This paper presents a mathematical model that describes the transmission dynamics of schistosomiasis for humans, snails, and the free living miracidia and cercariae. The model incorporates the treated compartment and a preventive factor due to water sanitation and hygiene (WASH) for the human subpopulation. A qualitative analysis was performed to examine the invariant regions, positivity of solutions, and disease equilibrium points together with their stabilities. The basic reproduction number, R 0 , is computed and used as a threshold value to determine the existence and stability of the equilibrium points. It is established that, under a specific condition, the disease-free equilibrium exists and there is a unique endemic equilibrium when R 0 > 1 . It is shown that the disease-free equilibrium point is both locally and globally asymptotically stable provided R 0 < 1 , and the unique endemic equilibrium point is locally asymptotically stable whenever R 0 > 1 using the concept of the Center Manifold Theory. A numerical simulation carried out showed that at R 0 = 1 , the model exhibits a forward bifurcation which, thus, validates the analytic results. Numerical analyses of the control strategies were performed and discussed. Further, a sensitivity analysis of R 0 was carried out to determine the contribution of the main parameters towards the die out of the disease. Finally, the effects that these parameters have on the infected humans were numerically examined, and the results indicated that combined application of treatment and WASH will be effective in eradicating schistosomiasis.
This paper considers the current global issue of containing the coronavirus pandemic as an optimal control problem. The goal is to determine the most advantageous levels of effectiveness of the various control and preventive measures that should be attained in order to cost effectively drive the epidemic towards eradication within a relatively short time. Thus, the problem objective functional is constructed such that it minimizes the prevalence as well as the cost of implementing the various control measures subject to a model for the disease transmission dynamics which incorporates the existing controls. The optimality system of the model is derived based on Pontryagin's maximum principle while the resulting system is solved numerically using the Runge-Kutta fourth order scheme with forward-backward sweep approach. Findings from our results show that the new cases and the prevalence of the disease can be remarkably reduced in a cost effective way, if the specified optimal levels of effectiveness of the various preventive and control measures are upheld continuously for at least a month. Moreover, the results also show that the disease can be eventually eradicated if these effectiveness levels are sustained over a reasonable length of time.
The novel coronavirus (COVID-19) pandemic continues despite series of control measures implemented to curtail it. Therefore, it is pertinent to study how the various proposed control measures can be effectively combined in order to stem the alarming spread of the disease and its attendant consequences. In this paper, a deterministic model for the transmission dynamics of the disease, which incorporates the impacts of the various implemented control measures, is presented. Based on the proposed model, the disease’s basic reproduction number (R0) was derived and the equilibrium solutions were determined. It was shown that whenever R0 \(\le\) 1, the model has only the disease-free equilibrium which is globally stable while in circumstances where R0 > 1, there exists an endemic equilibrium which is globally asymptotically stable. When the latter equilibrium state exists, the former becomes unstable. In addition, the model parameters were estimated using Nigeria’s demographic and COVID-19 surveillance data. The model is simulated for different scenarios of the disease outbreak and the results suggest that the disease will die out quickly in the population if about half of the population adhere to personal protection, about half of exposed individuals are efficiently traced and about half of symptomatic individuals are promptly isolated and treated.
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