asymptotically stable whenever [Formula: see text]. The findings suggest that, for [Formula: see text] to be less than unity, effective use of ivermectin drug for treatment, distribution of treated nets and cloths, etc., is necessary for the minimization and possible elimination of onchocerciasis infection.
A mathematical model describing the epidemic interactions between humans and blackflies in the transmission of onchocerciasis is considered. In this model, the onchocerciasis infected human individuals are divided into two classes of infected humans with high and low microfilarial output incorporating saturated treatment function, which caters for high saturation of onchocerciasis disease. We analyze the model feasible region and obtain the basic reproduction number [Formula: see text] using the next generation matrix method. Also, we obtain the onchocerciasis-free and onchocerciasis endemic equilibrium solutions and show that if [Formula: see text] is less than unity, the onchocerciasis-free equilibrium is locally and globally asymptotically stable. Furthermore, we employed a Lyapunov function to analyze the global asymptotic stability of the onchocerciasis — endemic equilibrium whenever [Formula: see text] is greater than unity. In addition, data on mass drug distribution of ivermectin drug to combat onchocerciasis prevalence in Ekiti state of Nigeria were fitted to the model. Graphical results reveal that consistent annual or bi-annual distribution of ivermectin drug in the region is effective in reducing the disease menace. Further simulations also show that public health control measures are needed to minimize infectious contact between humans and blackflies which leads to onchocerciasis infections and visual blindness.
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