Transmission Model for Plasmodium Vivax Malaria: Conditions for Bifurcation
Plasmodium vivax malaria differs from P. falciparum malaria in that a person suffering from P. vivax infection can suffer relapses of the disease. This is due the parasite being able to remain dormant in the liver of the patients where it is able to re-infect the patient after a passage of time. During this stage, the patient is classified as being in the dormant class. The model to describe the transmission of P. vivax malaria consists of a human population divided into four classes, the susceptible, the infected, the dormant and the recovered. The effect of a time delay on the transmission of this disease is studied. The time delay is the period in which the P. vivax parasite develops inside the mosquito (vector) before the vector becomes infectious (i.e., pass on the infection). We analyze our model by using standard dynamic modeling method. Two stable equilibrium states, a disease free state E0 and an endemic state E1, are found to be possible. It is found that the E0 state is stable when a newly defined basic reproduction number G is less than one. If G is greater than one the endemic state E1 is stable. The conditions for the endemic equilibrium state E1 to be a stable spiral node are established. For realistic values of the parameters in the model, it is found that solutions in phase space are trajectories spiraling into the endemic state. It is shown that the limit cycle and chaotic behaviors can only be achieved with unrealistic parameter values.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1327698Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1877
 PCC. Garnhan., Malaria parasites of man: life-cycles and morphology (excluding unltrastructure) IN W.H. Wernsdorfer and I. McGregor (Eds), Malaria, Churchill Livingstone, Edinburg, 1988.
 WHO, World Malaria Situation in 1994: Weekly Epidemiological Record, Geneva, 1997.
 B. Sina, "Focus on Plasmodium Vivax," Trends in Parasitology., vol.18, pp.287-289, 2002.
 R.M. Anderson, and R.M. May, Infectious Disease of Humans., Dynamics and Control.Oxford U. Press, Oxford, 1991.
 A. Kammanee, N. Kanyamee, and I.M. Tang , "Basic Reproduction Number for the Transmission of Plasmodium Vivax Malaria," Southeast Asian Journal of Tropical Medicine and Public Health ., vol. 32, pp.702-706, 2001.
 R. Ross, The preventation of Malaria, 2th ed. Murray, London, 1911.
 G. MacDonald, The epidemiology and control of malaria. Oxford University Press, London, 1957.
 L. Esteva, and C. Vargus, "Analysis of a dengue disease trasmission model," Mathematical Bioscience., vol. 150, pp.131-151, 1998.
 J.E. Marsden, and M. McCracken, The Hopf Bifurcation and its application. Springer-Verlag: New York, 1976.
 S.Ruan, and J.Wei, "On the zeros of a third degree exponential polynomial with application to a delay model for control of testosterone secretion," IMA J. Math. Appl. Med. Biol., vol. 18, pp. 41-52, 2001.
 Q.L.A. Khan, and D. Greenhalgh, "Hopf bifurcation in epidemic models with a time delay in vaccination. IMA," J. Math. Appl. Med. Bio., vol. 16, pp. 113-142, 1998.
 J. Tam, "Delay effect in a model for virus replication.," IMA J. Math. Appl. Med. Biol., vol. 16, no. 29, 1999.