### 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, 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.

Original language | English (US) |
---|---|

Pages (from-to) | 24-45 |

Number of pages | 22 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 67 |

Issue number | 1 |

DOIs | |

State | Published - 2006 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Applied Mathematics*,

*67*(1), 24-45. https://doi.org/10.1137/050638941

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

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 67, no. 1, pp. 24-45. https://doi.org/10.1137/050638941

}

TY - JOUR

T1 - Bifurcation analysis of a mathematical model for malaria transmission

AU - Chitnis, Nakul

AU - Cushing, Jim M

AU - Hyman, J. M.

PY - 2006

Y1 - 2006

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

UR - http://www.scopus.com/inward/record.url?scp=33847746157&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33847746157&partnerID=8YFLogxK

U2 - 10.1137/050638941

DO - 10.1137/050638941

M3 - Article

VL - 67

SP - 24

EP - 45

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 1

ER -