【Practical Exercise】Establishment and Analysis of MATLAB Epidemic Models

# 1. Fundamentals of Epidemic Models

Epidemic models are a mathematical framework used to describe and predict the spread of diseases within populations. They are based on an understanding of epidemiological principles and statistical methods, aiding in the comprehension of disease transmission patterns, influencing factors, and the effectiveness of control measures.

Epidemic models typically include the following basic elements:

***Population Structure:** Describing the characteristics and behaviors of different individuals within a population, such as age, gender, and immunity status.
***Transmission Dynamics:** Describing the way diseases spread among people, including parameters such as contact rate, infection rate, and recovery rate.
***Environmental Factors:** Considering environmental factors that affect disease transmission, such as population density, sanitation conditions, and socioeconomic status.

# 2. Establishing Epidemic Models in MATLAB

### 2.1 Types and Selection of Epidemic Models

Depending on their complexity and the level of detail in describing disease transmission mechanisms, epidemic models can be classified into several categories:

- **Deterministic Models:** Assume that the disease transmission process is deterministic and can be described using differential or difference equations.
- **Stochastic Models:** Consider the randomness in the disease transmission process and describe it using probability distributions and stochastic processes.
- **Individual-Based Models:** Treat individuals in the population as modeling entities, describing the state changes and interactions of individuals.
- **Network Models:** View individuals in the population as nodes in a network and describe the spread of disease through the network.

When establishing an epidemic model in MATLAB, it is necessary to choose an appropriate model type based on the characteristics of the disease transmission and the research objectives. For instance, deterministic models can be used for highly infectious diseases; stochastic models for diseases with random transmission; individual-based models for diseases requiring consideration of individual heterogeneity; and network models for diseases requiring consideration of network structure effects.

### 2.2 Implementation of Epidemic Models in MATLAB

#### 2.2.1 Setting Model Parameters

Epidemic model parameters generally include:

- **Infection Rate:** Describing the speed at which a disease spreads through the population.
- **Recovery Rate:** Describing the speed at which individuals recover from the infected state to the susceptible state.
- **Mortality Rate:** Describing the speed at which individuals die from the infected state.
- **Birth Rate:** Describing the birth speed of new individuals in the population.
- **Death Rate:** Describing the death speed of individuals in the population.

These parameters can be obtained through literature research, epidemiological investigations, or historical data estimates.

#### 2.2.2 Solving Model Equations

Epidemic model equations are typically differential or difference equations. In MATLAB, functions like `ode45` or `ode23` can be used to solve differential equations, and functions like `diff` or `diffeq` can be used to solve difference equations.

% Solving Differential Equations
y0 = [100, 0, 0];   % Initial conditions
t = 0:0.1:100;     % Time range
[t, y] = ode45(@(t, y) [
     -beta * y(1) * y(2),
     beta * y(1) * y(2) - gamma * y(2),
     gamma * y(2) - delta * y(3)
], t, y0);

% Solving Difference Equations
y0 = [100, 0, 0];   % Initial conditions
t = 0:1:100;       % Time range
y = zeros(length(t), 3);
y(1, :) = y0;
for i = 2:length(t)
     y(i, 1) = y(i-1, 1) - beta * y(i-1, 1) * y(i-1, 2);
     y(i, 2) = y(i-1, 2) + beta * y(i-1, 1) * y(i-1, 2) - gamma * y(i-1, 2);
     y(i, 3) = y(i-1, 3) + gamma * y(i-1, 2) - delta * y(i-1, 3);
end

The above code demonstrates the solution process for differential and difference equations. By solving the model equations, the dynamic changes in the spread of a disease within a population can be obtained.

# 3. Analysis of Epidemic Models in MATLAB

### 3.1 Sensitivity Analysis o