Dynamics System in the SEIR-SI Model of the Spread of Malaria with Recurrence (original) (raw)
Related papers
SEIRS Mathematical Model for Malaria with Treatment
Mathematical Modelling and Applications, 2020
In this paper a deterministic mathematical model for the spread of malaria in human and mosquito populations are presented. The model has a set of eight non-linear differential equations with five state variables for human and three for mosquito populations respectively. Susceptible humans can be infected when they are bitten by an infectious mosquito. They then progress through the exposed, infectious, treatment and recovered or immune classes before coming back to the susceptible class. Susceptible mosquitoes can become infected when they bite infectious humans, and once infected they move through exposed and infectious class. However, mosquitoes once infected will never recover from the disease during their lifetime. That is, infected mosquitoes will remain infectious until they die. Formula for the basic reproduction number R 0 is established and used to determine whether the disease dies out or persists in the populations. It is shown that the disease-free equilibrium point is locally asymptotically stable using the magnitude of Eigen value and Routh-Hurwitz stability Criterion. Result and detailed discussion of the analysis as well as the simulation study is incorporated in the text of the paper lucidly.
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.
2019
In this study we have develop a basic deterministic mathematical model to investigate SEIR Model and Simulation for controlling malaria Diseases Transmission without Intervention Strategies. The model has seven non linear differential equations which describe the spread of malaria with three state variables for mosquitoes populations and four state variables for humans population and to introduce the model without intervention strategies. The models are analyzed qualitatively to determine criteria for control of a malaria transmission, 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 locally asymptotically stable when the R 0 1, the basic malaria model manifest ones 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 reproduc...
Relapse Effect on the Dynamics of Malaria in Humans and Mosquitoes: A Mathematical Model Analysis
IOSR Journals , 2019
In this paper we consider nonlinear dynamical system to study the dynamics of malaria with relapse effect in both human and mosquito population. The total population is divided in to six compartments in which human population into three compartments and mosquito population into two compartments. We found the dynamical system has disease free equilibrium point and endemic equilibrium point.
Numerical simulation of malaria transmission model considering secondary infection
Communications in Mathematical Biology and Neuroscience, 2020
Malaria is an infectious disease caused by Plasmodium and transmitted through the bite of female Anopheles mosquitoes. This article constructs a mathematical model to understand the spread of malaria by considering the vector-bias phenomenon in the infection process, secondary infection, and fumigation as a means of malaria control. The model is constructed as a SIRI-UV model based on six-dimensional non-linear ordinary differential equations. Analysis of the equilibrium points with their local stability and sensitivity analysis of the basic reproduction number R 0 is shown analytically and numerically. Based on the analytical studies, two types of equilibrium points were obtained, namely the disease-free equilibrium points and the endemic equilibrium points. We find that the disease-free equilibrium is locally stable if R 0 < 1. Our proposed model shows the possibility of a forward bifurcation, backward bifurcation, or forward bifurcation with hysteresis. To support the interpretation of the model, a numerical simulation for the sensitivity of R 0 and some autonomous simulations conducted to see how the change of parameter will affect the dynamics of our model. Simulation results show that the increasing of mortality rate on mosquitoes due to fumigation will increase the probability that malaria is eliminated.
MATHEMATICAL MODELLING OF CAUSATIVE EFFECT AND CONTROL OF MALARIA
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
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.
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.
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.
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.