DIMACS Summer School Tutorial on Dynamic Models of Epidemiological Problems

June 24 - 27, 2002
Rutgers University, Piscataway, NJ

Organizers:

Carlos Castillo-Chavez,, Cornell University, Biometrics, cc32@cornell.edu

Herbert Hethcote, University of Iowa, Mathematics, herbert-hethcote@uiowa.edu

Pauline van den Driessche, University of Victoria, Mathematics and Statistics, pvdd@math.uvic.ca

Abstracts (in order of the schedule):

1.

Three Basic Epidemiology Models I

Herbert Hethcote
University of Iowa

This introductory lecture develops the three basic types of
epidemiological models for infectious diseases, formulates them as
systems of nonlinear differential equations, identifies the thresholds
and equilibrium points, and describes their dynamical behavior by
analyzing local and global stability of these models. The goal is to
introduce notation, terminology, and results that will be generalized
in later lectures on more advanced models.  Intuitive interpretations
will be given for the basic reproduction number, the contact number,
and the infective replacement number.



2.

Three Basic Epidemiology Models II

Herbert Hethcote
University of Iowa

This continuation of the previous introductory lecture looks at
applications of the basic epidemiology models. The basic reproduction
number is estimated for many directly transmitted diseases, herd
immunity is defined, and the fraction that must be vaccinated to
obtain herd immunity is determined for diseases such as measles,
rubella, mumps, chickenpox, influenza, polio, and smallpox. These
diseases are compared with each other and the modeling results are
compared with national and international experience with these
diseases.



3.

Epidemiology Models with Immigration

Pauline van den Driessche
University of Victoria

Some models of disease transmission that include immigration of 
infectives and variable population size are constructed and 
analyzed. Threshold-like results are given, but in the presence 
of immigration of infectives there is no disease-free equilibrium. 
In a model for a disease that confers no immunity, the number of 
infectives is shown to tend to an endemic value.  In a simple model 
for HIV transmission in a prison system, a considerable reduction of 
infectives is suggested by the incorporation of screening and 
quarantining of infectives. Mathematical tools are presented as 
needed in the analysis.



4.

Thresholds and the Basic Reproduction Number RO

Pauline van den Driessche
University of Victoria

A general compartmental disease transmission model is formulated
as a system of ordinary differential equations, and a precise 
definition of the basic reproduction number RO is presented.
This parameter acts as a threshold, with the disease-free 
equilibrium being locally stable if RO <1, but unstable if 
RO >1. The results are illustrated by several specific models, 
for example, a treatment model for tuberculosis, a staged 
progression model for HIV/AIDS and a model that includes
vaccination.



5.

Epidemiology Models with Variable Population Size

Herbert Hethcote
University of Iowa

Basic epidemiology models often assume that births balance deaths, so
that the total population size is constant. But populations may be
growing or decreasing significantly due to differences in the natural
birth and death rates, an excess disease-related mortality, or
disease-related decreases in reproduction. In models with a variable
total population size, the persistence of the infectious disease may
slow the growth rate of a naturally growing population, lead to a
lower equilibrium population size, or even reverse the population
growth to a decay to extinction. Variable population size models of
SIS and SIR type are formulated and analyzed in order to demonstrate
the effects of the disease on the population size and the effects of
the population structure on the disease dynamics.



6.

Age Structured Models

Herbert Hethcote
University of Iowa

When modeling a disease in which vaccinations are given at different
ages, it is necessary to include both age and time as independent
variables. SEIR and MSEIR models are formulated with either continuous
age or discrete age groups.  Expressions for the basic reproduction
number are derived and Liapunov functions used to prove stability
below the thresholds. Values of the basic reproduction number and
contact number are estimated for various diseases including measles in
Niger, Africa and pertussis in the United States.



7.

Dynamical Models of Tuberculosis Transmission and Control 

Carlos Castillo-Chavez
Cornell University

The reemergence of Tuberculosis (TB) in the United States and
elsewhere has generated intense research on its transmission dynamics
and on potential methods of control.  These tutorial lectures will
focus on a review of recent tuberculosis models and some of the
questions that they addressed.

A discsussion of the epidemiology of Tuberculosis is followed by a
quick review of historical models for TB transmission before a simple
framework for modeling TB dynamics is introduced. Specific models
built on this framework are used to study the impact of various
factors on TB dynamics and its control (further discussion of these
topics will continue on the second lecture).

1. Demographic effects.  The modeling framework includes
density-dependent recruitment rates of various types that are used to
illustrate the impact of some demographic regimes on TB dynamics.

2. Fast and slow TB.  The impact of long and variable periods of
latency and progression on TB dynamics are studied.

3. Exogenous reinfections.  The impact of exogenous reinfections on TB
dynamics is addressed.

4. Age structure and vaccination strategies. A model with age
structured is instroduced and used to define the concept of optimal
vaccination strategy. Some recent results are presented and discussed.



8.

TB Cluster Models, Time Scales and Relations to HIV

Carlos Castillo-Chavez
Cornell University

The focus of this lecture is on a review of recent tuberculosis models
and some of the questions that they addressed. Specific models built
on the framework discussed in the first lecture on TB are modified to
study the impact of some additional factors on TB dynamics and its
control.

1. TB and HIV. A model for TB dynamics that incorporates the
demography of the US population and the recent impact of HIV is
considered. The possibility of reaching national goals on the
reduction of TB prevalence are discussed using this model.


2. Close and Casual Contacts. Models that incorporate local and
individual interactions are introduced in the context of the
transmission dynamics of tuberculosis (TB). The multi-level contact
structure implicitly assumes that individuals who are at risk of
infection from close contacts in generalized household (clusters) as
well as from casual (random) contacts in the general population.
Epidemiological time scales can be used to reduce the dimensionality
of the model and singular perturbation methods can be used to
corroborate the results of time-scale approximations. It can be shown
that quasi-steady assumptions (diabetic elimination of variables) are
valid.  The concept optimal average cluster or generalized household
size and its impact on TB dynamics is discussed.



9.

Vaccination Strategies: Rubella

Herbert Hethcote
University of Iowa

Rubella is a mild infectious disease, but children of women who get
rubella in the first trimester of pregnancy can have congenital
rubella syndrome. One method of controlling rubella is to vaccinate
young girls before they reach childbearing age. Another method is to
routinely vaccinate young children for measles, mumps and rubella. The
effects of these two strategies are compared in a cost-benefit
analysis using an age structured model. Then rubella vaccination
strategies are compared in some countries that vaccinate for
rubella. About half of the world including China, India, and Africa do
not currently vaccinate against rubella. A model for rubella
vaccination in China is presented.



10.

Vaccination Strategies: Chickenpox

Herbert Hethcote
University of Iowa

Two possible dangers of an extensive varicella vaccination program are
more varicella (chickenpox) cases in adults, when the complication
rates are higher, and an increase in cases of zoster (shingles). An
age-structured epidemiologic-demographic model with vaccination is
developed for varicella and zoster. Parameters are estimated from
epidemiological data. This mathematical and computer simulation model
is used to evaluate the effects of varicella vaccination programs.



11.

Epidemiology Models with Delay

Pauline van den Driessche
University of Victoria

In the dynamics of epidemics, time delays can be used to model 
some mechanisms. for example, the infectious period. A model
can then be formulated in terms of delay differential and/or
integral equations. Such models are more difficult to analyse 
than analogous ordinary differential equation models, but can
exhibit richer behavior. Some of the mathematical tools 
needed are explained, and the ideas are illustrated with an 
SIS model in a variable size population with delay corresponding 
to a constant infectious period



12.

Modeling the Spread and the Evolution of Influenza 

Carlos Castillo-Chavez
Cornell University

This tutorial lecture first gives a quick review of some of the basic
epidemiology of influenza at the population level. A review of models
for the spread of influenza and the questions that they adressed is
presented. The impact of population structure and crossimmunity on the
evolution of influenza is addressed.



13.

Vaccination Strategies: Pertussis

Herbert Hethcote
University of Iowa

Both disease-acquired and vaccine-acquired immunity to pertussis
(whooping cough) wane with time, so that several infections can occur
in an individual's lifetime. The severity of a pertussis infection
depends on how low the immunity has declined since the previous
vaccination, infection, or exposure. In the United States five DTP or
DTaP (diphtheria-tetanus-pertussis) vaccinations are recommended at
ages 2, 4, 6, 15-18 months, and 4-6 years. The new acellular pertussis
vaccine (aP) has fewer side effects, so that it is safe for
adults. New strategies for reducing pertussis incidence include: 1)
combining the aP vaccine with the current Td (tetanus-diphtheria)
booster that is now recommended every ten years, 2) giving the aP
vaccine to adolescents at age 12 years, 3) giving the aP vaccine to
young adults at age 20 years, and 4) giving the aP vaccine to adults
at age 50 years. The effects of these new vaccination strategies are
analysed using an age-structured model.



14.

Modeling HIV/AIDS

Carl Simon
University of Michigan

To model the spread of HIV, one must include stages of infection,
disease-related death, and non-random mixing formulations. We examine
such models and compute Ro for certain non-random mixing
patterns. We use these models to shed light on the importance of the
primary infection period and to describe how an HIV vaccine might
work.  



15.

Epidemics on Attractors

Abdul-Aziz Yakubu
Howard University

Discrete-time S-I-S epidemic models are capable of generating
complex (chaotic) dynamics, a property not shared by classical
continuous-time epidemic models. Models for epidemic processes
on attractors are presented. Thresholds for disease persistence
are computed and used in the study of global behavior of solutions
of simple epidemic processes. The potential role of delayed
recruitment (age-structure) on disease is explored via a simple
model that differentiates between adults and juveniles. 



16.

Epidemiology Models for Heterogeneous Populations

Herbert Hethcote
University of Iowa

The spread of infectious diseases is greatly influenced by the
patterns of encounters between people in the population, particularly
for sexually transmitted diseases. The overview here of heterogeneous
mixing in deterministic models includes discussions of the threshold
result, proportionate mixing and preferred mixing. It is shown how
data on seropositivity in socially-defined groups can be used to
estimate mixing matrices. Relationships are given between the basic
reproduction numbers for heterogeneously mixing populations and
"equivalent" homogeneously mixing populations. The mixing matrices
between age-related groups have often been based on ad hoc
assumptions, but proportionate or preferred mixing matrices could also
be used. Proportionate or preferred mixing formulations have the
advantage that the activity levels and mixing matrices can often be
estimated from epidemiological data.



17.

Modeling Peer Influence by Epidemiological Approaches

Carlos Castillo-Chavez
Cornell University

Epidemiological modeling approaches can be used to study the dynamics
of processes whose spread depends strongly on one-to-one contacts
between distinct types of individuals. Dynamical models for the spread
of social "diseases" such as cigarettes smoking, ecstasy use or for
the spread of fanatic ideologies are presented.  The concept of core
group introduced by Hethcote and Yorke plays a fundamental role in the
study of these socially-driven processes. Some surprising resulst are
found. For example, it is shown that peer pressure can drive a sudden
increase in the growth of the "epidemic." In fact, a small group of
"infectious" is capable of supporting dramatic rapid growth on disease
prevalence.  The analysis of these type of models demonstrate how
difficulty it is to eliminate social diseases once a "considerable"
number of people have become infected.



18.

Mathematical Models Coupling Epidemics and Population Genetics

Zhilan Feng
Purdue University

Several ODE models will be presented to demonstrate the interaction
between the prevalence of an infectious disease and the genetic
structure of a population. The dynamics of such models can usually be
separated into two time-scales with a faster time-scale for the
epidemics and a slower time-scale for the change in gene
frequencies. A particular model for malaria epidemics and sickle-cell
gene dynamics will be discussed.



19.

Disease Control Strategies for HIV in a Population with Genetic
   Heterogeneity

Christopher Kribs-Zaleta
University of Texas, Arlington

Recent studies have identified two alleles which may provide some
level of genetic resistance to HIV infection.  This paper discusses
joint work which investigates the interplay between two levels of
genetic resistance and three control strategies -- vaccination,
treatment, and education -- in a homosexually active population, with
a focus on their effects in the infection's basic reproductive number.



20.

Deterministic and Stochastic SIR Epidemic Models

Linda J.S. Allen
Texas Tech University

The deterministic ODE  models for  SIS and SIR epidemics
will be put in a stochastic framework. SIS and SIR
epidemics will be formulated as discrete-time Markov
chains, continuous-time Markov chains and stochastic
differential equations.  The expected duration of an
epidemic, occurrence of an epidemic, and size of an epidemic
will be discussed in terms of each of these formulations.
In particular, in models with variable host population size,
S+I+R not constant, it will be shown that the results depend
on the particular form of the birth and death rates.

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