A model for meningitis is developed by adding a class of carriers to the basic SIR model. This model is used to analyze the impact a vaccination program can have on the health of the population of epidemic prone countries. Analysis of the model shows the local stability of the disease free equilibrium, the existence of an endemic equilibrium and computation of the reproduction number, ℜ0 . Using a MATLAB program we simulate a time course of the model using parameters gathered from the World Health Organization. The numerical solution demonstrates that our reproduction number was correct. We thenconcluded that a high infection transmission rate requires a high vaccine rate.

This thesis discusses limit cycles and behavior of rumor models. The first part presents the deterministic Daley-Kendall model (DK) with arrivals and departures and comparison of the Susceptibles, Infectives and Removed (SIR) model and the DK model. The second result is a part of the qualitative analysis and the general behavior of extension of the Daley-Kendall model. Here we discuss how the halting rate of spreaders causes the model to change from a stable equilibrium or a stable limit cycle. In the third part we carry out model validation and use both synthetic data and real data sets and fit them to the numerical solutions of the extended Daley-Kendall model. Finally, we find the parameter estimates and standard errors. In this way we shall be able to decide whether the numerical solutions quantifying the relationships between the variables obtained from the qualitative analysis can be accepted as the best description of the data. We discuss sensitivity analysis results and traditional sensitivity functions.

<p><font face="CMR12"> <p align="left">This thesis deals with analysis and implementation of a positivity preserving numerical method for a vaccination model for the transmission dynamics of two HIVsubtypesnin a given community. The continuous model is analyzed for stability and equilibria. The qualitative information thus obtained is used while designing numerical method(s). Three numerical methods, namely, Implicit Finite Difference Method (IFDM), Non-standard Finite Difference Method (NSFDM) and the Runge-Kutta method of order four (RK4), are designed and implemented. Extensive numerical simulation are carried out to justify theoretical outcomes.</p> </font></p>

<p><font face="CMR12"> <p align="left">This thesis deals with analysis and implementation of a positivity preserving numerical method for a vaccination model for the transmission dynamics of two HIVsubtypesnin a given community. The continuous model is analyzed for stability and equilibria. The qualitative information thus obtained is used while designing numerical method(s). Three numerical methods, namely, Implicit Finite Difference Method (IFDM), Non-standard Finite Difference Method (NSFDM) and the Runge-Kutta method of order four (RK4), are designed and implemented. Extensive numerical simulation are carried out to justify theoretical outcomes.</p> </font></p>

Magister Scientiae - MSc / This thesis deals with analysis and implementation of a positivity preserving numerical method for a vaccination model for the transmission dynamics of two HIVsubtypesnin a given community. The continuous model is analyzed for stability and equilibria. The qualitative information thus obtained is used while designing numerical method(s). Three numerical methods, namely, Implicit Finite Difference Method (IFDM), Non-standard Finite Difference Method (NSFDM) and the Runge-Kutta method of order four (RK4), are designed and implemented. Extensive numerical simulation are carried out to justify theoretical outcomes

Within population biology, models are often analyzed for the net reproduction number or other generalized target reproduction numbers, which describe the growth or decline of the population based on specific mechanisms. This is useful in determining the strength and efficiency of control measures for inhibiting or enhancing population growth. The literature contains many algebraic and combinatorial approaches for deriving the net reproduction number and generalized target reproduction numbers from digraphs and associated matrices. Finding, categorizing, and counting the permutations of disjoint cycles, or cycles unions is a requirement of the Cycle Union approach by Lewis et al. (2019). These cycles and subsequent cycle unions can be found via the digraphs and associated matrices. We developed cycle counting patterns for targeting fertilities within Leslie Matrices, Lefkovitch Matrices, Sub-Diagonal Lower Triangle Transition Matrices, and Lower Triangle Transition Matrices to serve as a foundation for future work. Presented are the counting patterns and closed-form summations of the cycle unions.

A pandemic subtype of influenza A sometimes replaces (e.g., in 1918, 1957, 1968) but sometimes coexists (e.g., in 1977) with the previous seasonal subtype. This research aims to determine a condition for replacement or coexistence of influenza subtypes. We formulate a hybrid model for the dynamics of influenza A epidemics taking into account cross-immunity of influenza strains depending on the most recent seasonal infection. A combination of theoretical and numerical analyses shows that for very strong cross-immunity between seasonal and pandemic subtypes, the pandemic cannot invade, whereas for strong and weak cross-immunity there is coexistence, and for intermediate levels of cross-immunity the pandemic may replace the seasonal subtype. Cross-immunity between seasonal strains is also a key factor of our model because it has a major influence on the final size of seasonal epidemics, and on the distribution of susceptibility in the population. To determine this cross-immunity, we design a novel statistical method, which uses a theoretical model and clinical data on attack rates and vaccine efficacy among school children for two seasons after the 1968 A/H3N2 pandemic. This model incorporates the distribution of susceptibility and the dependence of cross-immunity on the antigenic distance of drifted strains. We find that the cross-immunity between an influenza strain and the mutant that causes the next epidemic is 88%. Our method also gives an estimated value 2.15 for the basic reproduction number of the 1968 pandemic influenza. Our hybrid model agrees qualitatively with the observed subtype replacement or coexistence in 1957, 1968 and 1977. However, our model with the homogeneous mixing assumption significantly over estimates the pandemic attack rate. Thus, we modify the model to incorporate heterogeneity in the contact rate of individuals. Using the determined values of cross-immunity and the basic reproduction number, this modification lowers the pandemic attack rate slightly, but it is still higher than the observed attack rates. / Graduate

Doctor of Philosophy / Department of Electrical and Computer Engineering / Caterina Scoglio / Vector-borne diseases not only cause devastating economic losses, they also significantly impact human health in terms of morbidity and mortality. From an economical and humane point of view, mitigation and control of vector-borne diseases are essential. Studying dynamics of vector-borne disease transmission is a challenging task because vector-borne diseases show complex dynamics impacted by a wide range of ecological factors. Understanding these factors is important for the development of mitigation and control strategies. Mathematical models have been commonly used to translate assumptions concerning biological (medical, demographical, behavioral, immunological) aspects into mathematics, linking biological processes of transmission and dynamics of infection at population level. Mathematical analysis translates results back into biology. Classical deterministic epidemic models do not consider spatial variation, assuming space is homogeneous. Spatial spread of vector-borne diseases observed many times highlights the necessity of incorporating spatial dynamics into mathematical models. Heterogeneous demography, geography, and ecology in various regions may result in different epidemiological characteristics. Network approach is commonly used to study spatial evolution of communicable diseases transmitted among connected populations. In this dissertation, the spread of vector-borne diseases in time and space, is studied to understand factors that contribute to disease evolution. Network-based models have been developed to capture different features of disease transmission in various environments. Network nodes represent geographical locations, and the weights represent the level of contact between regional pairings. Two competent vector populations, Aedes mosquitoes and Culex mosquitoes, and two host populations, cattle and humans were considered. The deterministic model was applied to the 2010 Rift Valley fever outbreak in three provinces of South Africa. Trends and timing of the outbreak in animals and humans were reproduced. The deterministic model with stochastic parameters was applied to hypothetical Rift Valley fever outbreak on a large network in Texas, the United States. The role of starting location and size of initial infection in Rift Valley fever virus spread were studied under various scenarios on a large-scale network. The reproduction number, defined as the number of secondary infections produced by one infected individual in a completely susceptible population, is typically considered an epidemic threshold of determining whether a disease can persist in a population. Extinction thresholds for corresponding Continuous-time Markov chain model is used to predict whether a disease can perish in a stochastic setting. The network level reproduction number for diseases vertically and horizontally transmitted among multiple species on heterogeneous networks was derived to predict whether a disease can invade the whole system in a deterministic setting. The complexity of computing the reproduction number is reduced because the expression of the reproduction number is the spectral radius of a matrix whose size is smaller than the original next generation matrix. The expression of the reproduction number may have a wide range of applications to many vector-borne diseases. Reproduction numbers can vary from below one to above one or from above one to below one by changing movement rates in different scenarios. The observations provide guidelines on executing movement bans in case of an epidemic. To compute the extinction threshold, corresponding Markov chain process is approximated near disease free equilibrium. The extinction threshold for Continuous-time Markov chain model was analytically connected to the reproduction number under some assumptions. Numerical simulation results agree with analytical results without assumptions, proposing a mathematical problem of proving the existence of the relationships in general. The distance of the extinction threshold were shown to be closer to one than the reproduction number. Consistent trends of probability of extinction varying with disease parameters observed through numerical simulations provide novel insights into disease mitigation, control, and elimination.

Leishmaniasis is a parasitic disease that affects humans and animals in more than 98 countries across the globe placing more than 1 billion people at risk for the disease and killing more than 20,000 people per year. In the United States the disease is enzootic within the hunting dog population and vertical transmission has been identified as the primary route of transmission in this population. In Brazil the disease is endemic in the human population and enzootic in the dog population with vector and vertical transmission having been reported. In many diseases reports have found there is increased disease severity when an individual is co-infected with another organism. Case reports have suggested this may also occur with tick borne diseases and leishmaniosis in dogs but there is limited longitudinal data to support this relationship. Even less is known and understood regarding the risk factors and basic reproduction number, number of secondary cases one infected individual can cause in a susceptible population, of leishmaniosis in regards to vertical transmission. The goal of the work presented in this thesis is to address host factors related to the transmission of L. infantum and the way in which co-infections affect the progression of the disease both in the U.S. and in Brazil. Understanding the risk factors associated with the transmission of the parasite Leishmania infantum, the causative agent of the disease, are necessary to controlling and potentially elimination the disease. Utilizing a large prospective cohort and both active and passive surveillance it was identified that leishmaniosis can be maintained in a population via vertical transmission at prevalence rates similar to other endemic countries, 20%. With this knowledge an additional study examining a longitudinal cohort and assessing the impact of tick borne disease co-infections upon disease transmission was performed. It was identified that dogs exposed to three or more tick borne diseases were 11x more likely to progress to clinical disease (Adjusted RR: 11.64 95% CI: 1.22-110.99 p-value: 0.03) than dogs with no tick borne disease exposures. Furthermore, dogs with Leishmania and tick borne disease were 5x more likely to die within the study (RR: 4.85 95% CI: 1.65-14.24 p-value: 0.0051). When examining this relationship in a cross-sectional study in Brazil it was found that dogs with multiple tick borne disease exposures had 1.68x greater risk of being positive for Leishmania (Adjusted RR: 1.68 95% CI: 1.09-2.61 p-value: 0.019). Using a retrospective cohort of dogs and information regarding their dam’s diagnostic status near the time of pregnancy risk factors associated with vertical transmission and the basic reproduction number were calculated. It was found that dogs who were born to dams that were ever diagnostically positive for exposure and/or infection with L. infantum were 13.84x more likely become positive for L. infantum within their lifetime (RR: 13.84 95% CI: 3.54-54.20 p-value < 0.0001). The basic reproduction number for vertically transmitted L. infantum within this cohort was 4.16. The results of these studies suggest that leishmaniosis can be maintained in a population through vertical transmission. Furthermore, the studies show the risk factors associated with vertical transmission relate to the mother’s diagnostic status at time of pregnancy. The results of the co-infection studies highlight the importance of tick prevention in order to reduce disease progression. With increased disease severity associated with increased transmission to potential vectors these studies underline the need for immunotherapies and prevention measures to reduce disease progression in order to reduce transmission. Furthermore, these studies highlight the need for public health control and prevention programs to address vertical transmission if elimination of the disease is to ever be successful.

Parasites may express proteins that mimic host proteins such that the host immune system cannot discriminate between host and parasite. An immune response to host proteins results in autoimmunity, and therefore, mechanisms to avert autoimmunity also limit the capacity of the immune system to respond to parasites that are mimics. In failing to elicit an immune response, parasites that are mimics appear to have a selective advantage and so it is unclear why all parasites do not evolve to be mimics. In this thesis, I demonstrate that next-generation methods can be used to perform an evolutionary invasion analysis. Subsequently, I use this method to identify evolutionarily stable strategies and to investigate three hypotheses for why all parasites are not mimics. These are: (1) that mimicry increases the risk of inducing autoimmunity and that hosts with autoimmunity are less likely to transmit the parasite; (2) that proteins which mimic host proteins have a reduced ability to perform the vital functions necessary for parasite replication; and (3) that owing to a feedback between the types of parasites that infect a host and the host's immune response, when parasites are likely to re-infect the same hosts, mimics are more likely to elicit an immune response. I show that each of these hypotheses explain why all parasites are not mimics. The key data needed to assess the applicability of these results is to quantify the number of secondary infections generated by hosts infected with parasites that are molecular mimics. This thesis motivates a new question in evolutionary epidemiology, furthers the field of evolutionary medicine and contributes novel methodologies for host-parasite multi-scale modelling in mathematical biology. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-04-05 10:27:20.49