Modeling Mosquito-borne Diseases: An adventure through ODEs and integral equations

In this work, we consider two types of mosquito-born disease modeling. First, we develop an ordinary differential equation (ODE) compartment model in order to investigate the effects of treatment on the spread of drug-resistant malaria. In order to investigate drug-resistance, we incorporate a drug-sensitive and drug-resistant parasite strain into a vector-borne disease model with treatment. In particular, we calculate reproduction and invasion numbers that will inform disease outcome and strain competition to be able to inform public health policies. We investigate the parameters associated with treatment in order to make recommendations for public health guidelines in the fight against malaria. Our results indicate that long-term population-level disease dynamics are insensitive to the effects of drug concentration in the blood on human susceptibility. That is, we see no correlation between pharmacokinetics and number of infections. On the other hand, we found that the parasite density, pharmacodynamics, in the human bloodstream significantly effects the magnitude of disease prevalence. We next investigate compartment stage durations. Many epidemiological ODE models inherently assume that the waiting times for each of the disease stages are exponentially distributed, which simplifies the model formulation and its analysis. However, this is not always the correct assumption and many methods have been developed to account for more biologically realistic waiting time distributions for each stage. We use malaria as a guiding example and formulate a two-strain vector-borne disease integral equation model with general waiting time distributions in order to more accurately capture the timing of within-human and between-human disease dynamics with treatment. We develop a novel numerical algorithm in order to simulate our model. Our results indicate that incorporating different assumptions on waiting times in each disease class significantly impact the outcome of disease persistence in populations.