Stavros Busenberg

Analysis of a disease transmission model in a population with varying size

An S → I → R → S epidemiological model with vital dynamics in a population of varying size is discussed. A complete global analysis is given which uses a new result to establish the nonexistence of periodic solutions. Results are discussed in terms of three explicit threshold parameters which respectively govern the increase of the total population, the existence and stability of an endemic proportion equilibrium and the growth of the infective population. These lead to two distinct concepts of disease eradication which involve the total number of infectives and their proportion in the population.

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.

Global behavior of an age-structured epidemic model

The global behavior of the general

Separable models in age-dependent population dynamics

s \to i \to s

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

age-structured epidemic model in a population of constant size is obtained. It is shown that there is a sharp threshold which determines the existence and global stability of an endemic state; hence, periodic solutions are ruled out. The threshold is identified as the spectral radius of a positive linear operator. The analysis employs the theory of semigroups and positive operator methods, and is based on the formulation of the problem as an abstract differential equation in a Banach space.

The dynamics of a model of a plankton-nutrient interaction

A class of population models is considered in which the parameters such as fecundity, mortality and interaction coefficients are assumed to be age-dependent. Conditions for the existence, stability and global attractivity of steady-state and periodic solutions are derived. The dependence of these solutions on the maturation periods is analyzed. These results are applied to specific single and multiple population models. It is shown that periodic solutions cannot occur in a general class of single population age-dependent models. Conditions are derived that determine whether increasing the maturation period has a stabilizing effect. In specific cases, it is shown that any number of switches in stability can occur as the maturation period is increased. An example is given of predator-prey model where each one of these stability switches corresponds to a stable steady state losing its stability via a Hopf bifurcation to a periodic solution and regaining its stability upon further increase of the maturation period.

Demography and 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.

A class of nonlinear diffusion problems in age-dependent population dynamics☆

A stability analysis is given for a model of plankton dynamics introduced by Wroblewski et al. (Global Biogeochem. Cycles 2, 199–218, 1988). The detailed dependence of the steady-states and their stability on the various model parameters is explicitly presented and analysed. It is shown that under certain conditions the coexistence of phytoplankton and zooplankton occurs in an orbitally stable oscillatory mode. A distinguished parameter is varied and the steady-states computed. The significance of the lack of stable steady-states leading to periodic population levels is investigated and related to certain oceanographic data.

The effect of integral conditions in certain equations modelling epidemics and population growth

Abstract For a demographic SIRS epidemic model with vertical transmission, the balance between recruitment of new susceptibles and the persistence of the disease is investigated.

Interaction of spatial diffusion and delays in models of genetic control by repression

IN THIS paper we study the existence, the uniqueness and the asymptotic behaviour of a class of nonlinear diffusion problems. These problems are motivated by models of age-dependent population dynamics which originate in the article by Gurtin & MacCamy [ll]. Only a few special cases of these models have been analyzed to date, with the most recent published work being that of MacCamy [17], Gurtin & MacCamy [12] and Langlais [15, 161. The cases treated in [12, 171 are restricted to special simple forms of the birth and death moduli that occur in these models and to a specific form of the nonlinear diffusion mechanism. On the other hand, Langlais [16] considers a different nonlinear diffusion mechanism and obtains a type of generalized solution for general forms of the birth and death moduli. In this work, we consider a general form of the nonlinear diffusion mechanism and we develop a method for establishing the existence and uniqueness of solutions of the resulting equations. Our method allows us to obtain classical solutions even with general forms of the birth and death moduli. We also describe the asymptotic behaviour of the solutions when the time variable becomes large, and apply our results to some specific examples. The equations that we study are motivated as follows. Consider a population that can disperse in a spatial domain R. For simplicity we take Q = J, where J is an open interval in R, but it will be clear from the context that more general regions Q of R” can be considered. Let ~(a, t, x) denote the number of individuals, per unit length and unit age, that are of age a at time t and are located at the position x E J. The total population, per unit length, at time t and at position x is given by