American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Size-structured population dynamics models and their numerical solutions
  • DCDS-B Home
  • This Issue
  • Next Article
    Convergence analysis of the numerical method for the primitive equations formulated in mean vorticity on a Cartesian grid
November  2004, 4(4): 1173-1202. doi: 10.3934/dcdsb.2004.4.1173

Modelling the dynamics of endemic malaria in growing populations

G.A. Ngwa 1,

1. 

The Abdus Salam International Centre for Theoretical Physics, Trieste 34100, Italy

Received  January 2003 Revised  March 2004 Published  August 2004

A mathematical model for endemic malaria involving variable human and mosquito populations is analysed. A threshold parameter $R_0$ exists and the disease can persist if and only if $R_0$ exceeds $1$. $R_0$ is seen to be a generalisation of the basic reproduction ratio associated with the Ross-Macdonald model for malaria transmission. The disease free equilibrium always exist and is globally stable when $R_0$ is below $1$. A perturbation analysis is used to approximate the endemic equilibrium in the important case where the disease related death rate is nonzero, small but significant. A diffusion approximation is used to approximate the quasi-stationary distribution of the associated stochastic model. Numerical simulations show that when $R_0$ is distinctly greater than $1$, the endemic deterministic equilibrium is globally stable. Furthermore, in quasi-stationarity, the stochastic process undergoes oscillations about a mean population whose size can be approximated by the stable endemic deterministic equilibrium.
Keywords: disease related deaths, Threshold parameter, diffusion approximation..
Mathematics Subject Classification: 92D30, 60J2.
Citation: 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
[1]

Kazuo Aoki, Pierre Charrier, Pierre Degond. A hierarchy of models related to nanoflows and surface diffusion. Kinetic & Related Models, 2011, 4 (1) : 53-85. doi: 10.3934/krm.2011.4.53
[2]

Vladimir Gaitsgory, Tanya Tarnopolskaya. Threshold value of the penalty parameter in the minimization of $L_1$-penalized conditional value-at-risk. Journal of Industrial & Management Optimization, 2013, 9 (1) : 191-204. doi: 10.3934/jimo.2013.9.191
[3]

Jakub Cupera. Diffusion approximation of neuronal models revisited. Mathematical Biosciences & Engineering, 2014, 11 (1) : 11-25. doi: 10.3934/mbe.2014.11.11
[4]

Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191
[5]

Wenzhang Huang, Maoan Han, Kaiyu Liu. Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Mathematical Biosciences & Engineering, 2010, 7 (1) : 51-66. doi: 10.3934/mbe.2010.7.51
[6]

Dominique Duncan, Thomas Strohmer. Classification of Alzheimer's disease using unsupervised diffusion component analysis. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1119-1130. doi: 10.3934/mbe.2016033
[7]

W. E. Fitzgibbon, J. J. Morgan. Analysis of a reaction diffusion model for a reservoir supported spread of infectious disease. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6239-6259. doi: 10.3934/dcdsb.2019137
[8]

José-Francisco Rodrigues, Lisa Santos. On a constrained reaction-diffusion system related to multiphase problems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 299-319. doi: 10.3934/dcds.2009.25.299
[9]

Liviu I. Ignat, Ademir F. Pazoto. Large time behaviour for a nonlocal diffusion - convection equation related with gas dynamics. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3575-3589. doi: 10.3934/dcds.2014.34.3575
[10]

Siwei Duo, Hong Wang, Yanzhi Zhang. A comparative study on nonlocal diffusion operators related to the fractional Laplacian. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 231-256. doi: 10.3934/dcdsb.2018110
[11]

Stephen Thompson, Thomas I. Seidman. Approximation of a semigroup model of anomalous diffusion in a bounded set. Evolution Equations & Control Theory, 2013, 2 (1) : 173-192. doi: 10.3934/eect.2013.2.173
[12]

Razvan C. Fetecau, Mitchell Kovacic, Ihsan Topaloglu. Swarming in domains with boundaries: Approximation and regularization by nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1815-1842. doi: 10.3934/dcdsb.2018238
[13]

Massimiliano Tamborrino. Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity. Mathematical Biosciences & Engineering, 2016, 13 (3) : 613-629. doi: 10.3934/mbe.2016011
[14]

Masaki Kurokiba, Toshitaka Nagai, T. Ogawa. The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system. Communications on Pure & Applied Analysis, 2006, 5 (1) : 97-106. doi: 10.3934/cpaa.2006.5.97
[15]

Huimin Liang, Peixuan Weng, Yanling Tian. Threshold asymptotic behaviors for a delayed nonlocal reaction-diffusion model of mistletoes and birds in a 2D strip. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1471-1495. doi: 10.3934/cpaa.2016.15.1471
[16]

Cyrill B. Muratov, Xing Zhong. Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 915-944. doi: 10.3934/dcds.2017038
[17]

Elio E. Espejo, Masaki Kurokiba, Takashi Suzuki. Blowup threshold and collapse mass separation for a drift-diffusion system in space-dimension two. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2627-2644. doi: 10.3934/cpaa.2013.12.2627
[18]

Kunimochi Sakamoto. Destabilization threshold curves for diffusion systems with equal diffusivity under non-diagonal flux boundary conditions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 641-654. doi: 10.3934/dcdsb.2016.21.641
[19]

Mostafa Bendahmane, Mauricio Sepúlveda. Convergence of a finite volume scheme for nonlocal reaction-diffusion systems modelling an epidemic disease. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 823-853. doi: 10.3934/dcdsb.2009.11.823
[20]

Naoufel Ben Abdallah, Raymond El Hajj. Diffusion and guiding center approximation for particle transport in strong magnetic fields. Kinetic & Related Models, 2008, 1 (3) : 331-354. doi: 10.3934/krm.2008.1.331

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (25)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences