Search

We formulated and analyzed a class of coupled partial and ordinary differential equation (PDE-ODE) model to study the spread of airborne diseases. Our model describes human populations with patches and the movement of pathogens in the air with linear diffusion. The diffusing pathogens are coupled to the SIR dynamics of each population patch using an integro-differential equation. Susceptible individuals become infected at some rate whenever they are in contact with pathogens (indirect transmission), and the spread of infection in each patch depends on the density of pathogens around the patch. In the limit where the pathogens are diffusing fast, a matched asymptotic analysis is used to reduce the coupled PDE-ODE model into a nonlinear system of ODEs, which is then used to compute the basic reproduction number and final size relation for different scenarios. Numerical simulations of the reduced system of ODEs and the full PDE-ODE model are consistent, and they predict a decrease in the spread of infection as the diffusion rate of pathogens increases. Furthermore, we studied the effect of patch location on the spread of infections for the case of two population patches. Our model predicts higher infections when the patches are closer to each other.

Early in a disease outbreak, it is important to be able to estimate the final size of the epidemic in order to assess needs for treatment and to be able to compare the effects of different treatment approaches. However, it is common for epidemics, especially of diseases considered dangerous, to grow much more slowly than expected. We suggest that by assuming behavioural changes in the face of an epidemic and heterogeneity of mixing in the population it is possible to obtain reasonable early estimates.

In the case of a disease spreading over a time-scale comparable to the average lifetime in a host population, when the infectiousness of individuals depends on the tine since the onset of infection...

To cope with annual seasonal influenza epidemics there is a program of vaccination before the “flu” season begins. Each year a vaccine is produced aimed at protecting against the three influenza strains considered most dangerous for the coming season. We formulate a model to add vaccination to the simple SIR model under the assumption that vaccination reduces susceptibility (the probability of infection if a contact with an infected member of the population is made).

Communicable diseases that are endemic (always present in a population ) cause many deaths). For example, in 2011 tuberculosis caused an estimated 1,400,000 deaths and HIV/AIDS caused an estimated 1,200,000 deaths worldwide. According to the World Health Organization there were 627,000 deaths caused by malaria, but other estimates put the number of malaria deaths at 1,2000,000. Measles, which is easily treated in the developed world, caused 160,000 deaths in 2011, but in 1980 there were 2,600,000 measles deaths. The striking reduction in measles deaths is due to the availability of a measles vaccine. Other diseases such as typhus, cholera, schistosomiasis, and sleeping sickness are endemic in many parts of the world. The effects of high disease mortality on mean life span and of disease debilitation and mortality on the economy in afflicted countries are considerable. Most of these disease deaths are in less developed countries, especially in Africa, where endemic diseases are a huge barrier to development.

Epidemic models with a general infective period distribution are formulated as functional differential equations, as are population models with a general life span distribution. The analysis of the local stability properties of equilibria of such models leads to a characteristic equation involving the Laplace transform of the infective period (or life span) distribution. We obtain conditions under which all roots of the characteristic equation are in the left half plane, implying asymptotic stability of equilibrium, for every infective period distribution. We also consider the converse problem of describing when instability can occur for specific infective period distributions.

Simple models for disease transmission that include immigration of infective individuals and variable population size are constructed and analyzed. A model with a general contact rate for a disease that confers no immunity admits a unique endemic equilibrium that is globally stable. A model with mass action incidence for a disease in which infectives either die or recover with permanent immunity has the same qualitative behavior. This latter result is proved by reducing the system to an integro-differential equation. If mass action incidence is replaced by a general contact rate, then the same result is proved locally for a disease that causes fatalities. Threshold-like results are given, but in the presence of immigration of infectives there is no disease-free equilibrium. A considerable reduction of infectives is suggested by the incorporation of screening and quarantining of infectives in a model for HIV transmission in a prison system.

In modelling sexually transmitted diseases, it is standard to divide the population into a very active core group and a larger but much less active noncore group. Transitions between these two groups may be affected by the prevalence of disease in the core group. There may be a time-lag in this effect, and we explore the relation between such a time-lag and the stability of an endemic equilibrium.

Antiviral treatment is one of the key pharmacological interventions against many infectious diseases. This is particularly important in the absence of preventive measures such as vaccination. However, the evolution of drug-resistance in treated patients and its subsequent spread to the population pose significant impediments to the containment of disease epidemics using treatment. Previous models of population dynamics of influenza infection have shown that in the presence of drug-resistance, the epidemic final size (i.e., the total number of infections throughout the epidemic) is affected by the treatment rate. These models, through simulation experiments, illustrate the existence of an optimal treatment rate, not necessarily the highest possible rate, for minimizing the epidemic final size. However, the conditions for the existence of such an optimal treatment rate have never been found. Here, we provide these conditions for a class of models covered in the literature previously, and investigate the combination effect of treatment and transmissibility of the drug-resistant pathogen strain on the epidemic final size. For the first time, we obtain the final size relations for an epidemic model with two strains of a pathogen (i.e., drug-sensitive and drug-resistant). We also discuss this model with specific functional forms of de novo resistance emergence, and illustrate the theoretical findings with numerical simulations.

Vertical transmission, the direct transfer of a disease from an infective parent to a new born offspring, is an important aspect of many diseases. Most of what is known for vertical transmission models assumes birth rates proportional to population size. We consider models with nonlinear population dynamics and finite carrying capacity and analyze the stability of equilibria in the special case in which the overall birth rate does not depend on infective population size.

Up to now in our study of continuous population models we have been assuming that x′(t), the growth rate of population size at time t, depends only on x(t), the population size at the same time t. However, there are situations in which the growth rate does not respond instantaneously to changes in population size. One of the first models incorporating a delay was proposed by Volterra (1926) to take into account the delay in response of a population’s death rate to changes in population density caused by an accumulation of pollutants in the past. Other causes of response delays which have been mentioned in the biological literature include differences in resource consumption with respect to age structure, migration and diffusion of populations, gestation and maturation periods, delays in behavioral response to environmental changes, and dependence of a population on a food supply that requires time to recover from grazing. In deriving a mathematical model to reflect a particular biological delay mechanism one must consider carefully how this mechanism affects the growth rate. One approach to modeling delays that has been used is formulation of a discrete model (or difference equation) and consideration of the delay in the time between steps. While this is appropriate for populations with a discrete reproduction cycle, such as many fish populations, it does not accurately model populations with continuous growth and time lags. The metered models studied in Section 2.5 allow for a continuous death process but involve a discrete reproduction stage.

In this chapter we shall consider populations with a fixed interval between generations or possibly a fixed interval between measurements. Thus, we shall describe population size by a sequence {x n }, with x 0 denoting the initial population size, x 1 the population size at the next generation (at time t 1), x 2 the population size at the second generation (at time t 2), and so on. The underlying assumption will always be that population size at each stage is determined by the population sizes in past generations, but that intermediate population sizes between generations are not needed. Usually the time interval between generations is taken to be a constant.

Populations may be structured by spatial location. There are two common different ways to include spatial location in a population. One way is by means of metapopulations, that is, populations of populations, with links between them such as a collection of towns and cities connected by a transportation network. The air transport subnetwork includes connecting links between distant communities, and we may study the dynamics of populations of different cities as a function of the flow of people between them and their own local dynamics in this framework. A metapopulation may be divided into patches, with each patch corresponding to a separate location. The corresponding models may be systems of ordinary differential equations, with the population size of each species in each patch as a variable. Thus metapopulation models are often systems of ordinary differential equations of high dimension. Some basic references are Hanski (1999), Hanski and Gilpin (1997), Levin, Powell and Steele (1993), Neuhauser (2001).

In this chapter we look at a population in which all individuals develop independently of one another while living in an unrestricted environment where no form of competition is possible. If the initial population size is small then a stochastic model is more appropriate, since the likelihood that the population becomes extinct due to chance must be considered. Deterministic models often provide useful ways of gaining sufficient understanding about the dynamics of populations whenever they are large enough. Furthermore, perturbations to large populations at equilibrium often generate over short time scales independent individual responses, which may be appropriately modeled by deterministic models. For example, the introduction of a single infected individual into a large disease-free population leads to the generation of secondary cases of infection, propagating a disease. The environment is free of interference competition, at least at the beginning of the outbreak, when a large population of susceptibles provides a virtually unlimited supply of hosts. The spread of disease in a large population of susceptibles may be thought of as an invasion process generated by independent contacts between a huge pool of susceptibles and a few infectious individuals.

We present continuous-time models for age-structured populations and disease transmission. We show how to use the method of characteristic lines to analyze the model dynamics and to write an age-structured population model as an integral equation model. We then extend to an agestructured SIR epidemic model. As an example we describe an age-structured model for AIDS, derive a formula for the reproductive number of infection, and show how important a role pair-formation plays in the modeling process. In particular, we outline the semi-group method used in an age-structured AIDS model with non-random mixing. We also discuss models for populations and disease spread with discrete age structure. 9.1 Why Age-Structured Models? In the simplest models for a single population all members are assumed to be interchangeable. However, even the simplest models for disease transmission include structuring the population by disease state (susceptible, exposed, infective, or removed). More advanced population models add some structure to the population such as specification of spatial location or age. Age is one of the most important characteristics in the modeling of populations and infectious diseases. Individuals with different ages may have different reproduction and survival capacities. Diseases may have different infection rates and mortality rates for different age groups [1]. Department of Mathematical Sciences, University of Alabama in Huntsville, 301, Sparkman Dr., Huntsville, AL 35899, USA li@math.uah.edu Department of Mathematics, University of British Columbia, 1984, Mathematics Road, Vancouver BC, Canada V6T 1Z2 brauer@math.ubc.ca

In this chapter we will consider populations of two interacting species with population sizes x(t) and y(t), respectively, modeled by a system of two first order differential equations: $$\begin{gathered} x' = F(x,y) \hfill \\ y' = G(x,y) \hfill \\ \end{gathered} $$ (5.1) .

Background. People change their behaviour during an epidemic. Infectious members of a population may reduce the number of contacts they make with other people because of the physical effects of their illness and possibly because of public health announcements asking them to do so in order to decrease the number of new infections, while susceptible members of the population may reduce the number of contacts they make in order to try to avoid becoming infected. Methods We consider a simple epidemic model in which susceptible and infectious members respond to a disease outbreak by reducing contacts by different fractions and analyze the effect of such contact reductions on the size of the epidemic. We assume constant fractional reductions, without attempting to consider the way in which susceptible members might respond to information about the epidemic. Results We are able to derive upper and lower bounds for the final size of an epidemic, both for simple and staged progression models. Conclusions The responses of uninfected and infected individuals in a disease outbreak are different, and this difference affects estimates of epidemic size.

We consider a simple model for a universally fatal disease with an infective period long enough to allow natural deaths during the infective period. The analysis of this model is considerably more complicated than the analysis of a model with an infective period short enough that the population dynamics are confined to the susceptible class. However, the basic result that in some circumstances the stability of an endemic equilibrium may depend on the distribution of infective periods is shared by both models.

In the formulation of models of S-I-R type for the spread of communicable diseases it is necessary to distinguish between diseases with recovery with full immunity and diseases with permanent removal by death. We consider models which include nonlinear population dynamics with permanent removal. The principal result is that the stability of endemic equilibrium may depend on the population dynamics and on the distribution of infective periods; sustained oscillations are possible in some cases.

The population sizes of two species competing for the same food supply or living space are often modelled by a pair of ordinary differential equations. If the growth rates of the two population sizes are linear in the population sizes, then coexistence in stable equilibrium implies qualified competition at equilibrium, in the sense that the effect of the competition is to increase the total population. Since experiments indicate that a stable equilibrium with unqualified competition is possible, this suggests that non-linear growth rates would give a more accurate model. An example is given of a model with non-linear growth rates which exhibits a stable equilibrium with unqualified competition.

The paper studies the effect of constant rate harvesting on population models described by ordinary differential equations. The dependence of limits of populations on the harvest rate, the possibility of extinction in finite time, and the effect of the introduction of a time-lag delay on stability are studied. Various models for one population are analysed, and a geometric technique for the study of two-population models is discussed.

In order to model populations which go through distinct stages, such as laboratory insect populations it is necessary to take the age structure of the population into account. The oscillations which have frequently been observed in the sizes of such populations cannot be explained by a model which ignores the effects of age structure. While it is feasible to use a fully age-structured model of McKendrickkvon Foerster type [3] or an analogous size-structured model [4], it is generally impossible to derive qualitative information about the behavior of solutions of such models. Accordingly, models have been proposed which subdivide the lifetime of an individual into two stages (larvae and adults), with different dynamics in each stage. Such models may be formulated as systems of differentialdifference equations, and several different types of systems of differentialdifference equations to model different types of larval competition have been proposed in [S]. Our purpose is to analyze the stability properties of the various models proposed in [S], using the standard method of determining equilibria, linearizing about an equilibrium, and locating the roots of the corresponding characteristic equation. By an equilibrium of a system of differential-difference equations we mean simply a constant solution; thus equilibria are found by setting the right side of each equation equal to zero (assuming that each equation of the system has been solved for the derivatives of the unknown functions). In cases where every constant is a solution of one of the equations in a system, it is necessary to replace that equation by an integrated form which incorporates the initial data in order to solve for equilibria. The linearization of a system about an equilibrium is obtained by using the deviations of the unknown functions from equilibrium as new variables, expanding by Taylor’s theorem, and retaining only the linear terms. The

We examine the qualitative effects of constant-rate stocking of either or both species in a predator-prey system. The hypotheses are made as mild as possible so that several types of systems with different qualitative alternatives may be studied.

A predator-prey System is modelled by a pair of ordinary differential equations, and the qualitative effects of prey nutrient enrichment and predator harvesting at a rate proportional to the predator population size are studied. Some theoretical analysis concerning the stability of equilibrium points and the existence of stable limit cycles are included. Three models are examined as examples, and for two of them computer simulations are included to illustrate the changes in qualitative behaviour under nutrient enrichment and increase of harvesting effort. The essential difference between this study and our previous work on constant-rate harvesting (Brauer et al. 1976) is that, here, extinction of predators in finite time is impossible although the predator population may tend to zero as l→∞

The global behaviour of a class of predator-prey systems, modelled by a pair of non-linear ordinary differential equations, under constant rate harvesting and/or stocking of both species, is presented. Theoretically possible structures and transitions are developed and validated by computer simulations. The results are presented as transition loci in the F-G (prey harvest rate-predator harvest rate) plane.

We consider simple compartmental models for infectious diseases with exposed and infective periods of fixed length. In particular, we examine the effect of incorporating nonlinear population dynamics into such models on threshold phenomena and the stability of endemic equilibria.

The roots of modern mathematical ecology lie in attempts to model population systems whose sizes were estimated from catch data, notably the hare and lynx data compiled by the Hudson’s Bay Company and the oscillation of fish populations in the Adriatic Sea [Volterra (1931)]. For more than 25 years the study of systems from which members are harvested has been of importance in the management of fisheries, see [Schaefer (1954)] or [Beverton and Holt (1957)] for the origins of such studies. Here, information about population size is used to estimate feasible catch sizes.

Publisher Summary This chapter discusses a nonlinear predator–prey problem. A problem that has been of some interest in biology is to describe the population of two species, one of which preys on the other. This occurs either in a predator–prey form or a host–parasite form. It is assumed that an increase in the prey population produces an increase in the predator population, while an increase in the predator population produces a decrease in the prey population. The predator–prey problemare first order differential equations with the functions f(x,y)and g(x,y), which describe the growth rates of the prey and predator species respectively linear in x and y. It is assumed that the existence of an equilibrium point in the first quadrant. When the functions f and g are linear, this equilibrium can be calculated explicitly as the intersection of the lines f = 0 and y = 0. In the nonlinear case, it is natural to expect that there should be an equilibrium point which is approximated by the equilibrium point of the linearized problem.