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Prions are proteins that cause fatal neurodegenerative diseases. The misfolded conformation adopted by prions can be transmitted to other normally folded proteins. Therapeutics to stop prion proliferation have been studied experimentally; however, it is not clear how the combination of different types of treatments can decrease the growth rate of prions in the brain. In this article, we combine the implementation of pharmacological chaperones and interferons to develop a novel model using a non-linear system of ordinary differential equations and study the quantitative effects of these two treatments on the growth rate of prions. This study aims to identify how the two treatments affect prion proliferation, both individually and in tandem. We analyze the model, and qualitative global results on the disease-free and disease equilibria are proved analytically. Numerical simulations, using parameter values from in vivo experiments that provide a pharmaceutically important demonstration of the effects of these two treatments, are presented here. This mathematical model can be used to identify and optimize the best combination of the treatments within their safe ranges.

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Lyme disease is one of the most prevalent and fastest growing vector-borne bacterial illnesses in the United States, with over 25,000 new confirmed cases every year. Humans contract the bacterium Borrelia burgdorferi through the bite of the tick Ixodes scapularis. The tick can receive the bacterium from a variety of small mammal and bird species, but the white-footed mouse Peromyscus leucopus is the primary reservoir in the northeastern United States, especially near human settlement. The tick's life cycle and behavior depend greatly on the season, with different stages of tick biting at different times. Reducing the infection in the tick-mouse cycle may greatly lower human Lyme incidence in some areas. However, research on the effects of various mouse-targeted interventions is limited. One particularly promising method involves administering vaccine pellets to white-footed mice through special bait boxes. In this study, we develop and analyze a mathematical model consisting of a system of nonlinear difference equations to understand the complex transmission dynamics and vector demographics in both tick and mice populations. We evaluate to what extent vaccination of white-footed mice can affect Lyme incidence in I. scapularis, and under which conditions this method saves money in preventing Lyme disease. We find that, in areas with high human risk, vaccination can eliminate mouse-tick transmission of B. burgdorferi while saving money.

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Zika virus (ZIKV) is a vector-borne disease that has rapidly spread during the year 2016 in more than 50 countries around the world. If a woman is infected during pregnancy, the virus can cause severe birth defects and brain damage in their babies. The virus can be transmitted through the bites of infected mosquitoes as well as through direct contact from human to human (e.g., sexual contact and blood transfusions). As an intervention for controlling the spread of the disease, we study a vaccination model for preventing Zika infections. Although there is no formal vaccine for ZIKV, The National Institute of Allergy and Infectious Diseases (part of the National Institutes of Health) has launched a vaccine trial at the beginning of August 2016 to control ZIKV transmission, patients who received the vaccine are expected to return within 44 weeks to determine if the vaccine is safe. Since it is important to understand ZIKV dynamics under vaccination, we formulate a vaccination model for ZIKV spread that includes mosquito as well as sexual transmission. We calculate the basic reproduction number of the model to analyze the impact of relatively, perfect and imperfect vaccination rates. We illustrate several numerical examples of the vaccination model proposed as well as the impact of the basic reproduction numbers of vector and sexual transmission and the effect of vaccination effort on ZIKV spread. Results show that high levels of sexual transmission create larger cases of infection associated with the peak of infected humans arising in a shorter period of time, even when a vaccine is available in the population. However, a high level of transmission of Zika from vectors to humans compared with sexual transmission represents that ZIKV will take longer to invade the population providing a window of opportunities to control its spread, for instance, through vaccination.

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We create and analyze a mathematical model to estimate the impact of condom-use and sexual behavior on the prevalence and spread of Sexually Transmitted Infections (STIs). STIs remain a significant public health challenge globally with a high burden of some Sexually Transmitted Diseases (STDs) in both developed and undeveloped countries. Although condom-use is known to reduce the transmission of STIs, there are a few quantitated population-based studies on the protective role of condom-use in reducing the incidence of STIs. The number of concurrent partners is correlated with their risk of being infectious by a STI such as chlamydia, gonorrhea, or syphilis. We define a Susceptible-Infectious-Susceptible (SIS) model that distributes the population by the number of concurrent partners. The model captures the multi-level heterogeneous mixing through a combination of biased (preferential) and random mixing between individuals with different risks, and accounts for differences in condom-use in the low- and high-risk populations. We use sensitivity analysis to assess the relative impact of high-risk people using condom as a prophylactic to reduce their chance of being infectious, or infecting others. The model predicts the STI prevalence as a function of the number of partners that a person has, and quantifies how this distribution changes as a function of condom-use. Our results show that when the mixing is random, then increasing the condom-use in the high-risk population is more effective in reducing the prevalence than when many of the partners of high-risk people have high risk. The model quantified how the risk of being infected increases for people who have more partners, and and the need for high-risk people to consistently use condoms to reduce their risk of infection.

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The recent H1N1 ("swine flu") pandemic and recent H5N1 ("avian flu") outbreaks have brought increased attention to the study of the role of animal populations as reservoirs for pathogens that could invade human populations. It is believed that pigs acquired flu strains from birds and humans, acting as a mixing vessel in generating new influenza viruses. Assessing the role of animal reservoirs, particularly reservoirs involving highly mobile populations (like migratory birds), on disease dispersal and persistence is of interests to a wide range of researchers including public health experts and evolutionary biologists. This paper studies the interactions between transient and resident bird populations and their role on dispersal and persistence. A metapopulation framework based on a system of nonlinear ordinary differential equations is used to study the transmission dynamics and control of avian diseases. Simplified versions of mathematical models involving a limited number of migratory and resident bird populations are analyzed. Epidemiological time scales and singular perturbation methods are used to reduce the dimensionality of the model. Our results show that mixing of bird populations (involving residents and migratory birds) play an important role on the patterns of disease spread.

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Ecstasy has gained popularity among young adults who frequent raves and nightclubs. The Drug Enforcement Administration reported a 500 percent increase in the use of ecstasy between 1993 and 1998. The number of ecstasy users kept growing until 2002, years after a national public education initiative against ecstasy use was launched. In this study, a system of differential equations is used to model the peer-driven dynamics of ecstasy use. It is found that backward bifurcations describe situations when sufficient peer pressure can cause an epidemic of ecstasy use. Furthermore, factors that have the greatest influence on ecstasy use as predicted by the model are highlighted. The effect of education is also explored, and the results of simulations are shown to illustrate some possible outcomes.