A Proposal on Bifurcation Analysis of a Mathematical Model for Malaria Transmission Under Treatment and Control
CHAPTER ONE
Objectives of the Study
The following objectives of the study will be ascertained following:-
- Carry out a detailed research on malaria transmission
- Study the metamorphosis of plasmodium parasite
- Construct a mathematical model for the transmission of malaria under treatment and control.
- Provide useful and analytical contributions that will help in exterminating malaria in our society.
- Pave way for further studies into the transmission dynamics of malaria.
CHAPTER TWO
Related literature review
Bifurcation analysis
Consider an ODE that depends on one or more parameters α
x˙ = f(x,α) ,
where, for simplicity, we assume α to be one parameter only. There is the possibility that under variation of α nothing interesting happens to Eqn. (3). There is only a quantitatively different behaviour (shifted equilibria, e.g.). Let us define Eqn. (3) to be structurally stable in the case there are no qualitative changes occurring. However, the ODE might change qualitatively. At that point, bifurcations will have occurred.
Transcritical bifurcation
Let us study the following example
x˙ = x(α − x)
In this case we have two equilibria
x = 0 , x = α ,
of which one is stable and the other one is unstable. The stability can be checked by taking the derivative f ′ (x) = α − 2x. For α < 0 the equilibrium x = 0 is stable, while for α > 0 this equilibrium is unstable. For the non-trivial equilibrium x 6= 0 the opposite is true. Note that the stable equilibrium functions as an attractor, while the unstable equilibrium functions as a separatrix. What happens at α = 0? Note, that we have only one equilibrium x = 0 then. This equilibrium is stable. This point we call a transcritical bifurcation. At this point, the two equilibria exchange stability (see Figure 1). From a biological perspective we associate this behaviour with the invasion of a species x. The transcritical bifurcation is said to be the invasion boundary of this species. For α < 0 the stable equilibrium is zero, which means the species does not survive (for infinite time; it might stick around a while though before it inevitably disappears). For α > 0 the stable equilibrium is larger than zero, which means a population can be maintained always.
CHAPTER ONE
RESEARCH DESIGN AND METHODOLOGY
The researcher will used experimental design, analysis of variance; transformations, model validation and residual analysis. Factorial design with fixed, random and mixed effects. The design was suitable for the study as the study sought to Bifurcation analysis of a mathematical model for malaria transmission under treatment and control of malaria
METHOD OF DATA ANALYSIS
The researcher will employ statistical inference. Statistical inference is a method of making decisions about the parameters of a population, based on random sampling. It helps to assess the relationship between the dependent and independent variables. The purpose of statistical inference to estimate the uncertainty or sample to sample variation.
References
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- Calistus N. Ngonghala et al – Persistent oscillations and backward bifurcation in a malaria model with varying human and mosquito population implications of control. Springer link journal of Mathematical Biology vol. 70, pp 1581 – 1622, 2015.
- Gasper Godson Mwanga – Mathematical modeling and optimal control of malaria PhD thesis presented to Lappeenranta University of Technology, Lappeenranta, Finland, Dec. 2014.
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- Nakul Chitnis – Modeling the spread of malaria (T – 7, Ms B284, Theoretical Division Los Almos National Laboratory Los Almos, Nm 87545).
- Nakul Chitnis et al – Bifurcation analysis of a mathematical model of malaria transmission (2006) SIAM journal of Applied Mathematics vol 67 pp 24 – 45.
- Nakul Chitnis et al, – Determining important parameters in the spread of malaria through the Sensitivity Analysis of a mathematical model (Bulletin of mathematical Biology (2008).
- Niger A.M and Gumel A.B (2008). Mathematical Analysis of the Role of Repeated Exposure on Malarial Transmission Dynamics, Differential Equation and Dynamical Systems. Vol.3,pp 251-287.
- The effect of incidence function in backward bifurcation for malaria model with temporary immunity (Elsevier Mathematical Biosciences vol. 265 pp 47 – 64, July, 2015)
- Understanding malaria – U.S Department of Health and Human Services. National Institute of Allergy and Infectious Diseases NIH publication N0 07 – 7139, Feb. 2007.
- Xiaomei Feng et al – Stability and backward bifurcation in a malaria transmission model with applications to the control of malaria in China 2015 Elsevier Inc.