| Epidemic dynamics is important to first fully understand the mechanism of the spread and the transmission dynamics of the disease. And then provide useful predictions and guidance so that better strategies can be established.April 2009, the outbreak of influenza A H1N1 is in Mexico, and then this influenza spread rapidly to more than 30 countries and regions in half a year. At the beginning of the spread of influenza H1N1, each country takes different predictions and control strategies. Some countries such as Chinese takes strict quarantine measures; the others do not, so that the disease in these two kinds of regions shows the different spread patterns and rules.In order to study the spread patterns and rules in these two kinds of regions, the second and third chapter in this paper established the SEIQR model with the quarantine compartment and SEIR model without quarantine one, respectively. The threshold R0 of these two models, which is the basic reproductive number of the model and determines whether a disease persists or goes extinct, is obtained. The global stability of the disease-free equilibrium of these models follows the LaSalle-Lyapunov theory. Using geometric method of global stability, the global stability of endemic equilibrium of the SEIQR model is obtained. By building a Lyapunov function, the epidemic equilibrium of the SEIR model is proved. So, the global behavior is determined.Infectious diseases can be easily transmitted from one country (or one region) to other countries (or one region). Thus, it is important to consider the effect of population dispersal on the spread of a disease. So we establish a patch (a country or region) transmission model in the last chapter in this paper. Study results show that susceptive dispersal may cause the spread of the H1N1 in one patch even though the dies out in each isolated patch and the dispersal of infectious individuals is barred. And population dispersal can delay the extinction of the disease in two patches.
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