American Institute of Mathematical Sciences

Advanced
2008, 5(4): 789-801. doi: 10.3934/mbe.2008.5.789

A malaria model with partial immunity in humans

Jia Li 1,

1. 

Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899

Received  December 2007 Revised  February 2008 Published  October 2008

In this paper, we formulate a mathematical model for malaria transmission that includes incubation periods for both infected human hosts and mosquitoes. We assume humans gain partial immunity after infection and divide the infected human population into subgroups based on their infection history. We derive an explicit formula for the reproductive number of infection, $R_0$, to determine threshold conditions whether the disease spreads or dies out. We show that there exists an endemic equilibrium if $R_0>1$. Using an numerical example, we demonstrate that models having the same reproductive number but different numbers of progression stages can exhibit different transient transmission dynamics.
Keywords: malaria, endemic equilibrium, compartmental models, reproductive number.
Mathematics Subject Classification: 34D20, 34D23, 92D30.
Citation: Jia Li. A malaria model with partial immunity in humans. Mathematical Biosciences & Engineering, 2008, 5 (4) : 789-801. doi: 10.3934/mbe.2008.5.789
[1]

Timothy C. Reluga, Jan Medlock, Alison Galvani. The discounted reproductive number for epidemiology. Mathematical Biosciences & Engineering, 2009, 6 (2) : 377-393. doi: 10.3934/mbe.2009.6.377
[2]

G.A. Ngwa. Modelling the dynamics of endemic malaria in growing populations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1173-1202. doi: 10.3934/dcdsb.2004.4.1173
[3]

Burcu Adivar, Ebru Selin Selen. Compartmental disease transmission models for smallpox. Conference Publications, 2011, 2011 (Special) : 13-21. doi: 10.3934/proc.2011.2011.13
[4]

Ariel Cintrón-Arias, Carlos Castillo-Chávez, Luís M. A. Bettencourt, Alun L. Lloyd, H. T. Banks. The estimation of the effective reproductive number from disease outbreak data. Mathematical Biosciences & Engineering, 2009, 6 (2) : 261-282. doi: 10.3934/mbe.2009.6.261
[5]

Andrea Franceschetti, Andrea Pugliese, Dimitri Breda. Multiple endemic states in age-structured $SIR$ epidemic models. Mathematical Biosciences & Engineering, 2012, 9 (3) : 577-599. doi: 10.3934/mbe.2012.9.577
[6]

Renhao Cui. Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2997-3022. doi: 10.3934/dcdsb.2020217
[7]

Qixuan Wang, Hans G. Othmer. The performance of discrete models of low reynolds number swimmers. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1303-1320. doi: 10.3934/mbe.2015.12.1303
[8]

Pierre Monmarché. Hypocoercive relaxation to equilibrium for some kinetic models. Kinetic & Related Models, 2014, 7 (2) : 341-360. doi: 10.3934/krm.2014.7.341
[9]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[10]

Elvio Accinelli, Bruno Bazzano, Franco Robledo, Pablo Romero. Nash Equilibrium in evolutionary competitive models of firms and workers under external regulation. Journal of Dynamics & Games, 2015, 2 (1) : 1-32. doi: 10.3934/jdg.2015.2.1
[11]

Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260
[12]

Rafael Granero-Belinchón, Martina Magliocca. Global existence and decay to equilibrium for some crystal surface models. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2101-2131. doi: 10.3934/dcds.2019088
[13]

Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021007
[14]

E. Almaraz, A. Gómez-Corral. On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2153-2176. doi: 10.3934/dcdsb.2018229
[15]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595
[16]

Hui Wan, Jing-An Cui. A model for the transmission of malaria. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 479-496. doi: 10.3934/dcdsb.2009.11.479
[17]

Wen Jin, Horst R. Thieme. Persistence and extinction of diffusing populations with two sexes and short reproductive season. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3209-3218. doi: 10.3934/dcdsb.2014.19.3209
[18]

Dimitri Breda, Stefano Maset, Rossana Vermiglio. Numerical recipes for investigating endemic equilibria of age-structured SIR epidemics. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2675-2699. doi: 10.3934/dcds.2012.32.2675
[19]

Hassan Tahir, Asaf Khan, Anwarud Din, Amir Khan, Gul Zaman. Optimal control strategy for an age-structured SIR endemic model. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021054
[20]

Huseyin Coskun. Nonlinear decomposition principle and fundamental matrix solutions for dynamic compartmental systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6553-6605. doi: 10.3934/dcdsb.2019155

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences