## Simulation of SIR model for COVID-19

In the present page we numerically solve the initial value proble of the SIR model for COVID-19 of the form \begin{alignat*}{4} \frac{dS}{dt} & = -\beta IS, & \quad \frac{dI}{dt} & = \beta IS - \gamma I, & \quad \frac{dR}{dt} & = \gamma I, \quad t\in[0,\infty), \\ S(0) & = S_0, & \quad I(0) & = I_0, & \quad R(0) & = R_0, & \quad \end{alignat*} where $$S(t)$$, $$I(t)$$ and $$R(t)$$ are real-value unknown functions of $$t \in \mathbb{R}$$, $$S_0$$, $$I_0$$ and $$R_0$$ are given initial data, and $$\beta$$ and $$\gamma$$ are positive constants. In mathematical epidemiology, $$t$$ is time, $$S(t)$$ is the number of susceptible people, $$I(t)$$ is the number of people infected, $$R(t)$$s the number of people who have recovered and developed immunity to the infection, $$\beta$$ the infection rate, and $$\gamma$$ is the recovery rate. This systems of ordinary differential equations is well-known as the SIR model for infection disease.

We briefly mention the basic properties of the SIR model.
• Set $$N:=S+I+R$$ which is the total population. Note that the total poputation $$S+I+R$$ is preserved since $\frac{d}{dt} (S+I+R) =0.$
• In what follows we assume that $$S_0$$, $$I_0$$ and $$R_0$$ are nonnegative. Then $$0 \leqq S_0, I_0, R_0 \leqq N$$. Note that $$S$$, $$I$$ and $$R$$ satisfy \begin{align*} S(t) & = S_0 \exp\left( -\beta \int_0^t I(\tau) d\tau \right), \\ I(t) & = I_0 \exp\left( \int_0^t \bigl\{ \beta S(\tau)-\gamma \bigr\} d\tau \right), \\ R(t) & = R_0 + \gamma \int_0^t I(\tau) d\tau. \end{align*} Thus $$S(t)$$, $$I(t)$$ must be nonnegative for all $$t\in[0,\infty)$$. Moreover, it follows that $$S(t)$$ is nonincreasing, and $$R(t)$$ is nondecreasing and nonnegative. In particular $$0 \leqq S(t), I(t), R(t) \leqq N$$ for all $$t\in[0,\infty)$$. This fact works as the a priori estimates of $$S$$, $$I$$ and $$R$$, and one can easily prove that the initial value problem in the future direction has a unique smooth solution globally in time.

• Similarly we have for any $$t,t_0\in[0,\infty)$$ \begin{align*} I(t) & = I(t_0) \exp\left(\gamma\int_{t_0}^t \left\{\dfrac{\beta S(\tau)}{\gamma}-1\right\}d\tau\right) \\ & \leqq I(t_0) \exp\left(\gamma\int_{t_0}^t \left\{\dfrac{\beta N}{\gamma}-1\right\}d\tau\right). \end{align*} The following function and constant $\rho_e(t):=\dfrac{\beta S(t)}{\gamma}, \quad \rho_0:=\dfrac{\beta N}{\gamma}$ are said to be the effective reproduction number and the basic reproduction number respectively. Note that $$\rho_e(t) \leqq \rho_0$$, and that the signature of $$\rho_e(t)-1$$ shows the increase/decrease of the number of active cases $$I(t)$$. For this reason $$\rho_e(t)$$ is the most important function in mathematical epidemiology.
We are concerned with the simulation of SIR model for COVID-19 below. Set $$T(t)=I(t)+R(t)$$ which is the total cases at the time $$t$$. A unit of time here is supposed to be day, and set the initial date, say, today as $$t=0$$. Visit the government website dealing with the statistics about COVID-19, and get the total cases $$T(0)$$, the daily new cases $$T(0)-T$$, the total number of discharged individuals $$R(0)$$ and the daily new numbers of discharged individuals $$R(0)-R(-1)$$. Set $$S(0):=N-T(0)$$ and $$I(0):=T(0)-R(0)$$. See, e.g., South Korea, Hong Kong, Okinawa Osaka Tokyo. Note that a daily increase of a quantity can be seen as a differentiation of the quantity on the day. So we set the infection and recovery rates by $\beta := \frac{T(0)-T(-1)}{(T(0)-R(0))(N-T(0))}, \quad \gamma := \frac{R(0)-R(-1)}{T(0)-R(0)}.$ since $\frac{dT}{dt} = \beta(T-R)(N-T), \quad \frac{dR}{dt} = \gamma(T-R).$ Then we have $\rho_e(0) = \frac{T(0)-T(-1)}{R(0)-R(-1)}.$ By using Julia Language we draw a solution to the initial value problem for the SIR model with data including area, population $$N$$, staring date, ending date, $$T(0)$$, $$T(0)-T(-1)$$, $$R(0)$$ and $$R(0)-R(-1)$$. See sir1.jl for the detail.

The behavior of solutions to the SIR model depends sensitively on $$\rho_e(0)$$, and the above simulation follows the daily change of $$\rho_e(0)$$. So we take the average of some $$d$$ days for $$\beta$$ and $$\gamma$$ instead of the values determined only by the data of the initial date. We replace them by $\beta := \frac{T(0)-T(-d)}{d\bigl(T(0)-R(0)\bigr)(N-T(0))}, \quad \gamma := \frac{R(0)-R(-d)}{d\bigl(T(0)-R(0)\bigr)}.$ for some $$d=1,2,3,\dotsc$$. Then the effective reproduction number of the initial date becomes $\rho_e(0) = \frac{T(0)-T(-d)}{R(0)-R(-d)}.$ By using Julia Language we draw a solution to the initial value problem for the SIR model with data including area, population $$N$$, staring date $$d0$$, ending date $$d1$$, the date of the begining of taking average $$d2=d0-d$$, $$T(0)$$, $$T(-d)$$, $$R(0)$$ and $$R(-d)$$. If we fix the area, $$N$$, $$d_1$$ (or $$d1-d0$$), $$d2$$, $$T(-d)$$ and $$R(-d)$$, we can repeat the simulation only by inputting $$d0$$, $$T(0)$$ and $$R(0)$$. See sir2.jl for the detail.   