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.

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 study explores the mathematical modelling of measles transmission dy- namics in Nigeria, with a specific focus on assessing the impact of a single-dose vaccina- tion strategy. Given the resurgence of measles outbreaks, especially in regions with low vaccination coverage, this research aims to develop a robust model that can simulate dis- ease transmission and evaluate vaccination strategies. The primary objective of the study is to understand how varying levels of vaccination coverage, vaccine efficacy, and immunity waning affect the disease dynamics. A modified SEIR (Susceptible-Exposed-Infectious- Recovered) model was used, incorporating additional compartments for individuals vac- cinated with one dose, as well as a factor for immunity waning. Data from Nigeria’s Measles Situation Report (April 2024) informed the parameter values, initial population distributions, and vaccination rates within the model, providing a real-world context. The study employed numerical simulations using MATLAB to analyse the effects of vaccina- tion rates, immunity waning, and other epidemiological parameters on measles transmis- sion. The results reveal that high vaccination coverage specifically, achieving coverage rates above 80% with the single-dose strategy significantly reduces the disease prevalence, indicating effective outbreak prevention. However, the simulations also show that im- munity waning can increase susceptibility, suggesting a potential need for booster dose to sustain long-term immunity in the population. It recommends that public health authorities prioritize reaching at least 90% vaccination coverage with two doses and consider booster doses if immunity waning proves significant. These insights provide a foundation for en- hancing measles control efforts, informing policy decisions, and guiding future research on infectious disease dynamics in Nigeria and similar settings.