Zhien Ma and Jianquan Li

Summary. Infectious diseases have been a ferocious enemy since time immemorial. To prevent and control the spread of infectious diseases, epidemic dynamics has played an important role on investigating the transmission of infectious diseases, predicting the developing tendencies, estimating the key parameters from data published by health departments, understanding the transmission characteristics, and implementing the measures for prevention and control. In this chapter, some basic ideas of modelling the spread of infectious diseases, the main concepts of epidemic dynamics, and some developing tendencies in the study of epidemic dynamics are introduced, and some results with respect to the spread of SARS in China are given.

Infectious diseases are those caused by pathogens (such as viruses, bacteria, epiphytes) or parasites (such as protozoans, worms), and which can spread in the population. It is well known that infectious diseases have been a ferocious enemy from time immemorial. The plague spread in Europe in 600 A.C., claiming the lives of about half the population of Europe (Brauer and Castillo-Chavez 2001). Although human beings have been struggling indomitably against various infections, and many brilliant achievements earmarked in the 20th century, the road to conquering infectious diseases is still tortuous and very long. Now, about half the population of the world (6 billion people) suffer the threat of various infectious diseases. For example, in 1995, a report of World Health Organization (WHO) shows that infectious diseases were still the number one of killers for human beings, claiming the lives of 52 million people in the world, of which 17 million died of various infections within that single year (WHO). In the last three decades, some new infectious diseases (such as Lyme diseases, toxic-shock syndrome, hepatitis C, hepatitis E) emerged. Notably, AIDS emerged in 1981 and became a deadly sexually transmitted disease throughout the world, and the newest Severe Acute Respiratory Syndrome (SARS) erupted in China in 2002, spreading to

31 countries in less than 6 months. Both history and reality show that, while human beings are facing menace from various infectious diseases, the importance of investigating the transmission mechanism, the spread rules, and the strategy of prevention and control is increasing rapidly, and such studies ar an important mission to be tackled urgently.

Epidemic dynamics is an important method of studying the spread rules of infectious diseases qualitatively and quantitatively. It is based largely on the specific properties of population growth, the spread rules of infectious diseases, and related social factors, serving to construct mathematical models reflecting the dynamical property of infectious diseases, to analyze the dynamical behavior qualitatively or quantitatively, and to carry out simulations. Such research results are helpful to predict the developing tendency of infectious diseases, to determine the key factors of spread of infectious diseases, and to seek the optimum strategy of preventing and controlling the spread of infectious diseases. In contrast with classic biometrics, dynamical methods can show the transmission rules of infectious diseases from the mechanism of transmission of the disease, so that we may learn about the global dynamical behavior of transmission processes. Incorporating statistical methods and computer simulations into epidemic dynamical models could make modelling methods and theoretical analyses more realistic and reliable, enabling us to understand the spread rules of infectious diseases more thoroughly.

The purpose of this article is to introduce the basic ideas of modelling the spread of infectious diseases, the main concepts of epidemic dynamics, some development tendencies of analyzing models of infectious diseases, and some SARS spreading models in China.

2.2 The fundamental forms and the basic concepts of epidemic models

2.2.1 The fundamental forms of the models of epidemic dynamics

Although Bernouilli studied the transmission of smallpox using a mathematical model in 1760 (Anderson and May 1982), research of deterministic models in epidemiology seems to have started only in the early 20th century In 1906, Hamer constructed and analyzed a discrete model (Hamer 1906) to help understand the repeated occurrence of measles; in 1911, the Public Health Doctor Ross analyzed the dynamical behavior of the transmission of malaria between mosquitos and men by means of differential equation (Ross 1911); in 1927, Kermack and McKendrick constructed the famous compart-mental model to analyze the transmitting features of the Great Plague which appeared in London from 1665 to 1666. They introduced a "threshold theory", which may determine whether the disease is epidemic or not (Kermack and McKendrick 1927, 1932), and laid a foundation for the research of epidemic dynamics. Epidemic dynamics flourished after the mid-20th century, Bailey's book being one of the landmark books published in 1957 and reprinted in 1975 (Baily 1975).

In order to formulate the transmission of an epidemic, the population in a region is often divided into different compartments, and the model formulating the relations between these compartments is called compartmental model.

In the model proposed by Kermack and McKendrick in 1927, the population is divided into three compartments: a susceptible compartment labelled S, in which all individuals are susceptible to the disease; an infected compartment labelled I, in which all individuals are infected by the disease and have infectivity; and a removed compartment labelled R, in which all individuals are removed from the infected compartment. Let S(t),I(t), and R(t) denote the number of individuals in the compartments S, I, and R at time t, respectively. They made the following three assumptions:

1. The disease spreads in a closed environment (no emigration and immigration), and there is no birth and death in the population, so the total population remains constant, K, i.e., S(t) +1(t) + R(t) = K.

2. An infected individual is introduced into the susceptible compartment, and contacts sufficient susceptibles at time t, so the number of new infected individuals per unit time is pS(t), where p is the transmission coefficient. The total number of newly infected is pS(t)I(t) at time t.

3. The number removed (recovered) from the infected compartment per unit time is ^I(t) at time t, where 7 is the rate constant for recovery, corresponding to a mean infection period of 1. The recovered have permanent immunity.

For the assumptions given above, a compartmental diagram is given in Fig. 2.1. The compartmental model corresponding to Fig. 2.1 is the following:

Since there is no variable R in the first two equations of (1), we only need to consider the following equations

I3SI yI

Fig. 2.1. Diagram of the SIR model without vital dynamics

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