Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory

The study of infectious diseases represents one of the oldest and richest areas in mathematical biology. Infectious diseases have fascinated mathematicians for a century, and with good reason. Most seductive, of course, is the possibility of using mathematics to make a positive contribution to the world. In the study of infectious diseases, the essential elements are quickly grasped, and well-captured within mathematical representations. As new epidemics, from AIDS to bovine spongiform encephalopathy (mad-cow) to foot-and-mouth, make their appearances on the world stage, mathematical models are essential to inform decision-making. Governments and health agencies turn to the leading modelers for advice, and news services seek them out for the clarity they can bring. It is a rare opportunity for relevance for those who spend so much of their time in otherwise abstract and esoteric exercises.

When I first met Fred Brauer, he was warning a class of first-semester calculus students about the dangers of applying techniques without thinking. As an example, he wrote on the chalkboard the expression\( \frac{{\sin x}}{n} \). He then proceeded to cancel the n’s from numerator and denominator: , leaving six.

It was fifty years ago that Kenneth Cooke (Ken) started as an instructor at the State College of Washington while studying for his PhD degree at Stanford University. From there he went to Pomona College, Claremont, where he advanced through the ranks, and where he is now Professor Emeritus. During this time he has been Chairman of the Department, and he has held research positions and visited many institutions, including Brown University, IMA Minnesota, Cornell University, England, Italy, Brazil and Mexico.

For the basic versions of the SIR and SIRS epidemic models estimates for the maximal prevalence are computed in terms of the basic reproduction number and other relevant quantities. Maximal prevalence is studied as a function of the rate of loss of immunity. Total size and total cost (total infected time) of the epidemic are estimated. For SIR models with demographic renewal it is investigated whether prevalence is a monotone function of the renewal rate.

Life occurs in stages. Human (earthly) life is the stage between conception and death, clearly divided into two separate stages by birth. There are more stages that are less clearly separated: childhood, youth, adulthood, senescence. Invertebrate animals may have more distinctly separated stages, like egg, larva (or instar), pupa, imago. There may be even several larval stages. Infectious diseases take their course through various stages: latent period, infectious period without symptoms (the two together usually form the incubation period), infectious period with symptoms, and often an immunity period. With some diseases, HIV/AIDS e.g., the infectious period can be further subdivided according to the progress of the disease.

A fully stochastic model is used to study the phenomena of recurrence and extinction associated with childhood infections. This model is also used to investigate the properties of the corresponding deterministic model. We conclude that recurrent epidemics have such intricate behaviour that they cannot be studied with deterministic models; recognition of demographic stochasticity is necessary to understand them. The damping associated with the oscillations appearing in the deterministic model is shown to be a measure of the stochastic variability of the time between successive outbreaks. Furthermore, chaotic deterministic models for recurrent epidemics are shown to be unrealistic. The mechanism that drives chaos in these models is an unjustified mathematical approximation introduced in going from the stochastic to the deterministic formulation.

This paper considers SIR and SIS epidemics among a population partitioned into households. This heterogeneity has important implications for the threshold behaviour of epidemics and optimal vaccination strategies. It is shown that taking into account household structures when modelling public health problems is valuable. An overview of households models is given, including a determination of threshold parameters, the probability of a global epidemic and some new results on vaccination strategies for SIS households epidemics. Simulation and numerical studies are presented which exemplify the results discussed.

Ideal vaccine effect statistics should reflect biologically relevant parameters in an appropriate model of vaccine actions upon infection in the host. Ideal vaccine effectiveness statistics should reflect the effect of vaccination on the entire population or upon segments of that population such as vaccinated and unvaccinated individuals. The most commonly used vaccine effect statistic does not meet these ideals. It is one minus the risk in the vaccinated over the risk in the unvaccinated. These risks are sometimes calculated for disease and sometimes for infection. In this paper, we consider only infection. We label this statistic α and the risks in the vaccinated and unvaccinated populations on which it is based as R _v and R _u , respectively:

We develop simple epidemic models of co-circulating strains of an infectious disease in which the strains interact immunologically via cross-protective acquired immune responses. Two limiting forms of cross-protective immunity are explored: reduction of infectivity on infection with a strain that against which a degree of cross-protective immunity exists from prior excposure to a heterologous strain, and reduction of susceptibility to infection after exposure to the second strain. After developing a generic model framework capable of representing both forms of action, we show that model formulation can be simplified for some simple cross-immunity structures in the case of infectivity reduction. We then discuss equilbria and stability properties of the generic model, before investigating in detail the special case of allele-based cross-immunity, where antigenic relatedness depends on the number of alleles shared between two strains of a haploid pathogen. For this system, we present conditions for the stability of the symmetric and boundary equilibria in the case of purely infectivity-mediated cross-immunity, and illustrate numerically the wide range of complex limit cycle or chaotic dynamics that dominate a large region of parameter space. Finally, we describe the similarities between the dynamics exhibited by systems with each form of immunity action, and discuss biological applications of such models.

The aim of this paper is to provide an overview of existing models where multiple strains are coupled by cross-immunity. We discuss their differences and similarities, and propose a method to abstract some universal properties intrinsic to the coupling structure. More precisely, the coupling structure of a multiple-strain system can be organized as a matrix that is often invariant under many symmetry operations. Symmetries are known to constrain the behaviour of dynamical systems in many ways. Some symmetry effects are intuitive, but sometimes they can be rather subtle. Given that the assumptions and mechanisms of coupling strains are expected to have a major influence in determining the behaviour of the system, methods and techniques for abstracting their effects are valuable.

Evolutionary issues are very relevant in the comprehension of emerging or re-emerging disease (Ewald 1994): it seems likely that benign diseases may become, by evolutionary changes in virulence or in the capability of evading immune response, a new threat for the health of humans and animals, and thus be considered ‘emerging diseases’ (Dieckmann et al. 2000).

Disease models with recovery rates depending on disease-age and with exponentially distributed natural life spans have been studied by H. Thieme, C. Castillo-Chavez, and others. Here, we formulate S-I-R models in which both disease recovery and natural life spans have arbitrary distributions. We focus on the relation between the basic reproductive number, the mean life span, and the mean age at infection. The S-I model, with no recovery, is analyzed completely and partial stability results are obtained for models with mean infective period much shorter than mean life span.

Several AIDS cohort studies observe that the incubation period between HIV infection and AIDS onset can be shorter than 3 years in about 10% seropositive individuals, or longer than 10 years in about 10–15% individuals. On the other hand, many individuals remain seronegative even after multiple exposures to HIV. These distinct outcomes have recently been correlated with some mutant genes in HIV co-receptors (e.g., CCR5, CCR2 and CXCR4). For instance, the mutant alleles Δ32 and m303 of CCR5 may provide full protection against HIV infection in homozygotes and partial protection in heterozygotes; moreover, infected heterozygotes may progress more slowly than individuals who have no mutant alleles. Frequencies of these mutant alleles are not very low in Caucasian populations, therefore their effects may not be insignificant. To investigate the impact of such heterogeneity on the spread of HIV, Hsu Schmitz (2000a,2000b), based on available data, proposes a specific type of models where susceptibles are classified as having no, partial or full natural resistance to HIV infection and infecteds as rapid, normal or slow progressors. She also applies the models to CCR5-Δ 32 mutation in San Francisco gay men. This manuscript sketches her models with focus on the basic model without treatment and an extended model with treatment in certain proportion of newly infected individuals. The same example of CCR5-Δ 32 in San Francisco gay men is used, but some parameters are estimated in different ways. The results are very similar to those in Hsu Schmitz (2000a,2000b) with the following two main conclusions: 1) without any intervention, HIV infection will continue to spread in this population and the epidemic is mainly driven by the normal progressors; 2) treating only a certain proportion of newly infected individuals with currently available therapies is unlikely to eradicate the disease. Additional interventions are thus necessary for disease control.

The recruitment of new suceptibles into a core group of sexually-active individuals may depend on the current levels of infection within a population. We extend the formalism of Hadeler and Castillo-Chavez (1995), that includes prevalence dependent recruitment rates, to include age structure within core and noncore populations. Some mathematical results are stated but only a couple of proofs are included since our objetives are to highlight the modeling process and the dynamic possibilities. This paper concludes with an example where endemic distributions can be supported when the basic reproductive number R₀ is less than one. Systems that are capable of supporting multiple attractors are more likely to support disease re-emergence. This model is likely to support stable multiple attractors when R₀ < 1.

Mathematical models for Tuberculosis with linear and logistic growth rates are considered. The global dynamic structure for the logistic recruitment model is analyzed with the help of a strong version of the Poincaré-Bendixson Theorem. The nature of the global dynamics of the same model with a linear recruitment rate is established with the use of explicit threshold quantities controlling the absolute and relative spread of the disease and the likelihood of extinction or persistence of the total population. The classification of planar quadratic systems helps rule out the existence of closed orbits (limit cycles).

The dynamics of many epidemic models for infectious diseases that spread in a single host population demonstrate a threshold phenomenon. If the basic reproduction number R₀ is below unity, the disease-free equilibrium P₀ is globally stable in the feasible region and the disease always dies out. If R₀ > 1, a unique endemic equilibrium P_* is globally asymptotically stable in the interior of the feasible region and the disease will persist at the endemic equilibrium if it is initially present. In this paper, this threshold phenomenon is established for two epidemic models of SEIR type using two recent approaches to the global-stability problem.

A general SIS model with chronological age and infection age structures is formulated. We analyze the global dynamics of the model with a constructive iteration procedure. The basic reproductive number R ₀ is calculated using the next generation operator approach. R ₀ plays a sharp threshold role in determining the global dynamics, i.e., the endemic steady-state is globally asymptotically stable if R ₀ > 1, while the disease-free steady-state is globally asymptotically stable if R ₀ ≤ 1. The basic reproductive number is over estimated where the infection age is ignored.

In this paper, our main purpose is to investigate mathematical aspects of the Pease’s evolutionary epidemic model for type A influenza. First we formulate the Pease model as an abstract semilinear Cauchy problem and construct the semigroup solution. Next we prove existence and uniqueness of the endemic steady state and show endemic threshold phenomena. Subsequently by using semigroup approach, we investigate the local stability of endemic steady state. We prove that the endemic steady state is locally asymptotically stable if its prevalence is greater than fifty percent. Finally we discuss some possible extensions of the Pease’s model and open problems.