Christopher M. Kribs-Zaleta

A simple vaccination model with multiple endemic states.

A simple two-dimensional SIS model with vaccination exhibits a backward bifurcation for some parameter values. A two-population version of the model leads to the consideration of vaccination policies in paired border towns. The results of our mathematical analysis indicate that a vaccination campaign φ meant to reduce a diseases reproduction number R(φ) below one may fail to control the disease. If the aim is to prevent an epidemic outbreak, a large initial number of infective persons can cause a high endemicity level to arise rather suddenly even if the vaccine-reduced reproduction number is below threshold. If the aim is to eradicate an already established disease, bringing the vaccine-reduced reproduction number below one may not be sufficient to do so. The complete bifurcation analysis of the model in terms of the vaccine-reduced reproduction number is given, and some extensions are considered.

Model Parameters and Outbreak Control for SARS

Tool for estimating basic reproductive number for the SARS outbreak suggests need for multiple methods of control.

Vaccination strategies and backward bifurcation in an age-since-infection structured model.

We consider models for a disease with acute and chronic infective stages, and variable infectivity and recovery rates, within the context of a vaccination campaign. Models for SIRS and SIS disease cycles exhibit backward bifurcations under certain conditions, which complicate the criteria for success of the vaccination campaign by making it possible to have stable endemic states when R(0)<1. We also show the extent to which the forms of the infectivity and recovery functions affect the possibility of backward bifurcations. SIR and SI models examined do not exhibit this behavior.

Structured models for heterosexual disease transmission.

An SIS model for a heterosexually transmitted disease with core and non-core compartments and a generalized recovery function P(t) is analyzed. It exhibits R0 threshold behavior and leads to discussions of stability with respect to choice of P(t), and of the effects of allowing recruitment between core and non-core groups.

Influence of Vectors’ Risk-Spreading Strategies and Environmental Stochasticity on the Epidemiology and Evolution of Vector-Borne Diseases: The Example of Chagas’ Disease

Insects are known to display strategies that spread the risk of encountering unfavorable conditions, thereby decreasing the extinction probability of genetic lineages in unpredictable environments. To what extent these strategies influence the epidemiology and evolution of vector-borne diseases in stochastic environments is largely unknown. In triatomines, the vectors of the parasite Trypanosoma cruzi, the etiological agent of Chagas’ disease, juvenile development time varies between individuals and such variation most likely decreases the extinction risk of vector populations in stochastic environments. We developed a simplified multi-stage vector-borne SI epidemiological model to investigate how vector risk-spreading strategies and environmental stochasticity influence the prevalence and evolution of a parasite. This model is based on available knowledge on triatomine biodemography, but its conceptual outcomes apply, to a certain extent, to other vector-borne diseases. Model comparisons between deterministic and stochastic settings led to the conclusion that environmental stochasticity, vector risk-spreading strategies (in particular an increase in the length and variability of development time) and their interaction have drastic consequences on vector population dynamics, disease prevalence, and the relative short-term evolution of parasite virulence. Our work shows that stochastic environments and associated risk-spreading strategies can increase the prevalence of vector-borne diseases and favor the invasion of more virulent parasite strains on relatively short evolutionary timescales. This study raises new questions and challenges in a context of increasingly unpredictable environmental variations as a result of global climate change and human interventions such as habitat destruction or vector control.

Evaluating treatment of hepatitis C for hemolytic anemia management.

The combination therapy of antiviral peg-interferon and ribavirin has evolved as one of the better treatments for hepatitis C. In spite of its success in controlling hepatitis C infection, it has also been associated with treatment-related adverse side effects. The most common and life threatening among them is hemolytic anemia, necessitating dose reduction or therapy cessation. The presence of this side effect leads to a trade-off between continuing the treatment and exacerbating the side effects versus decreasing dosage to relieve severe side effects while allowing the disease to progress. The drug epoietin (epoetin) is often administered to stimulate the production of red blood cells (RBC) in the bone marrow, in order to allow treatment without anemia. This paper uses mathematical models to study the effect of combination therapy in light of anemia. In order to achieve this we introduce RBC concentration and amount of drug in the body as state variables in the usual immunological virus infection model. Analysis of this model provides a quantification of the amount of drug a body can tolerate without succumbing to hemolytic anemia. Indirect estimation of parameters allow us to calculate the necessary increment in RBC production to be > or =2.3 times the patients original RBC production rate to sustain the entire course of treatment without encountering anemia in a sensitive patient.

Modeling colony collapse disorder in honeybees as a contagion

Honeybee pollination accounts annually for over

Core recruitment effects in SIS models with constant total populations

14 billion in United States agriculture alone. Within the past decade there has been a mysterious mass die-off of honeybees, an estimated 10 million beehives and sometimes as much as 90% of an apiary. There is still no consensus on what causes this phenomenon, called Colony Collapse Disorder, or CCD. Several mathematical models have studied CCD by only focusing on infection dynamics. We created a model to account for both healthy hive dynamics and hive extinction due to CCD, modeling CCD via a transmissible infection brought to the hive by foragers. The system of three ordinary differential equations accounts for multiple hive population behaviors including Allee effects and colony collapse. Numerical analysis leads to critical hive sizes for multiple scenarios and highlights the role of accelerated forager recruitment in emptying hives during colony collapse.

Modeling nosocomial transmission of rotavirus in pediatric wards

We consider a set of SIS models for a heterosexually transmitted disease in which there is recruitment between core and non-core subpopulations as a function of prevalence of the disease. Behavior diverges from the traditional R0 threshold behavior and yields an extra pair of endemic equilibria in one case and a limit cycle in the other. Total at-risk population is constant.

Graphical analysis of evolutionary trade-off in sylvatic Trypanosoma cruzi transmission modes

Nosocomial transmission of viral and bacterial infections is a major problem worldwide, affecting millions of patients (and causing hundreds of thousands of deaths) per year. Rotavirus infections affect most children worldwide at least once before age five. We present here deterministic and stochastic models for the transmission of rotavirus in a pediatric hospital ward and draw on published data to compare the efficacy of several possible control measures in reducing the number of infections during a 90-day outbreak, including cohorting, changes in healthcare worker-patient ratio, improving compliance with preventive hygiene measures, and vaccination. Although recently approved vaccines have potential to curtail most nosocomial rotavirus transmission in the future, even short-term improvement in preventive hygiene compliance following contact with symptomatic patients may significantly limit transmission as well, and remains an important control measure, especially where resources are limited.