Annals of Communications in Mathematics

AbstractMalaria is one of the serious life-threatening diseases with negative effects on both the social and economic aspects of human life. Researching into its curtailment or eradication is necessary for elevating human health and social-economic status. In thisregard, this study focuses on the spatial non-linear mathematical model to investigate how vector control strategies are correlated with the dynamics of malaria transmission. The study employs a non-linear partial differential equations (NPDE) mathematical model to investigate malaria transmission. The model system incorporates human (host), mosquito (vector), and invasive alien plant populations. Some applicable epidemiological mathematical analyses were carried out on the model system, such as critical points, stability, the basic reproduction number, local asymptotic stability (LAS), bifurcation, global as- ymptotic stability (GAS), wave speed, and numerical analyses using relevant data were extensively analysed. Using the sharp threshold conditions imposed on the basic reproduction number, we were able to show that the model exhibited the backward bifurcation phenomenon and the DFE was shown to be globally asymptotic stable (GAS) under certain conditions. It was found that the invasive alien plants have significant effects on malaria transmission. This study suggests that mosquito repellent plants should be planted around the human environment to replace the invasive plants so as to reduce mosquito shelters andfeeding opportunities for mosquitoes.

AbstractAvian influenza is known as one of the respiratory diseases that causes high morbidity and mortality rate predominately among the immunodeficiency persons world- wide. Treatment and vaccination remain the optimal strategies in curbing the spread of avian infuenza infection.In this work, a mathematical model of the dynamics of influenza infection is formulated and was computed analytically and numerically. The analytic com- putation of the model is given in terms of the basic reproduction number, equilibria points and their stabilities. Thus, the disease dies out whenever the basic reproduction number is less than one. The disease free equilibrium (DFE) is locally asymptotically stable pro- vided R0 < 1 and unstable if otherwise. The endemic equilibrium only occurs whenever the disease threshold is greater than a unit. The endemic equilibrium, is locally, globally asymptotically stable under certain conditions. Numerical solution shows that vaccination and treatment of the susceptible and the infected individuals respectively have high impact for eradicating the disease. The non-linear incidence as a force of infection with param- eter, θ,Ψ1, u1 and u2 have great impact for reducing the pandemic of influenza disease. In conclusion, vaccination of susceptible individuals, isolation of exposed individuals and treatment of infected individuals are imperative for curbing the spread of an avian influenza infection. Modelling style or structure especially, the type of force of infection adopted for modelling an avian influenza disease depends on whether the disease, can easily be put under control.