Stability Analysis and Optimal Control of Endemic Malaria Disease Transmission Model with Cost-Effective Strategies (original) (raw)

Optimal Combinations of Control Strategies and Cost-Effectiveness Analysis of Dynamics of Endemic Malaria Transmission Model

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...

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...

A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria

Applied Mathematics and Computation, 2008

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.

Optimal Control Analysis of Malaria in the Presence of Non-Linear Incidence Rate

Applied and computational mathematics

We present the application of optimal control theory to a simple SI malaria model with non-linear incidence rate. The basic properties of the model, including the epidemic threshold are derived. The model is found to show transcritical bifurcation. We conclude from the study that an optimal controlled treatment strategy would ensure significant reduction in malaria incidence if fully adhered to.

Dynamics and Control Measures for Malaria Using a Mathematical Epidemiological Model

2018

Malaria is one of the most prevalent illness globally especially in the tropic and sub-tropic regions of the world. This work investigates the transmission dynamics of malaria disease and the different ways the disease can be controlled by formulating appropriate mathematical epidemiological model. To evaluate the impacts of control measures, we determine the important mathematical features of the model such as the basic reproduction number and analyze then accordingly. The disease free equilibrium and endemic equilibrium point of the model were derived and its stability investigated. For instance, our analysis showed that the disease free equilibrium point is stable when R0 < 1. Stability analyses of the endemic equilibrium is investigated using the centre manifold theorem. Numerical simulations were carried out using realistic parameter values to support our analytical predictions.

Modelling the Dynamics of Endemic Malaria Disease with Imperfect Quarantine and Optimal Control

Mathematical modeling and applications, 2021

Malaria is an infectious disease caused by Plasmodium parasite and it is transmitted among humans through bites of female Anopheles mosquitoes. In this paper, a new deterministic mathematical model for the endemic malaria disease transmission that incorporates imperfect quarantine and optimal control is proposed. Impact of various intervention strategies in the community with varying population at time t are analyzed using mathematical techniques. Further, the model is analyzed using stability theory of differential equations and the basic reproduction number is obtained from the largest eigenvalue of the next-generation matrix. Conditions for local and global stability of disease free, local stability of endemic equilibria and bifurcations are determined in terms of the basic reproduction number. The Center manifold theory is used to analyze the bifurcation of the model. It is shown that the model exhibit both a backward and a forward bifurcation. Reducing the biting rate of the quarantined people is advice able to minimize the spread of endemic malaria disease. The optimal control is designed by applying Pontryagins's Maximum Principle (PMP) with four control strategies namely, insecticide treated nets, screening, treatment and indoor residual spray. The best strategy to control endemic malaria disease is the combination that incorporated all four control strategies.

MATHEMATICAL MODELING OF MALARIA DISEASE WITH CONTROL STRATEGY

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.

Mathematical Analysis of Malaria Transmission Model with Nonlinear Incidences

2012

In this paper, an epidemic model with nonlinear incidences is proposed to describe the dynamics of malaria transmission. The stability of the system can be controlled by the threshold number 0  which governs the existence and stability of the endemic equilibrium. It is found that the disease-free equilibrium point is locally asymptotically stable when 0 1.   For 0 1   , the disease-free equilibrium becomes unstable and the endemic equilibrium is locally asymptotically stable using the general theory of competitive system and compound matrices. Numerical results are shown that the contribution of the nonlinear saturating incidence provides important guidelines for accessing control of malaria diseases. Keywords—Malaria, Nonlinear incidence, Basic reproduction number, Stability

Mathematical modeling of malaria transmission global dynamics: taking into account the immature stages of the vectors

Advances in Difference Equations

In this paper we present a mathematical model of malaria transmission. The model is an autonomous system, constructed by considering two models: a model of vector population and a model of virus transmission. The threshold dynamics of each model is determined and a relation between them established. Furthermore, the Lyapunov principle is applied to study the stability of equilibrium points. The common basic reproduction number has been determined using the next generation matrix and its implication for malaria management analyzed. Hence, we show that if the threshold dynamics quantities are less than unity, the mosquitoes population disappears leading to malaria disappearance; but if they are greater than unity, mosquitoes population persists and malaria also. Finally, numerical simulations are carried out to support our mathematical results.

Mathematical Model of Controlling the Spread of Malaria Disease Using Intervention Strategies

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.