Modelling the Dynamics of Endemic Malaria Disease with Imperfect Quarantine and Optimal Control (original) (raw)
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We formulate a deterministic model with two latent periods in the non-constant host and vector populations, in order to theoretically assess the potential impact of personal protection, treatment and possible vaccination strategies on the transmission dynamics of malaria. The thresholds and equilibria for the model are determined. The model is analysed qualitatively to determine criteria for control of a malaria epidemic and is used to compute the threshold vaccination and treatment rates necessary for community-wide control of malaria. In addition to having a disease-free equilibrium, which is locally asymptotically stable when the basic reproductive number is less than unity, the model exhibits the phenomenon of backward bifurcation where a stable disease-free equilibrium coexists with a stable endemic equilibrium for a certain range of associated reproductive number less than one. From the analysis we deduce that personal protection has a positive impact on disease control but to eradicate the disease in the absence of any other control measures the efficacy and compliance should be very high. Our results show that vaccination and personal protection can suppress the transmission rates of the parasite from human to vector and vice-versa. If the treated populations are infectious then certain conditions should be satisfied for treatment to reduce the spread of malaria in a community. Among the interesting dynamical behaviours of the model, numerical simulations show a backward bifurcation which gives a challenge to the designing of effective control measures.
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This article suggested and analyzed the transmission dynamics of malaria disease in a population using a nonlinear mathematical model. The deterministic compartmental model was examined using stability theory of differential equations. The reproduction number was obtained to be asymptotically stable conditions for the disease-free, and the endemic equilibria were determined. Moreso, the qualitatively evaluated model incorporates time-dependent variable controls which was aimed at reducing the proliferation of malaria disease. The optimal control problem was formulated using Pontryagin's maximum principle, and three control strategies: disease prevention through bed nets, treatment and insecticides were incorporated. The optimality system was stimulated using an iterative technique of forward-backward Runge-Kutta fourth order scheme, so that the impacts of the control strategies on the infected individuals in the population can be determined. The possible influence of exploring a single control, the combination of two, and the three controls on the spread of the disease was also investigated. Numerical simulation was carried out and pertinent findings are displayed graphically.
Asian Research Journal of Mathematics
In this study, a non-linear system of ordinary differential equation model that describes the dynamics of malaria disease transmission is formulated and analyzed. Conditions are derived from the existence of disease-free and endemic equilibria. The basic reproduction number R0 of the model is obtained, and we investigated that it is the threshold parameter between the extinction and persistence of the disease. If R0 is less than unity, then the disease-free equilibrium point is both locally and globally asymptotically stable resulting in the disease removing out of the host populations. The disease can persist whenever R0 is greater than unity and the conditions for the existence of both forward and backward bifurcation at R0 is equal to unity are derived. Sensitivity analysis is also performed and the important parameter that derive the disease dynamics is identified. Furthermore, optimal combinations of time dependent control measures are incorporated to the model, and we derived ...
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Pure and Applied Mathematics Journal, 2020
This paper proposes and analyses a basic deterministic mathematical model to investigate Simulation for controlling the spread of malaria Diseases Transmission dynamics. The model has seven non-linear differential equations which describe the control of malaria with two state variables for mosquito's populations and five state variables for human's population. To represent the classification of human population we have included protection and treatment compartments to the basic SIR epidemic model and extended it to SPITR model and to introduce the new SPITR modified model by adding vaccination for the transmission dynamics of malaria with four time dependent control measures in Ethiopia Insecticide treated bed nets (ITNS), Treatments, Indoor Residual Spray (IRs) and Intermittent preventive treatment of malaria in pregnancy (IPTP). The models are analyzed qualitatively to determine criteria for control of a malaria transmission dynamics and are used to calculate the basic reproduction R 0. The equilibria of malaria models are determined. In addition to having a disease-free equilibrium, which is globally asymptotically stable when the R 0 <1, the basic malaria model manifest one's possession of (a quality of) the phenomenon of backward bifurcation where a stable disease-free equilibrium co-exists (at the same time) with a stable endemic equilibrium for a certain range of associated reproduction number less than one. The results also designing the effects of some model parameters, the infection rate and biting rate. The numerical analysis and numerical simulation results of the model suggested that the most effective strategies for controlling or eradicating the spread of malaria were suggest using insecticide treated bed nets, indoor residual spraying, prompt effective diagnosis and treatment of infected individuals with vaccination is more effective for children.
Optimal Combinations of Control Strategies for Dynamics of Endemic Malaria Disease Transmission
Journal of Advances in Mathematics and Computer Science
In this study, a non-linear system of ordinary differential equation model that describe the dynamics of malaria disease transmission is derived and analyzed. Conditions are derived from the existence of disease-free and endemic equilibria. Basic reproduction number R0 of the model is obtained, and we investigated that it is the threshold parameter between the extinction and persistence of the disease. If R0 is less than unity, then the disease-free equilibrium point is both locally and globally asymptotically stable resulting in the disease removing out of the host populations. The disease can persist whenever R0 is greater than unity. At R0 is equal to unity, existence conditions are derived from the endemic equilibrium for both forward and backward bifurcations. Furthermore, optimal combinations of time dependent control measures are incorporated to the model, and we derived the necessary conditions of the optimal control using Pontryagins’s maximum principal theory. Numerical si...
Mathematical Modeling the Dynamics of Endemic Malaria Transmission with Control Measures
IOSR Journals , 2019
Malaria is an infectious disease caused by the Plasmodium parasite and is transmitted between humans through bites of female anopheles' mosquitoes. The disease continues to emerge in developing countries and remains as a global health challenge. In this paper, a mathematical model is formulated that insights in to some essential dynamics of malaria transmission with environmental management strategy for malaria vector control, insecticide treated bed net, indoor residual spray and treatment with antimalaria drugs as control strategies for humans so as to minimize the disease transmission or spread. The reproduction numbers with single and combined control strategies are calculated and they were compared with each other so as to find the one that benefits more the communities. Numerical simulation shows that among single controls strategies, insecticide treated bed net yields the best result. Furthermore, controlling results of two strategies are better than one; those of three are far better than two and so on. Also, the simulations with all four interventions showed that those results are the best among all possible combinations of intervention strategies. Furthermore, sensitivity analysis is performed and identified important parameters that drive the disease dynamics. Also, their relative importance to disease transmission as well as its prevalence is measured.
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We present an ordinary differential equation mathematical model for the spread of malaria in human and mosquito population. Susceptible humans can be infected when they are bitten by an infectious mosquito. They then progress through the exposed, infectious and recovered classes before reentering the susceptible class. Susceptible mosquitoes can become infected when they bite infectious or recovered humans, and once infected they move through the exposed and infectious classes. Both species follow logistic population model, with humans having immigration and disease-induced death. We define a reproductive number R0, for the number of secondary cases that one infected individual will cause through the duration of the infectious period. We find that the disease-free equilibrium is locally a asymptotically stable when R0 <1 and unstable when R0>1. We prove the existence of at least one endemic equilibrium point for all R0>1. In the absence of disease-induced death, we prove that the transcritical bifurcation at R0=1 is supercritical (forward). Numerical simulations shows that for larger values of the disease-induced death rate, a subcritical (backward) bifurcation is possible at R0=1
Modeling and Control of Malaria When Mosquitoes Are Used as Vaccinators
Mathematical Population Studies, 2015
From the idea of turning mosquitoes into vaccinators, a first model of the transmission of malaria based on standard incidence leads to express the basic reproduction number R 0 (w) and the effective reproduction number R(w) as a function of the vaccination rate w. The disease-free equilibrium is locally asymptotically stable if R 0 (w) < 1. A necessary and sufficient condition for backward bifurcation is derived. A unique endemic equilibrium exists if R 0 (w) > 1. A second model, based on mass action incidence, leads to express the basic reproduction number R m 0 ðwÞ. The disease-free equilibrium is both locally asymptotically stable and globally stable if R m 0 ðwÞ < 1. A unique endemic equilibrium exists if R m 0 ðwÞ > 1 and is locally asymptotically stable.
Computational and Mathematical Methods in Medicine
In this study, we propose and analyze a determinastic nonlinear system of ordinary differential equation model for endemic malaria disease transmission and optimal combinations of control strategies with cost effective analysis. Basic properties of the model, existence of disease-free and endemic equilibrium points, and basic reproduction number of the model are derived and analyzed. From this analysis, we conclude that if the basic reproduction number is less than unity, then the disease-free equilibrium point is both locally and globally asymptotically stable. The endemic equilibrium will also exist if the basic reproduction number is greater than unity. Moreover, existence and necessary condition for forward bifurcation is derived and established. Furthermore, optimal combinations of time-dependent control measures are incorporated to the model. By using Pontryagin’s maximum principal theory, we derived the necessary conditions of optimal control. Numerical simulations were condu...