Mathematical Modeling of Coronavirus III: SQDR model

SQDR model

Variables and Parameters

In SIR model, we can further split $I = Q + D$. Here $Q$ represents quarantined population and thus $D = I - Q$ is the most dangeours compartment: infected but not quarantined. After normalization by the total population, we have

• $S(t)$ : ratio of the susceptible population;
• $Q(t)$ : ratio of the quarantined population;
• $D(t)$ : ratio of the dangerous population;
• $R(t)$ : ratio of the recovered population

Parameters $(\alpha, \beta, \gamma)$ are

• $\alpha$ : the isolation rate

• $\beta$ : the infection rate

• $\gamma$ : the recovery rate

  • $\gamma_1$ recovery rate for $Q$
  • $\gamma_2$ recovery rate for $D$

   

Model

We list the system below

Compare to SIR model, the main change is: the product $IS$ is changed to $DS$ since part of infected people is quarantined and thus not reproductive anymore. $D$ is the chance of a healthy person to encounter an infected but not quarantined person. Multiply infection rate $\beta$ is the increasing rate of infected people.

Not all infected $I$ is quarantined. We further multiply $\alpha$ the isolation rate which is increasing rate of $Q$. The decreasing of $Q$ is due to the recovery. Small $\alpha$ could be due to the lack of test. Here we assume once the test is positive, the person will be quanrantized. So $\alpha$ can be also thought of the test ratio.

The last equation is easy to understand. Recovery rate is the contribution from both $Q$ and $D$.

The equation for $D$ can be obtained by imposing the conservation of population $(S+Q+D+R)'=0$. Among $\beta DS$ infected people, $(1-\alpha)$ of them are not isloated and become dangerous. Decreasing of $D$ is due to recovery.

The recovery rate $\gamma_1$ for $D$ is slightly smaller than the rate $\gamma_2$ for $Q$. As in the group $Q$, i.e. the test is positive, there is a chance to be hospitalized which is considered as recovered. The relation is $\gamma_1 = h + \gamma_2 (1-h) = \gamma_2 + (1-\gamma_2)h$, where $h$ is hosptilization ratio among all confirmed cases.

Consider the simplified case $\gamma = \gamma_1 = \gamma_2$, we can add equations for $Q$ and $D$ to get equation of $I$ and eliminate $Q = I - D$ from the system to obtain SIRD model

The first three equations are a variant of the SIR model and conservation of $S + I + R$ is still preserved. In the extreme case, $\alpha = 0$ , i.e. no one in $I$ is quanrantined, then it reduces to the SIR model.

Simplified Model

Small $\alpha$ will result less $Q$ and more $D$ and thus increase $I$. When $\alpha$ is small, the most effective way to reduce $I$ and $D$ is to reduce the infection rate $\beta$ by social distancing. We have talked this in SIR model.

When $\alpha = 1$, i.e., all infected people get qurantianed immedidately, from the fourth equation of $\eqref{SIRD}$ , the dangerous population $D$ will decay to zero exponentially and thus new infectious $\beta DS$ will also decay to zero exponentially. So an efficient control policy is to increasing the test and consequently isolation rate.

With effective control, $I$ will be tiny and $S\approx 1$. For example, the population of California is near $4\times 10^7$ (40 million). If in the end $4\times 10^4$ (40k) is infected, then $R_{\infty} = 10^{-3}$ and $S_{\infty} = 0.999\approx 1$.

Therefore we can safely use $S=1$ in the last equation of $\eqref{SIRD}$ to obtain one ODE for $D$:

Why not further merge the parameters into one like $\dot{D} = \rho D$ with $\rho = \beta -\alpha \beta - \gamma$?

The reason is that we can adjust $\alpha$ and $\gamma$ to some kernel functions and impose some delay into the equation. For example, $\alpha$ and $\gamma$ could be distribution functions over time.

Fitting data

We DO NOT know the parameters $(\alpha, \beta, \gamma)$. If we knew them, we can just solve the ODE and get prediction. So we should determine the parameters from a prior information and observed data.

What do we observe?

The number of daily confirmed cases which is $\alpha \beta D$. This number depends crucially on $\alpha$. We may have a large number of infected population $\beta D$ but due to small test ratio $\alpha$, we only observe small amount of confirmed cases.

After a critical time, if the test rate is increased, which dramatically increases $\alpha$, we may observe an increase of this number although the total infection number $I$ could be decreasing already.

We also know some critical time points.

For example, the time when the state enforces the stay-at-home order. And from the current date, we can identify the peak time. From the news or data, we know when the test is increased.

For example, for California, we can introduce the following critical time:

• peak time
• control time
• increasing test time

And chose different parameters among different time intervale. The infection rate $\beta$ is piecewise constant: before stay-at-home and after stay-at-home.

Steps to post-process data. Suppose we are give an array newcase recording daily confirmed cases.

1. Smooth the data. The real data is oscillated due to many stochastic effect. A simple method is the average between two neighbors

x
new(t) = (new(t) + new(t-1) + new(t+1))/3;
new(t) = (new(t) + new(t-1) + new(t+1))/3;

If we repeat the smoothing twice, it is a weighted average over 5 days.

Exercise. Figure out the formula corresponding to the code.

2. Identify the critical time: like the peak, the control time, and the decreasing time.
3. Apply polyfit to log(new) to figure out the rate $\rho$ in different stage.
4. Solve the simple ODE $\dot{y} = \rho y$ and compare the solution to the data.

Delay effect

For two integrable functions $f,g$, we intrdoce the convolution operator $*$

The new infected people $\beta D(t)$ may not be qurantined immedidately. The parameter $\alpha = \alpha(t)$ is a transition distribution and the new qurantined popluation is $\alpha(t)*\beta D(t)$.

The recovery is also delayed. A simple modification of recovery for $D$ is $\gamma_2D(t-7)$ with the averaged recover time $7$ days. The recovery for $Q$ is $\gamma_1(t)*\beta D(t) + \gamma_2 Q(t-7)$, where $\gamma_1(t)$ is the transition probability for an infected person to be hosipitalized.