SIR model solution script The SIR model for the spread of infectious diseases See  this introductory exposition for details of the specific model that will be solved here $$ \frac{dS}{dt} = -\alpha S I ~~~~~ \\ \frac{dI}{dt} = \alpha S I - \beta I \\ \frac{dR}{dt} = \beta I ~~~~~~~~~~~ $$ Here \(S(t), I(t), R(t)\) are functions that represent the respective population fractions that are susceptible, infected, recovered. It is assumed that \(S(t)+I(t)+R(t)=1\). The problem is solvable once initial conditions \(S(0), I(0), R(0)\) are specified. Interpreting the results A key goal of modeling is to assist in planning and decision-making. For example, in the case of an infectious disease, public health officials and medical facilities must be prepared to cope with the demand for medical supplies, doctors/nurses, hospital beds, etc. Here are some related questions to consider: What proportion of the total population will become infected? When does the infection reach its peak, and what is the peak number of infected people? (E.g., The sign of which derivative tells you whether new infections are increasing or decreasing?) If a vaccine exists, it is possible (according to this model) to vaccinate enough of the susceptible population to stop the spread of the disease. What is the minimum number of susceptible people that must be vaccinated to stop the spread? Would a quarantine policy help significantly reduce the number of people who catch the infection? How would you model and simulate the effect of a quarantine? Some extensions of the model to the current coronavirus infection (This is not a part of your lab, but it is just FYI) The SIR model is likely too simplistic to reliably predict how the current worldwide pattern of coronavirus (COVID-19) infections is evolving. For the coronavirus application, I think the primary shortcoming of this model is that we are trying to get too much out of it with too little information. A key parameter that determines the effectiveness of the SIR model is the coefficient \(a\) in the term \(-a\; S I\). This coefficient is a measure of contagiousness, which for COVID-19 is a function of many real-life variables that vary enormously across geographic regions and cultures. One way to adapt the SIR model to the COVID-19 application might be to treat each infected region of the world separately. With this strategy, the model parameters can be customized to each individual country or region. Here are some online resources with detailed data on COVID-19 infections around the world https://www.worldometers.info/coronavirus/ https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports/ Another interesting aspect of the coefficient \(a\) in \(-a\; S I\) is the model's overall sensitivity to its numerical value. For example, in the test cases given above, we used \(a=0.2\), which leads to the prediction that about 80% of the susceptible population will eventually get infected. If we increase \(a\) to \(0.4\), it predicts that nearly everyone will get infected. The code below shows results for that case.