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We develop and analyze a stage-progression compartmental model to study the emerging invasive nontyphoidal Salmonella (iNTS) epidemic in sub-Saharan Africa. iNTS bloodstream infections are often fatal, and the diverse and non-specific clinical features of iNTS make it difficult to diagnose. We focus our study on identifying approaches that can reduce the incidence of new infections. In sub-Saharan Africa, transmission and mortality are correlated with the ongoing HIV epidemic and severe malnutrition. We use our model to quantify the impact that increasing antiretroviral therapy (ART) for HIV infected adults and reducing malnutrition in children would have on mortality from iNTS in the population. We consider immunocompromised subpopulations in the region with major risk factors for mortality, such as malaria and malnutrition among children and HIV infection and ART coverage in both children and adults. We parameterize the progression rates between infection stages using the branching probabilities and estimated time spent at each stage. We interpret the basic reproduction number R0 as the total contribution from an infinite infection loop produced by the asymptomatic carriers in the infection chain. The results indicate that the asymptomatic HIV+ adults without ART serve as the driving force of infection for the iNTS epidemic. We conclude that the worst disease outcome is among the pediatric population, which has the highest infection rates and death counts. Our sensitivity analysis indicates that the most effective strategies to reduce iNTS mortality in the studied population are to improve the ART coverage among high-risk HIV+ adults and reduce malnutrition among children.

We develop a mathematical model for transmission of West Nile virus (WNV) that incorporates resident and migratory host avian populations and a mosquito vector population. We provide a detailed analysis of the model's basic reproductive number and demonstrate how the exposed infected, but not infectious, state for the bird population can be approximated by a reduced model. We use the model to investigate the interplay of WNV in both resident and migratory bird hosts. The resident host parameters correspond to the American Crow (Corvus brachyrhynchos), a competent host with a high death rate due to disease, and migratory host parameters to the American Robin (Turdus migratorius), a competent host with low WNV death rates. We find that yearly seasonal outbreaks depend primarily on the number of susceptible migrant birds entering the local population each season. We observe that the early growth rates of seasonal outbreaks is more influenced by the the migratory population than the resident bird population. This implies that although the death of highly competent resident birds, such as American Crows, are good indicators for the presence of the virus, these species have less impact on the basic reproductive number than the competent migratory birds with low death rates, such as the American Robins. The disease forecasts are most sensitive to the assumptions about the feeding preferences of North American mosquito vectors and the effect of the virus on the hosts. Increased research on the these factors would allow for better estimates of these important model parameters, which would improve the quality of future WNV forecasts.

The spread of an infectious disease is sensitive to the contact patterns in the population and to precautions people take to reduce the transmission of the disease. We investigate the impact that different mixing assumptions have on the spread an infectious disease in an age-structured ordinary differential equation model. We consider the impact of heterogeneity in susceptibility and infectivity within the population on the disease transmission. We apply the analysis to the spread of a smallpox-like disease, derive the formula for the reproduction number, R0, and based on this threshold parameter, show the level of human behavioral change required to control the epidemic. We analyze how different mixing patterns can affect the disease prevalence, the cumulative number of new infections, and the final epidemic size. Our analysis indicates that the combination of residual immunity and behavioral changes during a smallpox-like disease outbreak can play a key role in halting infectious disease spread; and that realistic mixing patterns must be included in the epidemic model for the predictions to accurately reflect reality.

We formulate differential susceptibility and differential infectivity models for disease transmission in this paper. The susceptibles are divided into n groups based on their susceptibilities, and the infectives are divided into m groups according to their infectivities. Both the standard incidence and the bilinear incidence are considered for different diseases. We obtain explicit formulas for the reproductive number. We define the reproductive number for each subgroup. Then the reproductive number for the entire population is a weighted average of those reproductive numbers for the subgroups. The formulas for the reproductive number are derived from the local stability of the infection-free equilibrium. We show that the infection-free equilibrium is globally stable as the reproductive number is less than one for the models with the bilinear incidence or with the standard incidence but no disease-induced death. We then show that if the reproductive number is greater than one, there exists a unique endemic equilibrium for these models. For the general cases of the models with the standard incidence and death, conditions are derived to ensure the uniqueness of the endemic equilibrium. We also provide numerical examples to demonstrate that the unique endemic equilibrium is asymptotically stable if it exists.

Mathematical models of HIV-1 infection can help interpret drug treatment experiments and improve our understanding of the interplay between HIV-1 and the immune system. We develop and analyze an age-structured model of HIV-1 infection that allows for variations in the death rate of productively infected T cells and the production rate of viral particles as a function of the length of time a T cell has been infected. We show that this model is a generalization of the standard differential equation and of delay models previously used to describe HIV-1 infection, and provides a means for exploring fundamental issues of viral production and death. We show that the model has uninfected and infected steady states, linked by a transcritical bifurcation. We perform a local stability analysis of the nontrivial equilibrium solution and provide a general stability condition for models with age structure. We then use numerical methods to study solutions of our model focusing on the analysis of primary HIV infection. We show that the time to reach peak viral levels in the blood depends not only on initial conditions but also on the way in which viral production ramps up. If viral production ramps up slowly, we find that the time to peak viral load is delayed compared to results obtained using the standard (constant viral production) model of HIV infection. We find that data on viral load changing over time is insufficient to identify the functions specifying the dependence of the viral production rate or infected cell death rate on infected cell age. These functions must be determined through new quantitative experiments.