Modeling and Stability Analysis of Host-parasite Population Dynamics
Mathematical Modelling and Applications
Volume 5, Issue 2, June 2020, Pages: 118-128
Received: Jan. 1, 2020; Accepted: May 5, 2020; Published: May 28, 2020
Views 68      Downloads 35
Geremew Kenassa Edessa, Department of Mathematics, Wollega University, Nekemte, Ethiopia
Purnachandra Rao Koya, Department of Mathematics, Wollega University, Nekemte, Ethiopia
Article Tools
Follow on us
In this study, a mathematical model is developed to explore the population dynamics of two host species. Both the hosts depend on the same resources and the availability of such resources is limited in nature. If the host populations increase abnormally the limited natural resources will be used up. Hence, the concept of parasite is brought in to the picture to regulate the host populations. The parasite is a mechanism that reduces the host populations. However, on one hand if the parasite attacks more the hosts may extinct and on the other hand if the parasite do not attack then the host populations may increase and resource may be used up. Hence, the parasite is expected to maintain a balance so that neither the host populations nor the resources extinct. Here, both the hosts are classified in to susceptible and infected and hence the model comprises of four populations: Susceptible Host–1, Infected Host–1, Susceptible Host–2 and Infected Host–2. Thus, the mathematical model comprises of a system of four first order non-linear ordinary differential equations. Mathematical analysis of the model is conducted. Positivity and boundedness of the solution have been verified and thus shown that the model is physically meaningful and biologically acceptable. Equilibrium points of the model are identified and stability analysis is conducted. Simulation study is conducted in order to support the mathematical analysis using software packages Mat lab and DeDiscover.
Modeling, Hosts, Parasite-mediated Interactions, Stability, Numerical Simulation
To cite this article
Geremew Kenassa Edessa, Purnachandra Rao Koya, Modeling and Stability Analysis of Host-parasite Population Dynamics, Mathematical Modelling and Applications. Vol. 5, No. 2, 2020, pp. 118-128. doi: 10.11648/
Copyright © 2020 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
AlyssaLois M. (2016). Non-native parasite enhances susceptibility of host to native predators, Springer-Verlag, Berlin, Heidelberg.
Anderson R. M., May R. M. (1981). The population dynamics of micro-parasites and their invertebrate hosts. Phil. Trans. R. Soc. Lond. 291, 451–524.
Anderson, R. and May R. Infectious Diseases of Humans, Dynamics and Control. Oxford University Press, Oxford (1991).
Benjamin J. Z., Buckling A. (2012). The mode of host–parasite interaction shapes co-evolutionary dynamics and the fate of host cooperation. Proc. R. Soc. Lond., B 279, 3742–48.
Bowers R. G., Boots M., Begon M. (1994). Life-history trade-offs and the evolution of pathogen resistance: competition between host strain. Proc. R. Soc. Lond., B 257, 247–53.
Chakra M. A., Hilbe C. and Treutlen A.(2014). Plastic behaviors in hosts promote the emergence of retaliatory parasites, Sci. Rep., 4, 4251.
D. Adak and N. Bairagi. (2014). Dynamical behavior of a predator-prey-parasite model with nonlinear incidence rate, J. Biol. Syst., 1 (1).
Diekmann O., Heesterbeek J. A. P. (2000). Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley, New York, p. 365.
Dobson A. P. (2004). Population dynamics of pathogens with multiple host species. Am. Nat., 164, S64–S78.
Fanghong Zhang and CunchengJin. (2017). Analysis of an eco-epidemiological model with Disease in the prey and predator. 6 (1), 22-28.
GeremewKenassa Edessa, BokaKumsa, Purnachandra Rao Koya. (2018) Modeling and Simulation Study of the PopulationDynamics of Commensal-Host-Parasite System. American Journal of Applied Mathematics. Vol. 6, No. 3, pp. 97-108.
GeremewKenassa Edessa, BokaKumsa, Purnachandra Rao Koya (2018). Dynamical behavior of Susceptible prey–Infected prey–Predator Populations. IOSR Journal of Mathematics (IOSR-JM) 14.4 PP: 31-41.
Hatcher P. E., Moore J., Taylor J. E., Tinney G. W. and Paul N. D. (2004). Phytohormones and plant–herbivore–pathogen interactions: integrating the molecular with the ecological. Ecology, 85, 59–69.
Hatcher P. E. and Paul N. D. (2000) Beetle grazing reduce natural infection of Rumexobtusifoliusby fungal pathogens, New Phytol, 146, 325–33.
Hatcher P. E., Paul N. D., Ayres P. G. & Whittaker J. B. (1994). The effect of an insect herbivore and a rust fungus individually and combined in sequence on the growth of two Rumex species. New Phytol, 128, 71–78.
Holt R. D. & Dobson A. P. (2006). Extending the principles of community ecology to address the epidemiology of host pathogen systems. In: Disease Ecology: Community Structure and Pathogen Dynamics. Oxford University Press, Oxford, pp. 6–27.
Holt R. D. & Hochberg M. E. (1998). The coexistence of competing parasites. Part II–Hyper parasitism and food chain dynamics. J. Theor. Biol., 193, 485–95.
Holt R. D. (1977) Predation, apparent competition, and the structure of prey communities. Theoretical Population Biology, 12: 197–229.
Rainey P. B. (2002). Antagonistic co-evolution between a bacterium and a bacterio phage. Proc. R. Soc. Lond. B 269, 931–36.
Sand land G. J., Minchella D. J. (2004). Life-history plasticity in hosts exposed to differing resources and parasitism. Can. J. Zool., 82, 1672–77.
Sen P., Das k. (2017). Simultaneous Effects of Prey Defence and Predator Infection on a Predator Prey System. Ann. Bio. Sci., 5 (1): 37-46.
Siekmann. I., (2013). On competition in ecology, epidemiology and eco - epidemiology. Ecol. Complex,
Sule H., MuhamadR., and Omar D. (2014). Parasitism rate host stage preference and functional response of Tamarixiaradiata on Diaphorinacitri. Int. J. Agric. Biol. 16, 783–88.
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
Tel: (001)347-983-5186