Annals of Communications in Mathematics

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.

AbstractA spatial mathematical model to study the impact of vector control strategies on the dynamics of malaria transmission and its analysis is considered in this paper. The resulting model equations are divided into homogeneous and non-homogeneous equations. The homogeneous equations are solved to determine their disease-free equilibrium (DFE) and their stability. A basic reproduction number was determined from the DFE. It was found that when basic reproduction number is less one, the disease will die out, when the basic reproduction number is exactly one, the model undergoes a backward bifurcation, when the basic reproduction number is exactly zero, the model undergoes forward bifur- cation and whenever the basic reproduction is greater than one, the disease will persist in the population. A quantitative sensitivity analysis of the model parameters was also conducted through the disease’s basic reproduction number to determine the parameters that are sensitive to malaria transmission. A travelling wave equation and solutions were also provided for a possible understanding of the behaviour of mosquitoes’ mobility in the human environment. Finally, we carried out a simulation of our formulated partial differ- ential model and quantitatively assessed and investigated the twin effect of the presence of invasive plants and the spatial dispersion of vectors on malaria dynamics. Sensitivity analysis was also carried out, and the quantitative effect of diffusion and advection on the wave front was demonstrated. The speed of the disease propagation by using travelling wave solutions of the model was also investigated numerically.

AbstractThis paper provides a mathematical analysis of a host vectors disease model with the influence of available hospital resources. We derive the basic reproduction num- ber Rh 0 of the model. We prove the existence of a unique disease-free equilibrium, which is stable when the basic reproduction number Rh 0 is less than 1, indicating that the disease can be eradicated under these conditions. However, when Rh 0 exceeds 1, the system ex- hibits multiple endemic equilibria, leading to the possible persistence of the disease into the population. The study also reveals the existence of bifurcations, indicating qualitative changes in the system’s dynamics depending on certain critical parameter values. A sensi- tivity analysis of the parameters is carried out to assess the most influential parameters in managing the epidemic.