# SIR equations solver

This webpage uses the Runge-Kutta-Fehlberg fourth-order method with fifth-order error checking (RKF45) to approximate the solution to the SIR equations with the $$\delta$$ parameter to account for quarantine effects: \begin{aligned} \dfrac{dS}{dt} &= -\dfrac{\beta I (1-\delta)S}{N} \\ \dfrac{dI}{dt} &= \dfrac{\beta I(1-\delta)S}{N} - \gamma I \\ \dfrac{dR}{dt} &= \gamma I. \end{aligned} Where $$S$$ is the number of susceptible persons, $$I$$ is the number of infected persons and $$R$$ is the number of recovered persons. $$\beta$$ is a parameter that pertains to the average number of contacts per person per time and the rate of transmission for the disease. $$\gamma$$ is the inverse of the average time a person is infected with the disease. $$N$$ is the total population. My original model had $$\gamma I$$ multiplied by $$1-\delta$$, but as quarantine should not affect how long it takes for people to recover, it should not affect this term.
Parameter Value Explanation
A parameter that pertains to how many contacts there are per person and how easily the disease spreads from an infected person to an infected person.
A parameter that is a measure of how quickly people recover from the disease.
A parameter with values from 0 to 1 pertaining to how effective quarantine measures are at slowing the disease outbreak. If $$\delta = 0$$, the measures are either non-existent or completely ineffective. If $$\delta = 1$$, all infected persons are immediately, as soon as they become infected, quarantined.
Starting time for the simulation in days.
End time for the simulation in days. Default is $$14\times 7 = 98$$, which is the number of days in 14 weeks.
Value of $$S$$ at $$t_0$$.
Value of $$I$$ at $$t_0$$.
Value of $$R$$ at $$t_0$$.
Error tolerance. $$\epsilon \lt$$ 1e-12 usually freezes the webpage up.
Initial guess for step size.