Improved Epidemic Dynamics Model and Its Prediction for COVID-19 in Italy

: The outbreak of coronavirus disease 2019 (COVID-19) has become a global public health crisis due to its high contagious characteristics. In this article, we propose a new epidemic-dynamics model combining the transmission characteristics of COVID-19 and then use the reported epidemic data from 15 February to 30 June to simulate the spread of the Italian epidemic. Numerical simulations showed that (1) there was a remarkable amount of asymptomatic individuals; (2) the lockdown measures implemented by Italy e ﬀ ectively controlled the spread of the outbreak; (3) the Italian epidemic has been e ﬀ ectively controlled, but SARS-CoV-2 will still exist for a long time; and (4) the intervention of the government is an important factor that a ﬀ ects the spread of the epidemic. of the disease signiﬁcantly. This paper only takes Italy’s “closure measures” into account; in practice, it still needs to be comprehensively considered according to the local epidemic situation.


Introduction
In December 2019, a new type of coronavirus pneumonia (COVID-19) broke out in Wuhan, China. Since then, there have been cases of infected individuals in various regions of the world. On 30 January 2020, the WHO listed COVID-19 as an "emergency public health event of international concern" [1]. According to the official statistics of the World Health Organization, as of 30 June 2020, the outbreak of COVID-19 has caused 10,193,723 confirmed cases and 503,867 fatalities globally [2]. The epidemic-prevention situation is extremely severe, and the number of confirmed cases in South Korea, Iran, the United States, and other countries continues to increase, while in Europe, Italy has become the "severe disaster country" of the epidemic. The collapse of the medical system [3] and the high proportion of the elderly population [4] make the mortality rate in Italy much higher than in other regions. According to the official report of the Italian Ministry of Health, as of 30 June 2020, the outbreak of COVID-19 has caused 240,578 confirmed cases and 34,767 fatalities in Italy, and the mortality rate is approximately 14% [5]. In view of the rapid spread of the epidemic nationwide, the Italian government took proactive prevention and control measures. On 10 March 2020, Italy implemented a lockdown measure nationwide.
Scientific and effective calculation of the spread tendency of the epidemic is crucial to the decision on epidemic prevention and restriction. It should rely on theoretically perfect models, reasonable parameters, and accurate predictions. The mathematical models generally advocated to describe the spread of infectious diseases include the Susceptible-Infectious-Recovered model (the SIR model), Susceptible-Exposed-Infectious-Recovered model (the SEIR model), Figure 1. Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic-dynamics 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.

SEIR_QJD Model
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 ( ), 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 ( ) Q E 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 Q E and J will be isolated. Note: we assume that the quarantined individuals cannot contact the susceptible ones; namely, it is not contagious.  Symptom Onset: Individuals in states , E Q E will become symptomatic after the infectious incubation period and will transition to states I and J at the rate of λ .  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. 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 ( ), 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 ( ), 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 Q E and J will be isolated. Note: we assume that the quarantined individuals cannot contact the susceptible ones; namely, it is not contagious.  Symptom Onset: Individuals in states , E Q E will become symptomatic after the infectious incubation period and will transition to states I and J at the rate of λ .
 Recovery: Individuals in states I , A , I and J will recover at rates of , respectively. 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. • Symptom 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 λ. • Recovery: Individuals in states I A , I, and J will recover at rates of γ A , γ I , and γ J , respectively.
• Death: Individuals in states I and J will die at rates of δ I and δ J , 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:

Estimation of Model Parameters
Given the values of parameters l, β, w I , λ, ρ, α, γ A , γ I , γ J , δ I , δ J and the initial conditions , D(t 0 ) of each group, the evolution of the epidemic can be simulated by Equations (1)- (8). The initial state t 0 represents the time when the first batch of infected people appeared (according to roughly 40% of asymptomatic individuals among infected ones [14], we assume that five patients are symptomatic and three patients are asymptomatic). Because the first cases discovered in Italy were all imported cases, we assume that E(t 0 ) = 0, E Q (t 0 ) = 0, and J(t 0 ), R(t 0 ), D(t 0 ) can be obtained from the reporting data. Besides, we set the initial value of the susceptible E(t 0 ) to unknown and estimate it together with other model parameters as described below. Since the number of exposures available to susceptible people is limited, we do not consider the total population of Italy as susceptible.
Since the SEIR_QJD model contains a relatively large number of parameters, the model could be overfitted while the available data are limited, which will affect the results and mislead the epidemic judgment. The proposed model consists of 11 parameters: l, β, w I , λ, ρ, α, γ A , γ I , γ J , δ I , δ J , of which γ A , γ I , γ J , δ I , δ J are related to the clinical characteristics of the disease and can be prefixed by relevant studies. Since the official report data show that R(t) and D(t) are only related to the diagnosed individuals J(t), we can roughly estimate them from the related data with the approximate equation The remaining parameters will be estimated according to the model. By observing the trend chart of γ J and δ J , in Figures 3 and 4, respectively, it can be found that the change in γ J over time is not significant, while that of δ J decreases continuously over time. Meanwhile, in order to reflect the effects of government interventions and medical investment, we will consider β and δ J as time-varying parameters, that is, β(t) = 1 {t<T 1 +t } β 0 + 1 {t>T 1 +t } cβ 0 and δ J = f (t); see Figure 4 for specific functions of f (t). The time T 1 is set to be the time when Italy implemented the nationwide lockdown measure, t is the lag time of the implementation, and the time T 2 is set to the end of Italy's lockdown measure.  The likelihood function is obtained by assuming that daily confirmed cases are independent Poisson random variables [10], that is:  Table 1.   The likelihood function is obtained by assuming that daily confirmed cases are independent Poisson random variables [10], that is:  Table 1. The likelihood function is obtained by assuming that daily confirmed cases are independent Poisson random variables [10], that is: where ∆J is the newly confirmed cases per day and λ i is the functions of model parameters l, β, w I , ρ, a, S(0) based on the differential Equations (1)- (8). We use the Markov chain Monte Carlo (MCMC) algorithm to fit the model based on Italian epidemic data from 15 February to 30 June (see Appendix A), where noninformative uniform distributions are chosen as the prior distributions; the specific information is in Table 1. Table 1. SEIR_QJD parameter setting (Italy).

S(t 0 )
The initial value of susceptible individuals 5 × 10 5 ∼ 1 ×  λ The speed of individuals from exposed to infected, which is the reciprocal of the incubation period

Empirical Analysis
1. Fitting Effect: Based on the cumulative number of confirmed cases in Italy as of 30 June 2020, we used the SEIR_QJD model for data fitting. The fitting effect of the model is measured by calculating the average deviation between the actual and simulated cumulative number of confirmed cases ( Table 2). The curve fitting of Italian epidemic cases is shown in Figure 5.  The infectiousness of symptomatic individuals relative to incubation ones 2.0 2.0 [12] The speed of individuals from exposed to infected, which is the reciprocal of the incubation period 1/5.3 1/5.9 The proportion of asymptomatic individuals among all infected ones 0.10 ∼ 0.60 0.4099 The diagnosed speed of symptomatic individuals 0.0 ∼ 0.5 0.0788 , The recovery rate of infected individuals 0.1 0.1 [9] The recovery rate of diagnosed individuals 0.01887 0.01887 , The mortality rate of infected and diagnosed individuals ( ) ( )

Empirical Analysis
1. Fitting Effect: Based on the cumulative number of confirmed cases in Italy as of 30 June 2020, we used the SEIR_QJD model for data fitting. The fitting effect of the model is measured by calculating the average deviation between the actual and simulated cumulative number of confirmed cases ( Table 2). The curve fitting of Italian epidemic cases is shown in Figure 5. This paper evaluates the fitting effect of the model by calculating the deviation between the simulated value and the real value ( Table 2). The average deviation of the model in the past 90 days was 0.86%, and the maximum deviation was controlled within 2.95%. Overall, the fitting effect of the model was very good and, to a great extent, largely simulated the development of the Italian Cumulative incidence Prevalence Incidence Figure 5. Simulation of epidemic-situation development trend. Simulation of the epidemic spread in Italy. Fitting curves: cumulative incidence (a), prevalence (b), incidence (c). This paper evaluates the fitting effect of the model by calculating the deviation between the simulated value and the real value ( Table 2). The average deviation of the model in the past 90 days was 0.86%, and the maximum deviation was controlled within 2.95%. Overall, the fitting effect of the model was very good and, to a great extent, largely simulated the development of the Italian epidemic.
2. The duration of the epidemic is relatively long. With the change in government control, people's awareness of epidemic prevention, medical level, and other factors, the contact rate between susceptible and infected people will also change accordingly. This paper describes the change by constructing a time-varying function of the contact rate. During the implementation of the nationwide "lockdown measures" in Italy, the time-varying parameters changed significantly (adjustment coefficient c_1 was 0.4364), while after the closure measures, the change of time-varying parameters was insignificant, indicating that the epidemic situation in Italy was basically under control and there was no secondary outbreak. Furthermore, according to the model (1-8), we used the regeneration matrix to get the formula of the regeneration number: The regeneration numbers of the three stages were R c0 = 2.9527, R c1 = 0.7064, and R c2 = 0.8204, respectively, indicating that the government's prevention and control measures suppressed the spread of the epidemic effectively.
3. COVID-19 is a disease with strong spreading ability, which can be effectively controlled under good public health conditions and quarantine measures. In order to analyze the impact of the government's response on the epidemic, this paper simulates four different situations by changing the time of the government interventions, which is based on time-varying parameters ( Figure 6). The analysis showed that the government's timely intervention measures could reduce the scale of the disease significantly. This paper only takes Italy's "closure measures" into account; in practice, it still needs to be comprehensively considered according to the local epidemic situation.

Region
Evaluation 2. The duration of the epidemic is relatively long. With the change in government control, people's awareness of epidemic prevention, medical level, and other factors, the contact rate between susceptible and infected people will also change accordingly. This paper describes the change by constructing a time-varying function of the contact rate. During the implementation of the nationwide "lockdown measures" in Italy, the time-varying parameters changed significantly (adjustment coefficient c_1 was 0.4364), while after the closure measures, the change of time-varying parameters was insignificant, indicating that the epidemic situation in Italy was basically under control and there was no secondary outbreak. Furthermore, according to the model (1-8), we used the regeneration matrix to get the formula of the regeneration number: The regeneration numbers of the three stages were 3. COVID-19 is a disease with strong spreading ability, which can be effectively controlled under good public health conditions and quarantine measures. In order to analyze the impact of the government's response on the epidemic, this paper simulates four different situations by changing the time of the government interventions, which is based on time-varying parameters ( Figure 6). The analysis showed that the government's timely intervention measures could reduce the scale of the disease significantly. This paper only takes Italy's "closure measures" into account; in practice, it still needs to be comprehensively considered according to the local epidemic situation. 4. The proportion of asymptomatic and undetected patients in Italy is approximately 40%, which accounts for a relatively high proportion of total infected people. Therefore, the government needs to strengthen the detection of close contacts of cases.
Overall, the model has good generalizability and can change the parameters according to the different regions to simulate the spread of the epidemic and provide scientific judgment for epidemic prevention and control decisions.  4. The proportion of asymptomatic and undetected patients in Italy is approximately 40%, which accounts for a relatively high proportion of total infected people. Therefore, the government needs to strengthen the detection of close contacts of cases. Overall, the model has good generalizability and can change the parameters according to the different regions to simulate the spread of the epidemic and provide scientific judgment for epidemic prevention and control decisions.

Discussion
In response to the characteristics of coronavirus disease 2019 (COVID-19), the infectiousness during the incubation period, the specificity of a large proportion of asymptomatic infections, and the strong government interventions, we proposed a SEIR_QJD epidemic-dynamics model that comprehensively considers the characteristics of infectious diseases and the impact of overall prevention and control measures to make it closely related to the reality.
According to the analysis results, the government intervention in Italy has effectively contained the spread of the epidemic. At the same time, after the lifting of the blockade, the Italian epidemic has not rebounded, indicating that the epidemic was controlled effectively. However, there is still a long time until the epidemic is completely over, so the government should continue to strengthen the prevention and control of the epidemic to avoid a second outbreak. In addition, due to the existence of a large number of asymptomatic infected people, which has increased the difficulty of controlling the epidemic, the government should strengthen the detection of the individuals who are asymptomatic and strengthen the isolation of confirmed patients and the medical observation of close contacts.
Recently, the number of new confirmed cases per day in Italy has shown a slight increase, so the maintenance of the current prevention and control measures is of great importance. Both the infection rate and isolation rate measure the ability of the virus to spread, so blocking transmission is the key. However, Italy's restrictions have now been lifted; we advise citizens to reduce public activities, strengthen self-protection, and wear masks when going out. People should seek medical treatment or self-quarantine after developing symptoms such as fever and dry cough so as not to spread the virus to others.

Conflicts of Interest:
The authors declare no conflict of interest.