A New Route to Periodic Oscillations in the Dynamics of Malaria Transmission
Calistus Ngonghala, Mathematics, West Virginia University (August 31, 2011)
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A a new SIS model for malaria that incorporates mosquito demography is developed and studied. This model differs from standard SIS models in that the mosquito population involved in disease transmission (adult female mosquitoes questing for human blood) are identified and accounted for. The main focus of this model is disease control. In the presence of the disease, we identified a trivial steady state solution, a nontrivial disease-free steady state solution and an endemic steady state solution and showed that the endemic steady state solution can be driven to instability via a Hopf bifurcation as a parameter is varied in parameter space. The model therefore captures natural oscillations known to exist in malaria prevalence without recourse to external seasonal forcing and/or delays. Besides the basic reproduction number, we also identified a second threshold parameter that is associated with mosquito demography. These two threshold parameters can be used for purposes of disease control. Analysis of our model also indicates that the basic reproduction number for malaria can be smaller than previously thought and that the model exhibits a backward bifurcation. Hence, simply reducing the basic reproduction number below unity may not be enough for disease eradication. The discovery of oscillatory dynamics and the re-interpretation of the basic reproduction number for malaria presents a novel and plausible framework for developing and implementing control strategies. Model results therefore indicate that accounting for mosquito demography is important in explaining observed patterns in malaria prevalence as well as in designing and evaluating control strategies, especially those interventions that are related to mosquito control.