Tech Reports Archive

We analyze the long term behavior of a system involved in cyclic competition similar to the rock-paper-scissors game. Previous studies have used cellular autonoma simulations to model the stochastic interactions and mean field equations to approximate this stochastic model. However, mean field approximation does not properly account for spatial correlations, leading to loss of spatial significance. We use pair approximations to model the local interactions with a system of differential equations. We then investigate the outcome of various initial conditions of the pair approximation model using numerical integration. Three categories of initial conditions are found that lead to three distinct behaviors: one species is present, all three species oscillate around the fixed point resulting in a heteroclinic cycle, or convergence to an interior fixed point.  We also explore the relative importance of initial conditions and lattice size on fixation probabilities for each species. We finally discuss the implications of these results in a biological context.

Place cells are excitable pyramidal neurons in the hippocampus. These neurons fire preferentially to a rat's location in the environment, which is referred to as the place field. Place cells also fire preferentially to a particular phase of the theta rhythm, which is an electroencephalograph recording of the hippocampus observed in rats during exploratory movement. As the rat moves through a place field, the particular phase of the theta rhythm at which the place cell fires has been observed to systematically precess, i.e. firing occurs at progressively earlier phases of the theta rhythm. The underlying neural mechanism of phase precession might be the basis of a temporal code utilized for further information processing in the hippocampus. We construct a network model of connected neurons in order to generate phase precession. Each neuron is modeled by considering a conductance based cell model.

During the spring of 2009, a novel H1N1 inuenza virus posed a serious threat worldwide.  However, seasonal influenza is still a public health concern since each year in the United States it kills 36,000 people, causes more than 200,000 hospitalizations, and infects up to 20% of the population. In this paper, we explore what could happen if both a seasonal influenza and H1N1 outbreak were to coincide. First, a simple SAIR model is considered to study the dynamics of a single influenza strain. Then, a more complex two-strain model is constructed to examine the dynamics of two coexisting strains. In the two-strain model, we incorporate a seasonal flu vaccine to study the impact of vaccination on these dynamics. Optimal control theory is applied to both the single-strain model and the two-strain model with the goal of reducing the overall morbidity during an outbreak. Two controls are introduced to the systems: social distancing and treatment of H1N1 infected individuals. Introducing the seasonal vaccine significantly reduces the number of seasonal influenza cases, yet moderately increases the number of H1N1 infections. In the two-strain model, the application of optimal controls has a substantial impact on the H1N1 and seasonal influenza dynamics.

Hepatitis C infection becomes chronic in most patients leading to end-stage liver disease. The standard-of-care treatment for HCV patients is only suboptimal. Several patients who exhibit undetectable viral load during, at the end of, or six months after cessation of therapy, observes relapse over variable time. This suggests that there may exist, a sub-clinical threshold, which governs the achievement of lasting cure. We propose an immunological model of hepatocytes and HCV to investigate this threshold behavior, where the infected hepatocytes are differentiated by the age of infection in them. The goal of the model is to obtain observed patient profiles if treatment is provided. Preliminary analysis of the model provides conditions for existence of two endemic equilibria for R0 < 1. Analysis of a reduced model without age of infection suggests backward bifurcation at R0 = 1. Artificial values of the parameters are taken to show bistability region for R0 < 1. However, it may not be possible to observe the patient profile in this region without treatment. The critical reproduction number below which the disease free equilibrium is stable may lead us to the decisive sub-clinical viral load threshold.

An SIS=SAS model of gonorrhea transmission in a men-seeking-men (MSM) system is presented in this paper to study the impact of education on the dynamics of gonorrhea prevalence. Education affects behaviors that may fall into two categories{prevention and disease status awareness. Stability conditions for the disease free equilibrium and endemic equilibrium are determined along with an analytic expression and nominal value for the basic reproductive number and the control reproductive number.  We carry out a complete analysis of global dynamics. Moreover, a time-dependent sensitivity analysis of the system and a sensitivity analysis of the control reproductive number is performed.

A discrete time SIR model to study the dynamics of influenza in a multi-city setting is introduced. Data from three cities in Mexico that include the rate of movement of individuals from city to city is used to explore the impact of “travel” on A/H1N1 outbreaks. The model is expanded in order to evaluate the potential impact of treatment and social distancing.

Chagas disease is a tropical parasitic disease that affects Latin America. The parasite causing Chagas, Trypanosoma cruzi, is transmitted by an insect vector of the subfamily Triatominae. Chagas is uncommon in the United States, but is becoming more common in the southern part of the country. This may be due to a more virulent strain type of the parasite moving northward from Mexico and invading the less virulent, native strain type. A region in southern Texas where there is a mixing of two Triatoma species is modeled with two different modeling frameworks. A mathematical deterministic model is created to describe the interactions between one host and one vector and determine the outcome of an invasion of a non-native strain type into the region. A stochastic, agent based model is created to determine the effect space and randomness may have on the interactions. Within the models, three modes of parasite transmission are considered to account for the different characteristics of each strain type.

We vary the horizontal transmission ability of the invasive strain and run simulations for an equivalent time period of 30000 days (82 years). We find that the horizontal transmission potential of the invasive strain must be about 1.5 times as great as the other in order for over 50% of the runs to end with the invasive strain dominating. However, in the ODE model, the horizontal transmission of the invasive need only be 1.056 times that of the other strain. We also determine from the ODE model that considering three modes of transmission and no migration of vectors into the region being modeled, it is impossible for the two strains to coexist.

There are many natural, physical, and biological systems that exhibit multiple time scales. For example, the dynamics of a population of ticks can be described in continuous time during their individual life cycle yet discrete time is used to describe the generation of offspring. These characteristics cause the population levels to be reset periodically. A similar phenomenon can be observed in a sociological college drinking model in which the population is reset by the incoming class each year, as described in the 2006 work of Camacho et al. With the latter as our motivation we analytically and numerically investigate the mechanism by which solutions in certain systems with this resetting characteristics stabilize. We analyze certain one-dimensional and two-dimensional nonlinear systems, and try to generalize our results to higher dimensions.

An SIR epidemiological household model is constructed and studied to understand the effects that clusters of unvaccinated individuals have on the disease dynamics of a population. The model contains two levels of mixing where individuals make more intra-household than inter-household contacts. Stochastic simulations, numerical solutions are utilized to explore the model. A new extension of the basic reproductive number that incorporates more spatial information by predicting the average number of tertiary infections caused by a single infected individual is introduced to describe the threshold behavior and severity of an epidemic. Using these methods we show that clustering of unvaccinated individuals always leads to more severe epidemics.

Autism's cause is unknown, but suggested causes are often attributed to genetic or environmental factors. This research examines whether advancing maternal age contributes to the increasing prevalence of autism. The model used to achieve this objective generates values that represent the proportion of offspring expected to be diagnosed with autism, provided their mother belongs to a specific age class. The age class consisting of ages 40-44 was most affected. From these values, projections were made about the prevalence of autism in future populations, specifically, for the United States and California. These projections predict a continued increase in the prevalence of autism.