Analysis of spatial dynamics and time delays in epidemic models

Reaction-diffusion systems and delay differential equations have been extensively used over the years to model and study the dynamics of infectious diseases. In this thesis we consider two aspects of disease dynamics: spatial dynamics in a reaction-diffusion epidemic model with nonlinear incidence rate, and a delayed epidemic model with combined effects of latency and temporary immunity. The first part of the thesis is devoted to the analysis of stability and pattern formation in an SIS-type epidemic model with nonlinear incidence rate. By considering the dynamics without spatial component, conditions for local asymptotic stability are obtained for general values of the powers of nonlinearity. We prove positivity, boundedness, invariant principle and permanence of our model. The next generation matrix method is used to derive the corresponding basic reproductive number R0, and the Routh-Hurwitz criterion is used to show that for R0 ≤ 1, the disease-free equilibrium is found to be locally asymptotically stable, for R0 > 1, a unique endemic steady state exists and is found to be locally asymptotically stable. In the presence of diffusion, Turing instability conditions are established in terms of system parameters. Numerical simulations are performed to identify the spatial regions for spots, stripes and labyrinthine patterns in the parameter space. Numerical simulations show that the system has complex and rich dynamics and can exhibit complex patterns, depending on the recovery rate r and the transmission rate β. We have discovered that whenever the transmission rate exceeds the recovery rate the system exhibits stripe patterns which correspond to a disease outbreak, and in the opposite case the system settles on spot patterns which imply the absence of disease outbreaks. Also, we find that increasing the power q can lead to epidemic outbreak even at lower values of the transmission rate β. All numerical simulations use an Implicit-Explicit (IMEX) Euler’s method, which computes diffusion terms in Fourier space and reaction terms in the real space. Numerical approximation of the model is benchmarked to prove stability of the numerical scheme, and the method is shown to converge with the correct order. Experimental order of convergence (EOC) and estimates for the error in both L2, H1 and maximum norms have also been computed. Also, we compare our results to those on infectious diseases and our model shows good predictions. In the second part of this thesis, we derive and analyse a delayed SIR model with bilinear incidence rate and two time delays which represent latency Τ1 and temporary immunity Τ2 periods. We prove both local and global stability of the system equilibria in the case when there are no time delays, i.e. both the latency and temporary immunity periods are set to zero. For the case when there is only latency (Τ1 > 1, Τ2 = 0) and the case when the two time delays are identical (Τ1 = Τ2 = Τ ), we show that the endemic steady state is always stable for any parameter values. For the case when there is only temporary immunity (Τ2 > 0, Τ1 = 0) and the case when there are both latency and temporary immunity in the system (Τ1 > 0, Τ2 > 0), we prove the existence of periodic solutions arising from the Hopf bifurcation. The endemic steady state undergoes Hopf bifurcation giving rise to stable periodic solutions. For the last two cases, we show interesting regions of (in)stability of the endemic steady state in the different parameter regimes. We find that by varying the transmission rate β, the natural death rate γ and the disease-induced death rate μ increase the regions of (in)stability. Also, we find that the dynamics of the system is richer when we have the two time delays in the model. Analytical results are supported by extensive numerical simulations, illustrating temporal behaviour of the system in different dynamical regimes. Finally, we relate our results to modelling infectious diseases and our results show good predictions of safety and epidemic outbreak

STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS IN ECOLOGY AND EPIDEMICS

Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and predictions in various areas of life sciences, such as population dynamics, epidemiology, immunology, physiology, and neural networks. The memory or time-delays, in these models, are related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period, and so on. In ordinary differential equations (ODEs), the unknown state and its derivatives are evaluated at the same time instant. In DDEs, however, the evolution of the system at a certain time instant depends on the past history/memory. Introduction of such time-delays in a differential model significantly improves the dynamics of the model and enriches the complexity of the system. Moreover, natural phenomena counter an environmental noise and usually do not follow deterministic laws strictly but oscillate randomly about some average values, so that the population density never attains a fixed value with the advancement of time. Accordingly, stochastic delay differential equations (SDDEs) models play a prominent role in many application areas including biology, epidemiology and population dynamics, mostly because they can offer a more sophisticated insight through physical phenomena than their deterministic counterparts do. The SDDEs can be regarded as a generalization of stochastic differential equations (SDEs) and DDEs.This dissertation, consists of eight Chapters, is concerned with qualitative and quantitative features of deterministic and stochastic delay differential equations with applications in ecology and epidemics. The local and global stabilities of the steady states and Hopf bifurcations with respect of interesting parameters of such models are investigated. The impact of incorporating time-delays and random noise in such class of differential equations for different types of predator-prey systems and infectious diseases is studied. Numerical simulations, using suitable and reliable numerical schemes, are provided to show the effectiveness of the obtained theoretical results.Chapter 1 provides a brief overview about the topic and shows significance of the study. Chapter 2, is devoted to investigate the qualitative behaviours (through local and global stability of the steady states) of DDEs with predator-prey systems in case of hunting cooperation on predators. Chapter 3 deals with the dynamics of DDEs, of multiple time-delays, of two-prey one-predator system, where the growth of both preys populations subject to Allee effects, with a direct competition between the two-prey species having a common predator. A Lyapunov functional is deducted to investigate the global stability of positive interior equilibrium. Chapter 4, studies the dynamics of stochastic DDEs for predator-prey system with hunting cooperation in predators. Existence and uniqueness of global positive solution and stochastically ultimate boundedness are investigated. Some sufficient conditions for persistence and extinction, using Lyapunov functional, are obtained. Chapter 5 is devoted to investigate Stochastic DDEs of three-species predator prey system with cooperation among prey species. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution to the model are established, by constructing a suitable Lyapunov functional. Chapter 6 deals with stochastic epidemic SIRC model with time-delay for spread of COVID-19 among population. The basic reproduction number ℛs0 for the stochastic model which is smaller than ℛ0 of the corresponding deterministic model is deduced. Sufficient conditions that guarantee the existence of a unique ergodic stationary distribution, using the stochastic Lyapunov functional, and conditions for the extinction of the disease are obtained. In Chapter 7, some numerical schemes for SDDEs are discussed. Convergence and consistency of such schemes are investigated. Chapter 8 summaries the main finding and future directions of research. The main findings, theoretically and numerically, show that time-delays and random noise have a significant impact in the dynamics of ecological and biological systems. They also have an important role in ecological balance and environmental stability of living organisms. A small scale of white noise can promote the survival of population; While large noises can lead to extinction of the population, this would not happen in the deterministic systems without noises. Also, white noise plays an important part in controlling the spread of the disease; When the white noise is relatively large, the infectious diseases will become extinct; Re-infection and periodic outbreaks can also occur due to the time-delay in the transmission terms

Amplitude death and restoration in networks of oscillators with random-walk diffusion

Systems composed of reactive particles diffusing in a network display emergent dynamics. While Fick's diffusion can lead to Turing patterns, other diffusion schemes might display more complex phenomena. Here we study the death and restoration of collective oscillations in networks of oscillators coupled by random-walk diffusion, which modifies both the original unstable fixed point and the stable limit-cycle, making them topology-dependent. By means of numerical simulations we show that, in some cases, the diffusion-induced heterogeneity stabilizes the initially unstable fixed point via a Hopf bifurcation. Further increasing the coupling strength can moreover restore the oscillations. A numerical stability analysis indicates that this phenomenology corresponds to a case of amplitude death, where the inhomogeneous stabilized solution arises from the interplay of random walk diffusion and heterogeneous topology. Our results are relevant in the fields of epidemic spreading or ecological dispersion, where random walk diffusion is more prevalent

Pattern formation in electrically coupled pacemaker cells : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū, New Zealand

Figures are re-used with permission.In this thesis we study electrical activity in smooth muscle cells in the absence of external stimulation. The main goal is to analyse a reaction-diffusion system that models the dynamical behaviour where adjacent cells are coupled through passive electrical coupling. We first analyse the dynamics of an isolated muscle cell for which the model consists of three first-order ordinary differential equations. The cell is either excitable, nonexcitable, or oscillatory depending on the model parameters. To understand this we reduce the model to two equations, nondimensionalise, then perform a detailed numerical bifurcation analysis of the nondimensionalised model. One parameter bifurcation diagrams reveal that even though there is no external stimulus the cell can exhibit two fundamentally distinct types of excitability. By computing two-parameter bifurcation diagrams we are able to explain how the cell transitions between the two types of excitability as parameters are varied. We then study the full reaction-diffusion system first through numerical integration. We show that the system is capable of exhibiting a wide variety of spatiotemporal behaviours such as travelling pulses, travelling fronts, and spatiotemporal chaos. Through a linear stability analysis we are able to show that the spatiotemporal patterns are not due to diffusion-driven instability as is often the case for reaction-diffusion systems. It is as a consequence of the nonlinear dynamics of the reaction terms and coupling effect of diffusion. The precise mechanism is not yet well understood, this will be subject of future work. We then examine travelling wave solutions in detail. In particular we show how they relate to homoclinic and heteroclinic solutions in travelling wave coordinates. Finally we review spectral stability analysis for travelling waves and compute the essential spectrum of travelling waves in our system

Pattern formation in three species food web model in spatiotemporal domain with Beddington–DeAngelis functional response

A mathematical model is proposed to study a three species food web model of preypredator system in spatiotemporal domain. In this model, we have included three state variables, namely, one prey and two first order predators population with Beddington-DeAngelis predation functional response. We have obtained the local stability conditions for interior equilibrium and the existence of Hopf-bifurcation with respect to the mutual interference of predator as bifurcation parameter for the temporal system. We mainly focus on spatiotemporal system and provided an analytical and numerical explanation for understanding the diffusion driven instability condition. The different types of spatial patterns with respect to different time steps and diffusion coefficients are obtained. Furthermore, the higher-order stability analysis of the spatiotemporal domain is explored