Second-order oscillatory problems have been found to be applicable in studying various phenomena in science and engineering; this is because these problems have the capabilities of replicating different aspects of the real world. In this research, a new hybrid method shall be formulated for the simulations of second-order oscillatory problems with applications to physical systems. The proposed method shall be formulated using the procedure of interpolation and collocation by adopting power series as basis function. In formulating the method, off-step points were introduced within the interval of integration in order to bypass the Dahlquist barrier, improve the accuracy of the method and also upgrade the order of consistence of the method. The paper further validated the some properties of the hybrid method derived and from the results obtained; the new method was found to be consistent, convergent and stable. The simulation results generated as a result of the application of the new method on some second-order oscillatory differential equations also showed that the new hybrid method is computationally reliable.

Modelling the impacts of lockdown and isolation on the eradication of COVID-19

A simple stochastic mathematical model is developed and investigated for the dynamics of measles epidemic. The model, which is a multi-dimensional diffusion process, includes susceptible individuals, latent (exposed), infected and removed individuals. Stochastic effects are assumed to arise in the process of infection of susceptible individuals. Using the best currently available parameter values, the intrinsic variability in response to a given initial infection is examined by solving the stochastic system numerically. The results of the simulation seem to agree with the historical pattern of measles in Nigeria. Keywords: Stochastic model, measles, epidemiology, Wiener process, Euler scheme.

A debilitating disease such as malaria has been a global health challenge for a long time. Mathematical models have provided insights on how best to control, prevent, or eradicate the disease. Most of the models used parameters to represent control strategies. The current research aims at using control strategies as disease compartments for both vector control and human-protection. By this, the long-time evolution of containment measures can be assessed. Consequently, a model of nine disease compartments is constructed using first-order ordinary differential equations. A qualitative analysis of the malaria-free equilibrium was carried out. The results indicate that the malaria-free equilibrium state is locally and globally asymptotically stable when $R_{0HV}<1$, which implies that the disease-free state will always be attained from any initial conditions. A bifurcation analysis also shows that the malaria-endemic and malaria-free equilibria cannot overlap. Numerical simulations show that vector control is more effective in the containment of malaria than human-protection, which confirms the findings of Ross 53 years ago. Numerical results also show that solutions attain a disease-free steady state with time, which agrees with the theoretical results. The implication of the findings is that more attention should be paid to vector control if malaria is to be eradicated.

A mathematical model for three-species interactions in a food chain, with the assumption that the interacting species are mobile, has been constructed using a combination of Holling’s type III and the BD functional responses. Conditions for the onset of diffusive instability were determined. The results indicate the possibility of a stable coexistence of the three interacting species in form of stable oscillations under the reflecting boundary conditions. Habitat segregation also occurs under these conditions. However, under the absorbing boundary conditions, the species experience damped oscillations leading to their extinction. The effects of cross-diffusion of the intermediate and the toppredator were also examined.

Malaria remains a global threat and the conventional methods used for combating the disease leave out mosquitoes that feed outdoors. This study addresses the challenge posed by such mosquitoes based on a tool called ivermectin drug which is lethal to mosquitoes that ingest bloodmeal containing a concentration of it. We formulated a mathematical model with three control tools (insecticide treated nets, treatment of infective individuals and ivermectin drug on livestock and humans) for the transmission and control of malaria under optimal condition. The model’s basic reproduction number, R0 was estimated and the local and global stability analyses of the disease-free and endemic equilibrium points of the model were carried out. Sensitivity analysis carried out showed that R0 is most sensitive to the mosquito biting rate and to the proportion of blood meal on human with cattle availability in such a way that any percent increase in the value of any of these parameters will lead to an equal percent increase in the value of R0. The result of an optimal control analysis based on three time dependent controls suggests that the combination of all three controls gives the best result followed by the strategy that combines the use of ivermectin drug and the treatment of infective human. Depending on available resources, any of these is recommended to be adopted in malaria intervention programmes because of their effectiveness on both the infective human and mosquito populations with the potential of contributing significantly to the disease elimination within a minimal time frame.