Qualitative analysis of malaria dynamics with nonlinear incidence function (original) (raw)
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Bifurcation Analysis of a Mathematical Model for Malaria Transmission
Siam Journal on Applied Mathematics, 2006
We present an ordinary differential equation mathematical model for the spread of malaria in human and mosquito populations. 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 a logistic population model, with humans having immigration and disease-induced death. We define a reproductive number, R 0 , 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 asymptotically stable when R 0 < 1 and unstable when R 0 > 1. We prove the existence of at least one endemic equilibrium point for all R 0 > 1. In the absence of disease-induced death, we prove that the transcritical bifurcation at R 0 = 1 is supercritical (forward). Numerical simulations show that for larger values of the disease-induced death rate, a subcritical (backward) bifurcation is possible at R 0 = 1.
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
GLOBAL STABILITY ANALYSIS OF MALARIA TRANSMISSION DYNAMICS WITH VIGILANT COMPARTMENT
Electronic Journal of Differential Equations, 2019
A deterministic compartmental model for the transmission dynamics of malaria incorporating vigilant human compartment is studied. The model is qualitatively analyzed to investigate its asymptotic behavior with respect to the equilibria. It is shown, using a linear Lyapunov function, that the disease-free equilibrium is globally asymptotically stable when the associated basic reproduction number is less than unity. When the basic reproduction number is greater than the unity, under certain specifications on the model parameters, we prove the existence of a globally asymptotically stable endemic equilibrium with the aid of a suitable nonlinear Lyapunov function.
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.
Global Stability of Malaria Transmission Dynamics Model with Logistic Growth
Discrete Dynamics in Nature and Society, 2018
Mathematical models become an important and popular tools to understand the dynamics of the disease and give an insight to reduce the impact of malaria burden within the community. Thus, this paper aims to apply a mathematical model to study global stability of malaria transmission dynamics model with logistic growth. Analysis of the model applies scaling and sensitivity analysis and sensitivity analysis of the model applied to understand the important parameters in transmission and prevalence of malaria disease. We derive the equilibrium points of the model and investigated their stabilities. The results of our analysis have shown that if R0≤1, then the disease-free equilibrium is globally asymptotically stable, and the disease dies out; if R0>1, then the unique endemic equilibrium point is globally asymptotically stable and the disease persists within the population. Furthermore, numerical simulations in the application of the model showed the abrupt and periodic variations.
Int. J. Pure Appl. Math., 2013
This paper presents a seven-dimensional ordinary differential equation modelling the transmission of Plasmodium falciparum malaria between humans and mosquitoes with non-linear forces of infection in form of saturated incidence rates. These incidence rates produce antibodies in response to the presence of parasite-causing malaria in both human and mosquito populations.The existence of region where the model is epidemiologically feasible is established. Stability analysis of the disease-free equilibrium is investigated via the threshold parameter (reproduction number R0) obtained using the next generation matrix technique. The model results show that the disease-free equilibrium is asymptotically stable at threshold parameter less than unity and unstable at threshold parameter greater than unity. The existence of the unique endemic equilibrium is also determined under certain conditions. Numerical simulations are carried out to confirm the analytic results and explore the possible behavior of the formulated model.
A NOTE ON STABILITY ANALYSIS OF COMPARTMENTAL MATHEMATICAL MODEL FOR THE SPREAD OF MALARIA
In this study, we present a compartmental model for the spread of malaria in a population where group of individuals were vaccinated. The purpose of this paper is to analyze the transmission dynamics of Malaria by using the compartmental model, including ordinary differential equations for human host and mosquito vector populations. A parallel system is obtained, which has two equilibriums: a disease-free equilibrium and an endemic equilibrium. The stability of the equilibrium points is verified by the basic reproduction numberR_0 . Asymptotically stable solution is obtained only for disease-free equilibrium and results are presented graphically.
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.
The effect of incidence function in backward bifurcation for malaria model with temporary immunity
Mathematical Biosciences, 2015
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • Malaria model with standard incidence rate exhibits the backward bifurcation. • Backward bifurcation does not occur in malaria model with nonlinear incidence rate. • Global stability analysis is based on constructing a Lyapunov function. • Sensitivity analysis of model parameters is helpful to design control strategies.
Mathematical analysis of the role of repeated exposure on malaria transmission dynamics
Differential Equations and Dynamical Systems, 2008
This paper presents a deterministic model for assessing the role of repeated exposure on the transmission dynamics of malaria in a human population. Rigorous qualitative analysis of the model, which incorporates three immunity stages, reveals the presence of the phenomenon of backward bifurcation, where a stable diseasefree equilibrium co-exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity. This phenomenon persists regardless of whether the standard or mass action incidence is used to model the transmission dynamics. It is further shown that the region for backward bifurcation increases with decreasing average life span of mosquitoes. Numerical simulations suggest that this region increases with increasing rate of re-infection of first-time infected individuals. In the absence of repeated exposure (re-infection) and loss of infection-acquired immunity, it is shown, using a non-linear Lyapunov function, that the resulting model with mass action incidence has a globallyasymptotically stable endemic equilibrium when the reproduction threshold exceeds unity.