Kenneth L. Cooke

Differential-Difference Equations

Publisher Summary A systematic development of the theory of differential–difference equations was not begun until E. Schimdt published an important paper about fifty years ago. The subsequent gradual growth of the field has been replaced, in the last decade or so, by a rapid expansion due to the stimulus of various applications. This chapter introduces the study of differential–difference equations and discusses some of the main features of the theory. The role of differential–difference equations is vital in some areas, such as engineering problem and fluid mechanics. In engineering problem, the problem of controlling the temperature in a reaction tank is addressed using differential difference equations. The temperature variation is reported because of random disturbances, inherent effects due to u being non-zero, and the operation of the control device. The chapter discusses the asymptotic behavior of solutions and the problem of stability.

Discrete delay, distributed delay and stability switches

In modelling in the biological, physical and social sciences, it is sometimes necessary to take account of time delays inherent in the phenomena. The inclusion of delays explicitly in the equations is often a simplification or idealization that is introduced because a detailed description of the underlying processes is too complicated to be modelled mathematically, or because some of the details are unknown. In these cases, it may be necessary to choose between a model with discrete or sharp delays and a model with distributed or continuous delay. A question of great importance is whether two models with parallel structure, one with discrete delay and one with distributed delay, will exhibit the same qualitative modes of behaviour. More generally, how does the qualitative behaviour depend on the form and magnitude of the delays? In this paper we shall examine certain aspects of this question. The paper is divided into two parts. In the first part (Sections 2-6), we examine how the stability properties of certain models change when the delay is increased. It has frequently been observed that stability of an equilibrium may be lost when delays are increased. Less frequently. it has been seen that further increase in the delay may result in restabilization. In this paper, we examine the possibilities for several simple equations: (1) a first order linear differential-difference equation; (2) a second order delayed friction model; (3) a second order equation with delayed restoring force; and (4) a population growth model of J. Cushing. In (2) and (3) and in a general equation including both, we show that there may be arbitrarily many switches from stability to instability to stability as the delay is increased, but in (1) this is not possible. In (4), the equation has distributed delay. and

The shortest route through a network with time-dependent internodal transit times

Abstract The shortest route problems so far analyzed fall seriously short of reality in that they assume that the time required between any two vertices (nodes) is constant, an assumption which is certainly not true in many physical and biological applications. This analysis demonstrates how to perform a shortest route iteration in the more realistic case where internodal time requirements are time-variable. In the paper a modified form of Bellmans iteration scheme [1] for finding the shortest route between any two vertices in a network is developed for application to our generalized case. It converges to the shortest path between any two vertices in a finite number of iterations, and for any initial starting time, provided that certain initial conditions are satisfied. One of the main points of this treatment has been to arrange the work so that when computing a new iterate one does not have to recompute previous iterates at more advanced time points.

Retarded differential equations with piecewise constant delays

Profound and close links exist between functional and functional differential equations. Thus, the study of the first often enables one to predict properties of differential equations of neutral type. On the other hand, some methods for the latter in the special case when the deviation of the argument vanishes at individual points have been used to investigate functional equations [ 11. Functional equations are directly connected with difference equations of a discrete (for example, integer-valued) argument, the theory of which has been very intensively developed in the book [2] and in numerous subsequent papers. Bordering on difference equations are the impulse functional differential equations with impacts and switching, loaded equations (that is, those including values of the unknown solution for given constant values of the argument), equations

Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission

This paper examines a multigroup epidemic model with variable population size. It is shown that even in the case of proportionate mixing, multiple endemic equilibria are possible. The significance of these results in the study of the dynamics of sexually transmitted diseases and theoretical biology is discussed.

MODELS OF VERTICALLY TRANSMITTED DISEASES WITH SEQUENTIAL-CONTINUOUS DYNAMICS

Publisher Summary This chapter discusses the models of vertically transmitted diseases with sequential-continuous dynamics. It presents the analysis of certain models of vertically transmitted diseases propagated by invertebrate vectors with discrete generations. There are a number of such diseases that are economically and sociologically important. The chapter discusses the example of Rocky Mountain spotted fever. The organism that causes this disease is Rickettsia rickettsi , which is transmitted to humans and other large mammals via contact with infected adult ticks. The models discussed in the chapter rely on the population dynamics of the tick vectors. The chapter presents a schematic description of these seasonal dynamics for one of these arthropod vectors, Dermacentor variabilis (American dog tick). The chapter presents a model to assess the influence of the two modes of transmission of the disease. The model collects a few salient parameters that affect the progress of this disease in the ticks and gives quantitative estimates of their relative influence and importance.

A periodicity threshold theorem for epidemics and population growth

Abstract Some infectious diseases have an incidence which is periodic in time. Contact rates may vary greatly during a year due to seasonal factors. In this paper we study the scalar delay integral equation x(t)= ʃ t t-τ f (s,x(s))ds , where f(t,x) is a continuous function which is periodic in t, f(t,0)=0, and where τ is a positive constant. This equation can be interpreted as a model for some infectious diseases with periodic contact rate, or as a growth equation for a single species population when the birth rate varies seasonally. The principal result is a threshold theorem: it is proved that if τ is small enough, all nonnegative solutions approach 0 as t→∞, whereas if τ is large enough, there exists a positive periodic solution with period equal to the period of f. Estimates on the critical size of τ are given in terms of ∂f ∂x (t,0) . A numerical example is given.

On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrome (AIDS) - Part 1: Single population models

In this study, we investigate systematically the role played by the reproductive number (the number of secondary infections generated by an infectious individual in a population of susceptibles) on single group populations models of the spread of HIV/AIDS. Our results for a single group model show that if R ⩽ 1, the disease will die out, and strongly suggest that if R > 1 the disease will persist regardless of initial conditions. Our extensive (but incomplete) mathematical analysis and the numerical simulations of various research groups support the conclusion that the reproductive number R is a global bifurcation parameter. The bifurcation that takes place as R is varied is a transcritical bifurcation; in other words, when R crosses 1 there is a global transfer of stability from the infection-free state to the endemic equilibrium, and vice versa. These results do not depend on the distribution of times spent in the infectious categories (the survivorship functions). Furthermore, by keeping all the key statistics fixed, we can compare two extremes: exponential survivorship versus piecewise constant survivorship (individuals remain infectious for a fixed length of time). By choosing some realistic parameters we can see (at least in these cases) that the reproductive numbers corresponding to these two extreme cases do not differ significantly whenever the two distributions have the same mean. At any rate a formula is provided that allows us to estimate the role played by the survivorship function (and hence the incubation period) in the global dynamics of HIV. These results support the conclusion that single population models of this type are robust and hence are good building blocks for the construction of multiple group models. Our understanding of the dynamics of HIV in the context of mathematical models for multiple groups is critical to our understanding of the dynamics of HIV in a highly heterogeneous population.

On the problem of linearization for state-dependent delay differential equations

The local stability of the equilibrium for a general class of statedependent delay equations of the form ẋ(t) = f ( xt, ∫ 0 −r0 dη(s)g(xt(−τ(xt) + s)) ) has been studied under natural and minimal hypotheses. In particular, it has been shown that generically the behavior of the state-dependent delay τ (except the value of τ) near an equilibrium has no effect on the stability, and that the local linearization method can be applied by treating the delay τ as a constant value at the equilibrium.

Endemic thresholds and stability in a class of age-structured epidemics

An age-structured epidemic model is analyzed when the fertility, mortality and removal rates depend on age. For certain general forms of the force of infection terms, endemic threshold criteria are derived and the stability of steady state solutions is determined. The relation between age-structured models of this type and catalytic curve models of epidemics is derived. The possibility of identifying vertically transmitted diseases from the catalytic curve is demonstrated.