Influenza project
Grading rubric
To earn credit, a project must meet the following criteria.
Criterion | Met | Not met |
---|---|---|
Develop a model that accurately represents the dynamics of the flu | ||
Analyze and interpret the model to accurately determine the dynamics of the flu | ||
Based on valid mathematical analysis, recommend an intervention strategy that will effectively eliminate the flu. |
Project receives credit | YES | NO |
Submitting project
Submit the following by the due date.
- This cover sheet
- Answers to the project questions (typed or handwritten)
Background
Influenza (or the flu) infects people across the globe, infecting between five and ten percent of susceptible adults and twenty to thirty percent of susceptible children each year. Worldwide, influenza causes three to five million cases of severe illness and about a quarter to a half million deaths, primarily among the very young, the elderly, or the chronically ill (WHO fact sheet 211). In order to determine the best strategies for controlling the spread of the disease, we will develop a model of the disease dynamics.
The overarching questions for this project are:
- How do the dynamics of the disease unfold in the short-term and in the long-term?
- How will different intervention strategies influence the outcome of the disease?
We will create a model of influenza among a population of $N=100,000$ individuals. Each person in this population will be considered to be in one of three classes: susceptible (meaning they are healthy but can contract the flu if exposed to the disease), infective (meaning they have the disease and can pass it on to others they contact), and removed (meaning they are no longer infective and also cannot contract the disease). In general, the removed class includes people that recovered from the disease as well as those who died. For simplicity, we won't model births or death, and total number of individuals in all three classes will stay fixed at $N$.
Let $t$ denote time in days. Let $S(t)$ be the number of susceptibles, $I(t)$ be the number of infectives and $R(t)$ be the number of removed individuals in day $t$. Since we assume there are $N=100,000$ total individuals at all times, we require that $S(t) + I(t) + R(t) = N$ for all time $t$. Our rule for the dynamics of $S$, $I$, and $R$ must respect that condition.
If an infective individual comes into contact with a susceptible individual, that susceptible individual might contract influenza. We assume that for any pair of individuals, if one is susceptible and the other is infective, there is a probability of $4 \times 10^{-6}=0.000004$ per day that the infective individual will pass on the flu to the susceptible individual. (Since we are treating every pair the same, we are ignoring the fact that one is more likely to contact and get the flu from family, neighbors, and friends.) We capture this fact with the infection rate parameter, $\beta = 4 \times 10^{-6} \text{ day}^{-1}= 0.000004 \text{ day}^{-1}$. (The unit $\text{day}^{-1}$ means per day. The parameter is the Greek character $\beta$, pronounced “beta.”)
An infective person, in addition to infecting susceptible people, also recovers with a probability of 0.2 per day, which we capture with the recovery rate parameter $\gamma=0.2 \text{ day}^{-1}$. (The parameter is the Greek character $\gamma$, pronounced “gamma.”) When they have recovered, they are in the removed class and can no longer infect people. They don't stay immune to the disease forever, however. They have a probability of $0.01$ per day of becoming susceptible again, which we capture with the immunity loss rate parameter $\alpha=0.01 \text{ day}^{-1}$. (The parameter is the Greek character $\alpha$, pronounced “alpha.”)