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# Lab 2: Modeling the Spread of Disease ## Goals - Implement and explore a simple model of disease spread. - Introduce stochasticity to the model. - Explore the effects of different intervention strategies. - Introduce spatial structure to the model. ## Background ### The Basic Idea We will mod...

# Lab 2: Modeling the Spread of Disease ## Goals - Implement and explore a simple model of disease spread. - Introduce stochasticity to the model. - Explore the effects of different intervention strategies. - Introduce spatial structure to the model. ## Background ### The Basic Idea We will model individuals as belonging to one of three states: - S: Susceptible. Healthy individuals who can contract the disease. - I: Infected. Individuals who have the disease and can transmit it. - R: Recovered. Individuals who have recovered from the disease and are now immune. ### The Model Assumptions: - Closed population (no births/deaths). - Homogeneous mixing. - Fixed length of infection. - Immunity after recovery. Model the transition between states using a system of ordinary differential equations (ODEs): $\qquad \frac{dS}{dt} = -\beta SI$ $\qquad \frac{dI}{dt} = \beta SI - \gamma I$ $\qquad \frac{dR}{dt} = \gamma I$ where - $\beta$ is the transmission rate. - $\gamma$ is the recovery rate. ## Part 1: Deterministic Model ### Tasks 1. **Implement the model.** Write a function to solve the system of ODEs above. The function should take as input: - Initial conditions: $S_0$, $I_0$, $R_0$ - Parameters: $\beta$, $\gamma$ - Time vector: t (the times at which to report the solution) It should return: - $S(t)$, $I(t)$, $R(t)$ 2. **Explore the model.** Use your function to explore the behavior of the model for different values of $\beta$ and $\gamma$. How does the fraction of the population that eventually gets infected depend on these parameters? 3. **Compute $R_0$.** The basic reproduction number, $R_0$, is the average number of new infections caused by a single infected individual in a completely susceptible population. Compute $R_0$ for your model as a function of $\beta$ and $\gamma$. How does the behavior of the model depend on whether $R_0 > 1$ or $R_0 < 1$?