Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission

• Published in: Mathematical biosciences Vol. 240; no. 1; pp. 45 - 62
• Main Authors: Ngonghala, Calistus N., Ngwa, Gideon A., Teboh-Ewungkem, Miranda I.
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 01.11.2012
• Subjects: Animals; Backward bifurcation; Computer Simulation; Culicidae - parasitology; Hopf bifurcation; Humans; Insect Vectors - parasitology; Malaria - epidemiology; Malaria - parasitology; Malaria - transmission; Malaria control; Models, Biological; Oscillations; Vector demography and life style; Vector demography and life style; Oscillations; Hopf bifurcation; Backward bifurcation; Malaria control
• ISSN: 0025-5564; 1879-3134
• DOI: 10.1016/j.mbs.2012.06.003

► We present a new framework for modeling malaria that emphasizes vector demography. ► Endemic steady state solutions can be driven to instability via a Hopf bifurcation. ► Oscillations in the SIS malaria model obtained without delay or external forcing. ► Two threshold parameters-one linked to the vector, the other to the disease. ► Our model exhibits the phenomenon of backward bifurcation. A deterministic ordinary differential equation model for the dynamics of malaria transmission that explicitly integrates the demography and life style of the malaria vector and its interaction with the human population is developed and analyzed. The model is different from standard malaria transmission models in that the vectors involved in disease transmission are those that are questing for human blood. Model results indicate the existence of nontrivial disease free and endemic steady states, which can be driven to instability via a Hopf bifurcation as a parameter is varied in parameter space. Our model therefore captures oscillations that are known to exist in the dynamics of malaria transmission without recourse to external seasonal forcing. Additionally, our model exhibits the phenomenon of backward bifurcation. Two threshold parameters that can be used for purposes of control are identified and studied, and possible reasons why it has been difficult to eradicate malaria are advanced.