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# A Model to Predict COVID-19 Epidemics with Applications to South Korea, Italy, and Spain

Our team has developed several differential equations models of COVID-19 epidemics [1-3] that use early reported case data from around the world to predict the future number of cases. These models incorporate three important elements of COVID-19: (1) the number of asymptomatic infectious individuals (with very mild or no symptoms), (2) the number of symptomatic reported infectious individuals (with severe symptoms), and (3) the number of symptomatic unreported infectious individuals (with less severe symptoms). They also decompose COVID-19 epidemics into three phases:

• Phase I, during which the number of cumulative reported cases increases linearly each day
• Phase II, during which the number of cumulative reported cases increases exponentially each day
• Phase III, during which the number of daily reported cases decreases each day.

The transitions between phases are generally difficult to determine, but one can estimate them from reported cases data as time progresses.

Our model consists of the following differential equations and initial conditions:

$\begin{cases} S'(t)=-\tau(t) S(t)[I(t)+U(t)], \quad S(t_0)=S_0, \\ E'(t)= \tau(t) S(t)[I(t)+U(t)]-\alpha E(t), \quad E(t_0)=E_0\\ I'(t)= \alpha E(t) - \nu I(t), \quad I(t_0)=I_0, \\ R'(t)=\nu_1 I(t)-\eta R(t), \quad R(t_0) = R_0, \\ U'(t)=\nu_2 I(t)-\eta U(t), \quad U(t_0) = U_0. \\ \end{cases} \tag 1$

Here, $$t \geq t_0$$ is time in days, $$t_0$$ is the beginning date of the epidemic, $$S(t)$$ is the number of individuals susceptible to infection at time $$t$$, $$E(t)$$ is the number of asymptomatic noninfectious (exposed or latent infected) individuals at time $$t$$, $$I(t)$$ is the number of asymptomatic but infectious individuals at time $$t$$, $$R(t)$$ is the number of reported symptomatic infectious individuals at time $$t$$, and $$U(t)$$ is the number of unreported symptomatic infectious individuals at time $$t$$.

The time-dependent transmission rate parameter is $$\tau(t)$$. Newly-infected noninfectious asymptomatic individuals $$E(t)$$ incubate for an average period of $$1 / \alpha$$ days. Asymptomatic infectious individuals $$I(t)$$ are infectious for an average period of $$1 / \nu$$ days. Reported symptomatic infectious individuals $$R(t)$$ are infectious for an average period of $$1 / \eta$$ days, as are unreported symptomatic infectious individuals $$U(t)$$. We assume that reported symptomatic infectious individuals $$R(t)$$ are reported and isolated immediately, and cause no further infections. One can also view the asymptomatic individuals $$I(t)$$ as having a low-level symptomatic state. All infections are acquired from either $$I(t)$$ or $$U(t)$$ infectious individuals. The fraction $$f$$ of asymptomatic infectious cases becomes reported symptomatic infectious, and the fraction $$1 - f$$ becomes unreported symptomatic infectious. The rate at which asymptomatic infectious cases become reported symptomatic is $$\nu_1 = f \, \nu$$, and the rate at which asymptomatic infectious cases become unreported symptomatic is $$\nu_2 = (1 - f) \, \nu$$, where $$\nu_1 + \nu_2 = \nu$$.

The cumulative number of reported cases $$CR(t)$$ at time $$t$$ is

$CR(t)=\nu_1 \int_{t_0}^t I(\sigma) d\sigma, \,\,\, t \geq t_0,$

the cumulative number of unreported cases $$CU(t)$$ at time $$t$$ is

$CU(t)=\nu_2 \int_{t_0}^t I(s) ds, \,\,\, t \geq t_0,$

and the daily number of reported cases $$DR(t)$$  at time $$t$$ is

$DR'(t)=\nu_1 \, I(t) - DR(t), \, \,\, t \geq t_0, \, \,\, DR(t_0)=DR_0.$

Figure 1 depicts a flow diagram of the model.

Figure 1. Compartments and flow chart of the model.

### Parameters

The fraction $$f$$ of total reported symptomatic infectious cases is unknown and varies from region to region. We assume that $$\eta =1/7$$, which means that the average period of infectiousness of both unreported and reported symptomatic infectious individuals is seven days. We also assume that $$\nu =1/6$$, which means that the average period of infectiousness of asymptomatic infectious individuals is six days. Finally, we assume that $$\alpha =1$$, which means that the average period of exposed individuals is one day. We can modify these values as further epidemiological information becomes available; as of early April, they were consistent with accepted values.

A COVID-19 epidemic transitions from phase I to phase II at time $$t_1>t_0$$. Before $$t_1$$, the cumulative number of reported cases increases linearly each day. After $$t_1$$, the cumulative number of reported cases increases exponentially each day. We estimate the value of $$t_1$$ from data pertaining to the cumulative reported cases. We then fit an exponentially growing curve $$CR(t)$$ to the cumulative reported cases data in an estimated time interval $$[t_1,t_2]$$, according to the following formula:

$CR(t) = \chi_1 \exp(\chi_2 \, t) - \chi_3, \, \,\, t _1 \leq t \leq t_2. \tag2$

We typically set $$\chi_3=1$$ but allow for other values. The initial value $$S_0$$ corresponds to the population of the reported cases data’s region. The other initial conditions are

$I_0 = \frac{\chi_2 \, \chi_3}{f (\nu_1 + \nu_2)}, \,\,\, E_0= \dfrac{\chi_2 +\nu}{\alpha}I_0, \,\,\, U_0= \dfrac{\nu_2}{\chi_2 +\eta} I_0. \tag3$

Furthermore, the value of $$t_0$$ $$($$when $$R(t_0) = CR(t_0) = 0)$$ for starting time $$t_0$$ of the epidemic is given by

$CR(t_0)=0 \Leftrightarrow \chi_1 \exp\left( \chi_2 t_0 \right)-\chi_3=0 \, \, \Rightarrow \, \, t_0=\dfrac{1}{\chi_2} \left(\ln\left(\chi_3 \right)-\ln\left(\chi_1 \right) \right). \tag4$

$\tau_0=\dfrac{(\chi_2+\alpha) E_0}{S_0[I_0+U_0]} =\dfrac{(\chi_2+\nu)(\chi_2+\alpha)(\chi_2+\eta)}{\alpha S_0(\chi_2+\eta+\nu_2)}, \tag5$

and the basic reproductive number is given by

${\cal R}_0 =\dfrac{(\chi_2+\nu)(\chi_2+\alpha)(\chi_2+\eta)}{\nu \, \alpha \, (\chi_2+\eta+\nu_2)} \left(1+ \dfrac{(1-f) \,\nu}{\eta}\right). \tag6$

We derive these formulas for $$I_0$$, $$E_0$$, $$U_0$$, $$t_0$$, $$\tau_0$$, and $$\mathcal{R}_0$$ in [1]; their values connect the phase II reported cases data to the parameterisation and initialisation of our differential equations model.

During phase II of the epidemic, $$\tau(t) \equiv \tau_0$$ is constant. When strong governmental measures like isolation, quarantine, and public closings are implemented, phase III begins. The timing of the implementation of these measures—and their subsequent impact on disease transmission—is complex. We use an exponentially decreasing time-dependent transmission rate $$\tau(t)$$ in phase III to incorporate these effects. The formula for $$\tau(t)$$, which has phase III beginning on day $$N$$, is

$\begin{cases} \tau(t) = \tau_0, \,\,\, 0 \leq \, t \, \leq \, N,\\ \tau(t) = \tau_0 \, \exp \left(- \mu \left( t - N \right) \right), \,\,\, N < t. \end{cases} \tag7$

We choose the date $$N$$ and intensity $$\mu$$ of the public measures so that the cumulative reported cases in the epidemic’s numerical simulation align with the cumulative reported case data at an identified date after day $$N$$. In this way, we can project forward the epidemic’s time path after the government-imposed public measures take effect.

### Applications

We apply our model to the COVID-19 epidemics in South Korea, Italy, and Spain. Figure 2 provides the parameters for these three countries.

Figure 2. We obtain the parameters $$\chi_1, \chi_2$$ by fitting $$\chi_1 \exp(\chi_2 \, t) - 1.0$$ to the cumulative reported cases data between the dates $$[t_1, t_2]$$ for each country: (1) $$t_1=$$ February 22 to $$t_2=$$ March 1 for South Korea, (2) $$t_1=$$ March 12 to $$t_2=$$ March 21 for Italy, and (3) $$t_1=$$ March 13 to $$t_2=$$ March 21 for Spain. The values $$I_0$$, $$U_0$$, $$\tau_0$$, $$t_0$$, and $$\mathcal{R}_0$$ are obtained via equations $$(3)$$-$$(6)$$. The parameters $$\nu =1/6$$, $$\eta =1/7$$, $$\alpha =1/1$$, $$\chi_3=1.0$$, and $$R_0=1.0$$ for all three countries.

COVID-19 Epidemic in South Korea: We divide the epidemic in South Korea into four stages (see Figure 3):

1. Before February 22: Phase I.
2. February 22 to March 1: Phase II.
3. March 2 to March 8: Phase III. The South Korean government implemented extensive testing, isolation, contact tracing of confirmed cases, and quarantine policies after February 20. These measures took effect in daily reports after March 2.
4. After March 8: The daily reported cases remained approximately the same each day, and the cumulative reported cases increased linearly. This stage corresponds to a new phase I, with a low-level background generation of reported cases each day.

Figure 3. Model simulation for South Korea. 3a. Cumulative reported cases. The shaded region $$=$$ phase II and the model turning point is March 5. 3b. Daily reported cases. The model turning point is March 2.

To account for this new phase, we modify model $$(1)$$ by replacing $$\tau(t)$$ with a novel transmission function $$\tau(t,S(t),I(t),U(t))$$ that depends on $$t$$, $$S(t)$$, $$I(t)$$, and $$U(t)$$ as follows:

$\begin{cases} \tau(t,S(t),I(t),U(t)) = \tau_0, \, \, \, t_0 \leq t \leq 27; \\ \tau(t,S(t),I(t),U(t)) = \tau_0 \, \exp \left(- 0.6 \left(t - 27 \right) \right) , \, \,\, 27 < t \leq 37; \\ \tau(t,S(t),I(t),U(t)) = 23.0 \, \tau_0 \exp \left(- 0.6 \left(37 - 27 \right) \right) \bigg( \frac{S(37) [ (I(37) + U(37) ] }{( S(t) [(I(t) + U(t)] } \bigg), \,\,\, 37 < t . \end{cases} \tag8$

We select the value $$23.0$$ to match the slope of the linear increasing cumulative reported cases data after day $$37$$. The equations and initial values remain the same, with the exception of this novel $$\tau$$ function. The formulas in $$(8)$$ connect the new phase I to the transmission rate in the model equations and the model outputs of $$E(t)$$, $$I(t)$$, $$U(t)$$, $$R(t)$$, $$CU(t)$$, $$CR(t)$$, and $$DR(t)$$. One can apply the form of $$(8)$$ to other examples that transition from phase III to a new phase I, corresponding to a linearly-increasing growth rate of cumulative reported cases. This new phase I can further transform to yet another phase I with a slower linearly increasing growth rate.

COVID-19 Epidemic in Italy: We divide the epidemic in Italy into three stages (see Figure 4):

1. Before March 12: Phase I.
2. March 12 to March 21: Phase II.
3. After March 24: Phase III. Beginning on March 1, the Italian government implemented extensive public regional lockdown measures, which were extended to all of Italy on March 10. These measures started to reduce the number of reported daily cases approximately two weeks later.

Figure 4. Model simulation for Italy. 4a. Cumulative reported cases. The shaded region $$=$$ phase II and the model turning point is March 31. 4b. Daily reported cases. The model turning point is March 26.

COVID-19 Epidemic in Spain: We divide the epidemic in Spain into three stages (see Figure 5):

1. Before March 13: Phase I.
2. March 13 to March 21: Phase II.
3. After March 28: Phase III. The Spanish government implemented partial shutdown measures on March 13 and imposed a general state of alarm for all of Spain on March 14. These measures started to reduce the number of reported daily cases approximately two weeks later.

Figure 5. Model simulation for Spain. 5a. Cumulative reported cases. The shaded region $$=$$ phase II and the model turning point is April 4. 5b. Daily reported cases. The model turning point is March 30.

### Concluding Thoughts

We have applied a new method [1-3] to predict a COVID-19 epidemic’s evolution in a particular geographical region, based on reported cases data from that region. Our model focuses on unreported cases, asymptomatic infectious cases, and division of the epidemic’s evolution through a succession of phases. Our method can be predictive when the epidemic is growing exponentially in phase II. Specifically, we demonstrate a technique to identify the exponentially increasing rate of cumulative reported cases in phase II [3]. When public measures to ameliorate the epidemic begin during this phase, we model these measures with a time-dependent exponentially decreasing transmission rate. These mitigations result in phase III: a subsequent reduction in daily reported cases. We determine the transition from phase II to phase III—which may require more than a week—in the model simulations.

The epidemic has attenuated in South Korea because of major measures that encourage social distancing. These measures involve surveillance, extensive testing, and isolation and contact tracing for reported and suspected cases. However, the cumulative number of reported cases in South Korea has not flattened; instead, it is growing linearly at a low rate. The epidemics in Italy and Spain have evidently passed the turning point, according to data about the daily reported cases. The cumulative reported cases may not flatten but instead continue to grow linearly at a low rate, as in South Korea.

Our model incorporates government and social distancing measures through the time-dependent transmission rate $$\tau$$. These measures should begin as early as possible and be as strong as possible. If such efforts cause the epidemic to substantially subside, the situation in South Korea indicates that a background level of daily cases may persist for an extended time. If countries reduce major distancing measures too early or too extensively, the epidemic can enter a new phase II and undergo another exponential increase in cumulative cases. Control of COVID-19 epidemics is possible, as evidenced by the situation in South Korea. The future of COVID-19 and its human toll is currently uncertain, and we hope that mathematical models will be of use.