# SEIR equations solver

This webpage uses the Runge-Kutta-Fehlberg fourth-order method with fifth-order error checking (RKF45) to approximate the solution to the SEIR equations with the $$\delta$$ parameter to account for quarantine effects: \begin{aligned} \frac{dS}{dt} & = \Lambda N - \mu S - \frac{\beta I (1-\delta)S}{N} \\ \frac{dE}{dt} & = \frac{\beta I (1-\delta) S}{N} - (\mu +a ) E \\ \frac{dI}{dt} & = a E - (\gamma +\mu ) I \\ \frac{dR}{dt} & = \gamma I - \mu R. \end{aligned} Where $$S$$ is the number of susceptible persons, $$E$$ is the number of exposed persons, $$I$$ is the number of infectious persons and $$R$$ is the number of recovered persons. $$a$$ is the inverse of the average incubation period. $$\beta$$ 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. $$\Lambda$$ is the birth rate. $$\mu$$ is the overall population death rate (not only including the disease death rate). $$N$$ 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
Inverse of the average incubation period.
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.
Birth rate.
Death rate.
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 $$E$$ 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.