# Bifurcation analysis of a mathematical model for malaria transmission

Nakul Chitnis, Jim M Cushing, J. M. Hyman

Research output: Contribution to journalArticle

212 Citations (Scopus)

### Abstract

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, 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 asymptotically stable when R 0 < 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 show that for larger values of the disease-induced death rate, a subcritical (backward) bifurcation is possible at R0 = 1.

Original language English (US) 24-45 22 SIAM Journal on Applied Mathematics 67 1 https://doi.org/10.1137/050638941 Published - 2006

### Fingerprint

Malaria
Bifurcation (mathematics)
Bifurcation Analysis
Mathematical Model
Mathematical models
Reproductive number
Transcritical Bifurcation
Backward Bifurcation
Immigration
Endemic Equilibrium
Logistic Model
Population Model
Asymptotically Stable
Equilibrium Point
Ordinary differential equations
Logistics
Ordinary differential equation
Numerical Simulation
Human
Computer simulation

### Keywords

• Bifurcation theory
• Disease-free equilibria
• Endemic equilibria
• Epidemic model
• Malaria
• Reproductive number

### ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics

### Cite this

Bifurcation analysis of a mathematical model for malaria transmission. / Chitnis, Nakul; Cushing, Jim M; Hyman, J. M.

In: SIAM Journal on Applied Mathematics, Vol. 67, No. 1, 2006, p. 24-45.

Research output: Contribution to journalArticle

title = "Bifurcation analysis of a mathematical model for malaria transmission",
abstract = "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, 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 asymptotically stable when R 0 < 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 show that for larger values of the disease-induced death rate, a subcritical (backward) bifurcation is possible at R0 = 1.",
keywords = "Bifurcation theory, Disease-free equilibria, Endemic equilibria, Epidemic model, Malaria, Reproductive number",
author = "Nakul Chitnis and Cushing, {Jim M} and Hyman, {J. M.}",
year = "2006",
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pages = "24--45",
journal = "SIAM Journal on Applied Mathematics",
issn = "0036-1399",
publisher = "Society for Industrial and Applied Mathematics Publications",
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T1 - Bifurcation analysis of a mathematical model for malaria transmission

AU - Chitnis, Nakul

AU - Cushing, Jim M

AU - Hyman, J. M.

PY - 2006

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N2 - 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, 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 asymptotically stable when R 0 < 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 show that for larger values of the disease-induced death rate, a subcritical (backward) bifurcation is possible at R0 = 1.

AB - 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, 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 asymptotically stable when R 0 < 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 show that for larger values of the disease-induced death rate, a subcritical (backward) bifurcation is possible at R0 = 1.

KW - Bifurcation theory

KW - Disease-free equilibria

KW - Endemic equilibria

KW - Epidemic model

KW - Malaria

KW - Reproductive number

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