Optimal Control of Epidemic Models Involving Rabies and West Nile Viruses

This research considers the application of Optimal Control theory to minimize the spread of viral infections in disease models. The population models under consideration are systems of ordinary differential equations and represent epidemics arising due to either rabies or West Nile virus. Optimal control strategies are analyzed using Pontryagin’s Maximum Principle and illustrated based upon computer simulations. The first model describes a population of raccoons and its interaction with the rabies virus, thus dividing the animals into four classes: susceptible, exposed, immune, and recovered (SEIR). The model includes a birth pulse during the spring of the year and an equation reflecting the dynamics of a potential vaccine. The vaccine equation contains a linear control variable representing the rate at which the vaccine is distributed. The goal is to minimize the number of infected raccoons and the cost of vaccine distributed. Due to linearity in the control, there is the possibility of a singular control and the generalized Legendre-Clebsch condition will be satisfied to obtain new necessary conditions for the singular case. A scenario with a limited amount of vaccine is also investigated. The system is modified to include a density-dependent death rate for each of the S, E, I, R classes, and the results of this model are compared with those of the non-density dependent model to determine how the different death rates affect control strategies. The second disease model considered describes the dynamics of mosquito, bird and human populations exposed to the West Nile virus. The mosquito and bird categories will be divided into susceptible and infected classes. In addition to these two groups, humans will also have the potential of entering the exposed, hospitalized and recovered classes. In this model, birth and death rates are assumed to be density-dependent. Two controls are applied with one control representing pesticide efforts to decrease the number of mosquitos and a second control representing prevention and repellant methods. The basic reproduction number is considered to justify the need for control. Approximations of the optimal solutions of the models are obtained using an iterative method. The numerical algorithm, Runge-Kutta of order four, is programmed in Matlab. Graphical results show the appropriate amount of control for various situations

The influence of biological rhythms on host–parasite interactions

Biological rhythms, from circadian control of cellular processes to annual cycles in life history, are a main structural element of biology. Biological rhythms are considered adaptive because they enable organisms to partition activities to cope with, and take advantage of, predictable fluctuations in environmental conditions. A flourishing area of immunology is uncovering rhythms in the immune system of animals, including humans. Given the temporal structure of immunity, and rhythms in parasite activity and disease incidence, we propose that the intersection of chronobiology, disease ecology, and evolutionary biology holds the key to understanding host–parasite interactions. Here, we review host–parasite interactions while explicitly considering biological rhythms, and propose that rhythms: influence within-host infection dynamics and transmission between hosts, might account for diel and annual periodicity in host–parasite systems, and can lead to a host–parasite arms race in the temporal domain

A computational investigation of seasonally forced disease dynamics

In recent years there has been a great increase in work on epidemiological modelling, driven partly by the increase in the availability and power of computers, but also by the desire to improve standards of public and animal health. Through modelling, understanding of the mechanisms of previous epidemics can be gained, and the lessons learnt applied to make predictions about future epidemics, or emerging diseases. The standard SIR model is in some sense quite a simplistic model, and can lack realism. One solution to this problem is to increase the complexity of the model, or to perform full scale simulation—an experiment in silico. This thesis, however, takes a different approach and makes an in depth analysis of one small improvement to the model: the replacement of a constant birth rate with a birth pulse. This more accurately describes the seasonal birth patterns observed in many animal populations. The combination of the nonlinearities of the SIR model and the strong seasonal forcing provided by the birth pulse necessitate the use of numerical methods. The model shows complex multi annual cycles of epidemics and even chaos for shorter infectious periods. The robustness of these results are proven with respect to a wide range or perturbations: in phase space, in the shape and temporal extent of the birth pulse and in the underlying model to which the pulsing is applied. To complement the numerics, analytic methods are used to gain further understanding of the dynamics in particular areas of the chosen parameter space where the numerics can be challenging. Three approximations are presented, one to investigate very small levels of forcing, and two covering short infectious periods.EThOS - Electronic Theses Online ServiceEngineering and Physical Sciences Research Council (EPSRC)GBUnited Kingdo

Outfoxing rabies: robust vaccination designs for disease elimination

Prediction of pathogen dynamics and the design of effective interventions to control and eliminate disease are key goals in epidemiology. While progress has been made towards the elimination of many infectious diseases, only two, smallpox and rinderpest, have been globally eradicated. Mass vaccination can greatly reduce the burden of vaccine-preventable diseases. However, there is relatively little scientific guidance on the optimal duration, frequency and placement of control interventions for achieving elimination. Such insights could greatly inform policy and practice. Rabies is a deadly and terrifying disease that exacts a heavy toll on human lives and national economies, with over 50,000 human deaths each year and many millions more requiring expensive life-saving post-exposure vaccines. Elimination of rabies is feasible through vaccination, and oral rabies vaccination (ORV) campaigns have eliminated fox rabies from Western Europe. However, scientific guidance could improve elimination efforts elsewhere, and is still needed for contingency planning to maintain rabies freedom and for emergency response to incursions. My thesis focuses on two pivotal questions in infectious disease ecology: what are the underlying determinants of disease persistence, and how can vaccination strategies be optimized to eliminate infection? To answer these questions, I analysed a rich and highly resolved spatial dataset of fox rabies cases and ORV efforts over three decades in Germany and neighbouring countries. The long-term, large-scale nature of these data provides a unique opportunity to improve our understanding of wildlife rabies dynamics in response to vaccination using novel spatial modeling techniques. In chapter 2, I create a metapopulation model of regional rabies dynamics that incorporates local transmission (within regions) and spatial coupling (between regions) using a hierarchical Bayesian state-space model. In chapter 3, I extend the model developed in chapter 2 to determine the best vaccination strategy, in terms of scale and duration of ORV efforts for three common epidemiological scenarios: {\bf endemic} circulation of rabies; {\bf high-risk} situations when rabies-free but neighbor endemic areas; and an {\bf endgame} scenario when only a single endemic foci remains. In chapter 4, I develop a space-time model of fox rabies dynamics and explore the effect of scale on estimates of transmission terms by aggregating rabies case data at different spatial resolutions. I then relate these estimates to the scaling of individual interactions to regional dynamics through population mixing. Collectively, the findings from this thesis contribute to our understanding of how infectious diseases persist and can be controlled through vaccination. The methods generated can be used to explore tradeoffs in the scale and duration of ORV efforts, and generate recommendations on the time horizon and investment required to achieve and maintain freedom from disease. The model developed in chapter 4 also presents the first steps to developing a highly resolved spatial model of local rabies dynamics. These findings have immediate application to the design of cordon sanitaires in Europe, and to strategies aiming to rapidly eliminate re-emergence in high-risk countries such as Greece and Turkey. The analytical and statistical framework developed in this thesis is also applicable to answering analogous questions for the elimination of dog-mediated rabies and for other vaccine preventable diseases

A computational investigation of seasonally forced disease dynamics

In recent years there has been a great increase in work on epidemiological modelling, driven partly by the increase in the availability and power of computers, but also by the desire to improve standards of public and animal health. Through modelling, understanding of the mechanisms of previous epidemics can be gained, and the lessons learnt applied to make predictions about future epidemics, or emerging diseases. The standard SIR model is in some sense quite a simplistic model, and can lack realism. One solution to this problem is to increase the complexity of the model, or to perform full scale simulation—an experiment in silico. This thesis, however, takes a different approach and makes an in depth analysis of one small improvement to the model: the replacement of a constant birth rate with a birth pulse. This more accurately describes the seasonal birth patterns observed in many animal populations. The combination of the nonlinearities of the SIR model and the strong seasonal forcing provided by the birth pulse necessitate the use of numerical methods. The model shows complex multi annual cycles of epidemics and even chaos for shorter infectious periods. The robustness of these results are proven with respect to a wide range or perturbations: in phase space, in the shape and temporal extent of the birth pulse and in the underlying model to which the pulsing is applied. To complement the numerics, analytic methods are used to gain further understanding of the dynamics in particular areas of the chosen parameter space where the numerics can be challenging. Three approximations are presented, one to investigate very small levels of forcing, and two covering short infectious periods

Fluorescent biomarkers demonstrate prospects for spreadable vaccines to control disease transmission in wild bats

Vaccines that autonomously transfer among individuals have been proposed as a strategy to control infectious diseases within inaccessible wildlife populations. However, rates of vaccine spread and epidemiological efficacy in real-world systems remain elusive. Here, we investigate whether topical vaccines that transfer among individuals through social contacts can control vampire bat rabies—a medically and economically important zoonosis in Latin America. Field experiments in three Peruvian bat colonies, which used fluorescent biomarkers as a proxy for the bat-to-bat transfer and ingestion of an oral vaccine, revealed that vaccine transfer would increase population-level immunity up to 2.6 times beyond the same effort using conventional, non-spreadable vaccines. Mathematical models showed that observed levels of vaccine transfer would reduce the probability, size and duration of rabies outbreaks, even at low but realistically achievable levels of vaccine application. Models further predicted that existing vaccines provide substantial advantages over culling bats—the policy currently implemented in North, Central and South America. Linking field studies with biomarkers to mathematical models can inform how spreadable vaccines may combat pathogens of health and conservation concern before costly investments in vaccine design and testing

The Drivers of Acute Seasonal Infectious Diseases.

Seasonality is a feature of all ecological systems. Earth's terrestrial and pelagic life has evolved in a highly seasonal abiotic environment with intra-annual variation in photoperiod, temperature, and precipitation, among many other abiotic and biotic factors. Seasonal aspects of mammals and birds include seasonally varying birth rates, seasonal changes in endocrine hormones, and seasonal variation in immunity. One area where seasonal biology is particularly salient is disease ecology. The mechanisms underlying the seasonality of communicable diseases are poorly understood. I propose that much of the unexplained seasonality observed in infectious disease dynamics could be attributed to seasonal biology, including (1) birth seasonality, (2) seasonal variation in immunity, and (3) seasonal cycles in parasite traits and parasite population parameters. In my dissertation, I present work on various aspects of seasonality. In Chapter II, I explored the seasonality of births in human populations and quantified the effects of birth seasonality on measles epidemics. In Chapter III, I reviewed circadian and circannual rhythms in host and parasite populations, and proposed both ecological and evolutionary models for integrating biological rhythms into the study of infectious diseases. In Chapters IV--V, I presented my in-depth ecological studies of poliovirus, a notoriously seasonal summertime infection. I explored geographical variation in polio's seasonality and tested whether human birth seasonality or transmission seasonality drove epidemics of this disease. In addition to studying polio seasonality, I revealed the connection between (i) polio's emergence and human demography, (ii) the geographical distribution of poliovirus and its persistence, and (iii) polio symptomatology and silent chains of transmission. Lastly, I highlighted the public health implications of seasonal transmission by measuring the efficacy of the two polio vaccines and discussing how seasonality can be utilized for vaccine interventions.PhDEcology and Evolutionary BiologyUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/113643/1/bakkerma_1.pd

Sarcoptic mange and the demography of the red fox, Vulpes vulpes

Vertebrate species are managed for many reasons, including their role as economically important predators or as carriers of disease. Successful management depends on the ability to predict the outcome of management actions on a species’ population dynamics. However, uncertainty in the models used to make such predictions can arise from multiple sources, including sampling error in vital rates, intraspecific demographic variation and unknown interspecific interactions. The red fox Vulpes vulpes provides a useful model organism for exploring such uncertainty, because management of this important predator and disease host is often ineffective, despite substantial sampling effort. By explicitly accounting for sampling error in survival and fecundity, confidence intervals for population growth rates were derived from published point estimates of red fox demographic data. Uncertainty in population growth rates was found to be high, requiring a quadrupling of sampling effort to halve the confidence intervals. Given the often poor justification for the choice of distribution used to model litter size, the influence of probability distributions on population model outcomes was tested. In this first comprehensive evaluation, estimates of quasi-extinction and disease control probabilities for three Canid species were found to be robust to litter size distribution choice. Demographic analyses of the red fox revealed a medium to fast life history speed and significant survival and fecundity contributions from juveniles to population growth. Intraspecific variation was detected within these spectra of demographic metrics: the first such demonstration for carnivores. Simulated data substitution between fox populations revealed that geographic proximity and similar levels of anthropogenic disturbance did not infer demographic similarity. Considering the sampling effort expended on the red fox, the species appears well-studied; yet, substantial limitations in data collection were identified. Compartment modelling of a sarcoptic mange outbreak in an urban fox population in Bristol, UK, revealed that disease transmission was frequency-dependent, consistent with contact rates being determined by social interactions rather than by population density. Individual-based modelling suggested that indirect transmission, genetic resistance and long-distance recolonisation were required to replicate the observed rapid spread of mange and subsequent population recovery. Thus, this first attempt to model mange dynamics in this canid provided novel insight into previously uncertain epidemiological and behavioural processes in the transmission of sarcoptic mange in the red fox

Field methods for rodent studies in Asia and the Indo-Pacific

Farm Management,