Time delayed models in population biology and epidemiology

Al-Darabsah, Isam (2018) Time delayed models in population biology and epidemiology. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

In this dissertation, we focus on the development and analysis of time-delayed mathematical models to represent real world applications in biology and epidemiology, especially then analyze the models using various theorems and methods in the literature, such as, the comparison principle and the method of fluctuations, to study qualitative features of the models including existence and uniqueness of solutions, boundedness, steady states, persistence, local, and global stability, with respect to the adult/basic reproduction number Ra/R₀, which is a key threshold parameter. Firstly, we discuss ecological models in Chapters 2-4. In Chapter 2, we derive a single species-fish model with three stages: juveniles, small adults and large adults with two harvesting strategies depending on the size and maturity. We study the population extinction and persistence with respect to Ra and find that the over-harvesting of large matured fish after a certain age can lead to population extinction under certain circumstances. Numerically, we investigate the influence of harvesting functions and discuss the optimal harvesting rates. In Chapter 3, we develop a model for the growth of sea lice with three stages such that the development age for non-infectious larvae to develop into infectious larvae relates to the size of adult population size. As a beginning, we describe the nonlinear dynamics by a system of partial differential equations, then, we transformed it into a system of delay differential equations with constant delay by using the method of characteristics and an appropriate change of variables. We address the system threshold dynamics for the established model with respect to the adult reproduction number, including the global stability of the trivial steady state, persistence, and global attractivity of a coexistence unique positive steady state. As a case study, we provide some numerical simulation results using Lepeophtheirus salmonis growth parameters. To explore the biological control of sea lice using one of their predators, "cleaner fish", we propose a model with predator-prey interaction at the adult level of sea lice in Chapter 4. Mathematically, we address threshold dynamics with respect to the adult reproduction number for sea lice Rs and the net reproductive number of cleaner fish Rf, including the global stability of the trivial steady state with Rs < 1, global attractivity of the predator-free equilibrium point when Rs > 1 and Rf < 1, persistence and coexistence of a unique positive steady state when Rs > 1 and Rf > 1. Furthermore, we discuss the local stability of the positive equilibrium point and investigate the Hopf bifurcation. Numerically, we compare between two cleaner fish species, goldsinny and ballan wrasse, as a case study. For epidemiological models, in Chapter 5, we propose an SEIRD model for Ebola disease transmission that incorporates both the transmission of infection between the living humans and from the infected corpses to the living individuals, with a constant latent period. Through mathematical analysis, we prove the globally stability of the disease-free and a unique endemic equilibria with respect to R₀. Moreover, we find that the long latent period or low transmission rate from infectious corpses may reduce the spread of Ebola. In Chapters 6, we consider the influence of seasonal fluctuations on disease transmission and develop a periodic infectious disease model where asymptomatic carriers are potential sources for disease transmission. We consider a general nonlinear incidence rate function with the asymptomatic carriage and latent periods. We implement a case study regarding the meningococcal meningitis disease transmission in Dori, Burkina Faso. Our numerical simulation indicates an irregular pattern of epidemics varying in size and duration, which is consistent with the reported data in Burkina Faso from 1940 to 2014. In summery, in population growth models, we find that the basic reproduction ratio depends on maturation time, indicating that this key parameter can play an important role in population extinction and persistence. In disease transmission model, we understand that latent period can play a positive role in eliminating or slowing a disease spread.

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