# Improved Epidemic Dynamics Model and Its Prediction for COVID-19 in Italy
## 2 SEIR_QJD Model  
The traditional SEIR model (Figure 1) only divides the population into four categories: susceptible (S), exposed (E), infected (I), and recovered (R), which could not describe the transmission law of COVID-19 well.  
In view of the uniqueness of the disease, we designed a new dynamic model to characterize COVID-19:  
1. Infected individuals during the incubation period are infectious [14].
2. There is a large proportion of asymptomatic infected individuals [15].
3. The rate of infection is different between symptomatic and asymptomatic individuals [15].  
The new dynamic model comprehensively covers the government's containment measures (lockdown of key epidemic areas, school suspension, and suspension of noncritical production activities) and the classification of infected people (symptomatic and asymptomatic). The model consists of eight objects in the proposed model: susceptible (S), exposed (E), quarantine-exposed (\(E_{Q}\)), asymptomatic infected (IA), symptomatic infected (I), diagnosed (J), recovered (R), and dead (D); we call it the SEIR_QJD for short, and the process is shown in Figure 2.  
For the purpose of conciseness, we denote \(S(t)\), \(E(t)\), \(E_{Q}(t)\), and so on as the amount in the corresponding states at time \(t\). The specific process is described as follows:  
* **Infection:** Every primary case (including \(E\), \(IA\), and \(I\) status) with infectiousness will transmit the virus to its secondary cases at time \(t\); that is, if a primary case contacts a susceptible individual, the susceptible individual will be infected with a certain probability.
* **Quarantine:** Due to the government's prevention and control measures (suspension, work stoppages, restrictions on the diagnosed individuals, etc.) and people's self-precautionary awareness (self-isolation), each individual in status \(E_{Q}\) and \(J\) will be isolated. Note: we assume that the quarantined individuals cannot contact the susceptible ones; namely, it is not contagious.
* **Sympton Onset:** Individuals in states \(E\), \(E_{Q}\) will become symptomatic after the infectious incubation period and will transition to states \(I\) and \(J\) at the rate of \(\lambda\).
* **Recovery:** Individuals in states \(I_{A}\), \(I\), and \(J\) will recover at rates of \(\gamma_{A}\), \(\gamma_{I}\), and \(\gamma_{J}\), respectively.  
Figure 1: Susceptible–Exposed–Infectious–Recovered (SEIR) epidemic-dynamics model.  
Figure 2: Process illustration.  
* **Death:** Individuals in states \(I\) and \(J\) will die at rates of \(\delta_{I}\) and \(\delta_{I}\), respectively. Note: as the mortality rates of asymptomatic and mild-symptomatic individuals are extremely low, the death of asymptomatic infections is not included. Meanwhile, we get the dynamic model defined by the following differential equations: \[\frac{dS(t)}{dt}=-S(t)l\beta(E(t)+\omega_{I}I(t)+I_{A}(t))-S(t)(1-l)\beta(E(t)+ \omega_{I}I(t)+I_{A}(t))\] (1) \[\frac{dE(t)}{dt}=S(t)(1-l)\beta(t)(E(t)+\omega_{I}I(t)+I_{A}(t))-\lambda E(t)\] (2) \[\frac{dE_{Q}(t)}{dt}=S(t)l\beta(t)(E(t)+\omega_{I}I(t)+I_{A}(t))-\lambda E _{Q}(t)\] (3) \[\frac{dI_{A}(t)}{dt}=\lambda\rho E(t)-\gamma_{A}I_{A}(t)\] (4) \[\frac{dI(t)}{dt}=\lambda(1-\rho)E(t)-(\delta_{I}+\gamma_{I}+\alpha)I(t)\] (5) \[\frac{dJ(t)}{dt}=\alpha I(t)+\lambda E_{Q}(t)-\Big{(}\delta_{I}+\gamma_{I} \Big{)}J(t)\] (6) \[\frac{dR(t)}{dt}=\gamma_{A}I_{A}(t)+\gamma_{I}I(t)+\gamma_{I}J(t)\] (7) \[\frac{dD(t)}{dt}=\delta_{I}I(t)+\delta_{I}J(t)\] (8)