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Kapitanov, G., Alvey, C., VogtGeisse, K., & Feng, Z. L. (2015). An AgeStructured Model For The Coupled Dynamics Of Hiv And Hsv2. Math. Biosci. Eng., 12(4), 803–840.
Abstract: Evidence suggests a strong correlation between the prevalence of HSV2 (genital herpes) and the perseverance of the HIV epidemic. HSV2 is an incurable viral infection, characterized by periodic reactivation. We construct a model of the coinfection dynamics between the two diseases by incorporating a timesinceinfection variable to track the alternating periods of infectiousness of HSV2. The model considers only heterosexual relationships and distinguishes three population groups: males, general population females, and female sex workers. We calculate the basic reproduction numbers for each disease that provide threshold conditions, which determine whether a disease dies out or becomes endemic in the absence of the other disease. We also derive the invasion reproduction numbers that determine whether or not a disease can invade into a population in which the other disease is endemic. The calculations of the invasion reproduction numbers suggest a new aspect in their interpretation – the class from which the initial disease carrier arises is important for understanding the invasion dynamics and biological interpretation of the expressions of the reproduction numbers. Sensitivity analysis is conducted to examine the role of model parameters in influencing the model outcomes. The results are discussed in the last section.

Towers, S., Vogt Geisse, K., ChiaChun, T., Han, Q., & Feng, Z. L. (2012). The Impact Of School Closures On Pandemic Influenza: Assessing Potential Repercussions Using A Seasonal Sir Model. Math. Biosci. Eng., 9(2), 413–430.
Abstract: When a new pandemic influenza strain has been identified, massproduction of vaccines can take several months, and antiviral drugs are expensive and usually in short supply. Social distancing measures, such as school closures, thus seem an attractive means to mitigate disease spread. However, the transmission of influenza is seasonal in nature, and as has been noted in previous studies, a decrease in the average transmission rate in a seasonal disease model may result in a larger final size. In the studies presented here, we analyze a hypothetical pandemic using a SIR epidemic model with time and agedependent transmission rates; using this model we assess and quantify, for the first time, the the effect of the timing and length of widespread school closures on influenza pandemic final size and average peak time. We find that the effect on pandemic progression strongly depends on the timing of the start of the school closure. For instance, we determine that school closures during a late spring wave of an epidemic can cause a pandemic to become up to 20% larger, but have the advantage that the average time of the peak is shifted by up to two months, possibly allowing enough time for development of vaccines to mitigate the larger size of the epidemic. Our studies thus suggest that when heterogeneity in transmission is a significant factor, decisions of public health policy will be particularly important as to how control measures such as school closures should be implemented.
Keywords: Pandemic influenza; epidemic model; dynamic systems

VogtGeisse, K., Lorenzo, C., & Feng, Z. L. (2013). Impact Of AgeDependent Relapse And Immunity On Malaria Dynamics. J. Biol. Syst., 21(4), 49 pp.
Abstract: An agestructured mathematical model for malaria is presented. The model explicitly includes the human and mosquito populations, structured by chronological age of humans. The infected human population is divided into symptomatic infectious, asymptomatic infectious and asymptomatic chronic infected individuals. The original partial differential equation (PDE) model is reduced to an ordinary differential equation (ODE) model with multiple age groups coupled by aging. The basic reproduction number R0 is derived for the PDE model and the age group model in the case of general n age groups. We assume that infectiousness of chronic infected individuals gets triggered by bites of even susceptible mosquitoes. Our analysis points out that this assumption contributes greatly to the R0 expression and therefore needs to be further studied and understood. Numerical simulations for n = 2 age groups and a sensitivity/uncertainty analysis are presented. Results suggest that it is important not only to consider asymptomatic infectious individuals as a hidden cause for malaria transmission, but also asymptomatic chronic infections (>60%), which often get neglected due to undetectable parasite loads. These individuals represent an important reservoir for future human infectiousness. By considering agedependent immunity types, the model helps generate insight into effective control measures, by targeting age groups in an optimal way.

VogtGeisse, K., Ngonghala, C. N., & Feng, Z. L. (2020). The Impact Of Vaccination On Malaria Prevalence: A VaccineAgeStructured Modeling Approach. J. Biol. Syst., 28(2), 475–513.
Abstract: A deterministic model for the effects on disease prevalence of the most advanced preerythrocytic vaccine against malaria is proposed and studied. The model includes two vaccinated classes that correspond to initially vaccinated and booster dose vaccinated individuals. These two classes are structured by timesinceinitialvaccination (vaccineage). This structure is a novelty for vectorhost models; it allows us to explore the effects of parameters that describe timed and delayed delivery of a booster dose, and immunity waning on disease prevalence. Incorporating two vaccinated classes can predict more accurately threshold vaccination coverages for disease eradication under multidose vaccination programs. We derive a vaccineagestructured control reproduction number R and establish conditions for the existence and stability of equilibria to the system. The model is bistable when R < 1. In particular, it exhibits a backward (subcritical) bifurcation, indicating that R = 1 is no longer the threshold value for disease eradication. Thus, to achieve eradication we must identify and implement control measures that will reduce R to a value smaller than unity. Therefore, it is crucial to be cautious when using R to guide public health policy, although it remains a key quantity for decision making. Our results show that if the booster vaccine dose is administered with delay, individuals may not acquire its full protective effect, and that incorporating waning efficacy into the system improves the accuracy of the model outcomes. This study suggests that it is critical to follow vaccination schedules closely, and anticipate the consequences of delays in those schedules.
