Intervention-Based Stochastic Disease Eradication

Disease control is of paramount importance in public health with infectious disease extinction as the ultimate goal. Although diseases may go extinct due to random loss of effective contacts where the infection is transmitted to new susceptible individuals, the time to extinction in the absence of control may be prohibitively long. Thus intervention controls, such as vaccination of susceptible individuals and/or treatment of infectives, are typically based on a deterministic schedule, such as periodically vaccinating susceptible children based on school calendars. In reality, however, such policies are administered as a random process, while still possessing a mean period. Here, we consider the effect of randomly distributed intervention as disease control on large finite populations. We show explicitly how intervention control, based on mean period and treatment fraction, modulates the average extinction times as a function of population size and rate of infection spread. In particular, our results show an exponential improvement in extinction times even though the controls are implemented using a random Poisson distribution. Finally, we discover those parameter regimes where random treatment yields an exponential improvement in extinction times over the application of strictly periodic intervention. The implication of our results is discussed in light of the availability of limited resources for control.Comment: 18 pages, 10 Figure

Modeling Spread of Polio with the Role of Vaccination

In this paper, we have proposed and analyzed a nonlinear mathematical model for the spread of Polio in a population with variable size structure including the role of vaccination. A threshold parameter, R , is found that completely determines the stability dynamics and outcome of the disease. It is found that if R 1, the disease free equilibrium is stable and the disease dies out. However, if R \u3e1, there exists a unique endemic equilibrium that is locally asymptotically stable. Conditions for the persistence of the disease are determined by means of Fonda’s theorem. Moreover, numerical simulation of the proposed model is also performed by using fourth order Runge - Kutta method. Numerically, it has been found that the system exhibits steady state bifurcation for some parameter values. It is concluded from our analysis that endemic level of infective population increases with the increase in rate of transmission of infection due to infective among susceptible class that further enhances because of transmission of infection due to latent hosts. A particular value of disease transmission coefficient r is found for which exposed and infective population dies out. It is found that periodic outbreak of the disease occurs when infection due to exposed and infective class occurs at the same rate. It is also observed from our analysis that although vaccination helps in eradicating polio by decreasing endemic equilibrium level yet careful administration of vaccination is desired because if vaccine is administered during incubation period, endemic equilibrium level increases and disease persists in the population

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

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

Effects of impulsive harvesting and an evolving domain in a diffusive logistic model

In order to understand how the combination of domain evolution and impulsive harvesting affect the dynamics of a population, we propose a diffusive logistic population model with impulsive harvesting on a periodically evolving domain. Initially the ecological reproduction index of the impulsive problem is introduced and given by an explicit formula, which depends on the domain evolution rate and the impulsive function. Then the threshold dynamics of the population under monotone or nonmonotone impulsive harvesting is established based on this index. Finally numerical simulations are carried out to illustrate our theoretical results, and reveal that a large domain evolution rate can improve the population survival, no matter which impulsive harvesting takes place. Contrary, impulsive harvesting has a negative effect on the population survival, and can even lead to the extinction of the population.Comment: 26 pages, 8 figure

Time Delayed Stage-Structured Predator-Prey Model with Birth Pulse and Pest Control Tactics

Normally, chemical pesticides kill not only pests but also their natural enemies. In order to better control the pests, two-time delayed stage-structured predator-prey models with birth pulse and pest control tactics are proposed and analyzed by using impulsive differential equations in present work. The stability threshold conditions for the mature prey-eradication periodic solutions of two models are derived, respectively. The effects of key parameters including killing efficiency rate, pulse period, the maximum birth effort per unit of time of natural enemy, and maturation time of prey on the threshold values are discussed in more detail. By comparing the two threshold values of mature prey-extinction, we provide the fact that the second control tactic is more effective than the first control method

Dynamics Days Latin America and the Caribbean 2018

This book contains various works presented at the Dynamics Days Latin America and the Caribbean (DDays LAC) 2018. Since its beginnings, a key goal of the DDays LAC has been to promote cross-fertilization of ideas from different areas within nonlinear dynamics. On this occasion, the contributions range from experimental to theoretical research, including (but not limited to) chaos, control theory, synchronization, statistical physics, stochastic processes, complex systems and networks, nonlinear time-series analysis, computational methods, fluid dynamics, nonlinear waves, pattern formation, population dynamics, ecological modeling, neural dynamics, and systems biology. The interested reader will find this book to be a useful reference in identifying ground-breaking problems in Physics, Mathematics, Engineering, and Interdisciplinary Sciences, with innovative models and methods that provide insightful solutions. This book is a must-read for anyone looking for new developments of Applied Mathematics and Physics in connection with complex systems, synchronization, neural dynamics, fluid dynamics, ecological networks, and epidemics