When using mathematics to study epidemics, oftentimes the goal is to determine when an infection can invade and persist within a population. The most common way to do so uses threshold quantities called reproductive numbers. An infection's basic reproductive number (BRN), typically denoted [Formula: see text], measures the infection's initial ability to reproduce in a naive population and is tied mathematically to the stability of the disease-free equilibrium. Next-generation methods have long been used to derive [Formula: see text] for autonomous continuous-time systems; however, many diseases exhibit seasonal behavior. Incorporating seasonality into models may improve accuracy in important ways, but the resulting non-autonomous systems are much more difficult to analyze. In the literature, two principal methods have been used to derive BRNs for periodic epidemic models. One, based on time-averages, is simple to apply but does not always describe the correct threshold behavior. The other, based on linear operator theory, is more general but also more complicated, and no detailed explanations of the necessary computations have yet been laid out. This paper reconciles the two methods by laying out an explicit procedure for the second and then identifying conditions (and some important classes of models) under which the two methods agree. This allows the use of the simpler method, which yields interpretable closed-form expressions, when appropriate, and illustrates in detail the simplest possible case where they disagree. Results show that seasonality alone cannot affect disease persistence, but must act in conjunction with non-hierarchical heterogeneity in the infected population, in order to do so.

Although the Zika virus is transmitted to humans primarily through the bite of infected female Aedes aegypti mosquitoes, it can also be sexually and vertically transmitted within both populations. In this study, we develop a new mathematical model of the Zika virus which incorporates sexual transmission in humans and mosquitos, vertical transmission in mosquitos, and mosquito to human transmission through bites. Analysis of this deterministic model shows that the secondary transmission routes of Zika increase the basic reproductive number (R0) of the virus by 5%, shift the peak time of an outbreak to occur 10% sooner, increase the initial growth of an epidemic, and have important consequences for control strategies and estimates of R0. Furthermore, sensitivity analysis show that the basic reproductive number is most sensitive to the mosquito biting rate and transmission probability parameters and reveal that the dynamics of juvenile mosquito stages greatly impact the peak time of an outbreak. These discoveries deepen our understanding of the complex transmission routes of ZIKV and the consequences that they may hold for public health officials.

Although dengue and Zika cocirculation has increased within the past 5 years, very little is known about its epidemiological consequences. To investigate the effect of dengue and Zika cocirculation on the spread of both pathogens, we create a deterministic dengue and Zika coinfection model, the first to incorporate altered infectivity of mosquitoes (due to coinfection). The model also addresses increased infectivity due to antibody-dependent enhancement (ADE) within the human population. Central to our analysis is the derivation and interpretation of the basic reproductive number and invasion reproductive number of both pathogens. In addition, we investigate how model parameters impact the persistence of each disease. Our results identify threshold conditions under which one disease facilitates the spread of the other and show that ADE has a greater impact on disease persistence than altered vector infectivity. This work highlights the importance of ADE and illustrates that while the endemic presence of dengue facilitates the spread of Zika, it is possible for high Zika prevalence to prevent the establishment of dengue. Keywords Invasion reproductive number • Copersistence • Zika • Dengue 1 Introduction With approximately 50-100 million cases annually, dengue is one of the most prevalent mosquito-borne diseases in the world (World Health Organization 2012). Over 40% of the world's population live in areas with high risk of dengue transmission and over B Omomayowa Olawoyin

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There are many cases within epidemiology where infections compete to persist within a population. In studying models for such cases, one of the goals is to determine which infections can invade a population and persist when other infections are already resident within the population. Invasion reproductive numbers (IRN), which are tied to the stability of boundary endemic equilibria, can address this question. By reinterpreting resident infections epidemiologically, this study extends methods for finding IRNs to periodic systems, and presents some examples which illustrate the often complex computations required. Results identify conditions under which a simple time-average can be used to derive IRNs, and apply the methods to examine how seasonal fluctuations in influenza incidence facilitate the year-round persistence of bacterial respiratory infections.

In 2012, more than three million students dropped out from high school. At this pace, we will have more than 30 million Americans without a high school degree by 2022 and relatively high dropout rates among Hispanic and African American students. We have developed and analysed a data-driven mathematical model that includes multiple interacting mechanisms and estimates of parameters using data from a specifically designed survey applied to a certain group of students of a high school in Chicago to understand the dynamics of dropouts. Our analysis suggests students' academic achievement is directly related to the level of parental involvement more than any other factors in our study. However, if the negative peer influence (leading to lower academic grades) increases beyond a critical value, the effect of parental involvement on the dynamics of dropouts becomes negligible.

The parasite Trypanosoma cruzi, spread by triatomine vectors, affects over 100 mammalian species throughout the Americas, including humans, in whom it causes Chagas' disease. In the U.S., only a few autochthonous cases have been documented in humans, but prevalence is high in sylvatic hosts (primarily raccoons in the southeast and woodrats in Texas). The sylvatic transmission of T. cruzi is spread by the vector species Triatoma sanguisuga and Triatoma gerstaeckeri biting their preferred hosts and thus creating multiple interacting vector-host cycles. The goal of this study is to quantify the rate of contacts between different host and vector species native to Texas using an agent-based model framework. The contact rates, which represent bites, are required to estimate transmission coefficients, which can be applied to models of infection dynamics. In addition to quantitative estimates, results confirm host irritability (in conjunction with host density) and vector starvation thresholds and dispersal as determining factors for vector density as well as host-vector contact rates.

Biodiversity is commonly believed to reduce risk of vector-borne zoonoses. However, researchers already showed that the effect of biodiversity on disease transmission is not that straightforward. This study focuses on the effect of biodiversity, specifically on the effect of the decoy process (additional hosts distracting vectors from their focal host), on reducing infections of vector-borne diseases in humans. Here, we consider the specific case of Chagas disease and use mathematical population models to observe the impact on human infection of the proximity of chickens, which are incompetent hosts for the parasite but serve as a preferred food source for vectors. We consider three cases as the distance between the two host populations varies: short (when farmers bring chickens inside the home to protect them from predators), intermediate (close enough for vectors with one host to detect the presence of the other host type), and far (separate enclosed buildings such as a home and hen-house). Our analysis shows that the presence of chickens reduces parasite prevalence in humans only at an intermediate distance under the condition that the vector birth rate from feeding on chickens is sufficiently low.