# Improved Epidemic Dynamics Model and Its Prediction for COVID-19 in Italy
## 3 Estimation of Model Parameters  
Given the values of parameters \(l,\beta,w_{I},\lambda,\rho,\alpha,\gamma_{A},\gamma_{I},\gamma_{J},\delta_{I}, \delta_{J}\) and the initial conditions \(\big{\{}S(t_{0}),E(t_{0}),E_{Q}(t_{0}),I(t_{0}),I_{A}(t_{0}),J(t_{0}),R(t_{0}),D(t_{0})\big{\}}\) 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: \(l,\beta,w_{I},\lambda,\rho,\alpha,\gamma_{A},\gamma_{I},\gamma_{J},\delta_{I}, \delta_{J}\), of which \(\gamma_{A},\gamma_{I},\gamma_{J},\delta_{I},\delta_{J}\) 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 \(\gamma_{J}\approx(R(t+1)-R(t))/I(t)\), \(\delta_{J}\approx(D(t+1)-D(t))/I(t)\). The remaining parameters will be estimated according to the model. By observing the trend chart of \(\gamma_{I}\) and \(\delta_{J}\), in Figures 3 and 4, respectively, it can be found that the change in \(\gamma_{J}\) over time is not significant, while that of \(\delta_{J}\) decreases continuously over time. Meanwhile, in order to reflect the effects of government interventions and medical investment, we will consider \(\beta\) and \(\delta_{J}\) as time-varying parameters, that is, \(\beta(t)=1_{[t<T_{1}+t^{\prime}]}\beta_{0}+1_{[t>T_{1}+t^{\prime}]}\varepsilon _{0}\) and \(\delta_{J}=f(t)\); 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, \(t^{\prime}\) 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:  
\[L(l,\beta,w_{l},\rho,a)=\sum_{i=1}^{k}\log\frac{e^{-\lambda_{i}}\lambda_{i}^{ \Delta l_{i}}}{\Delta l_{i}!} \tag{9}\]  
where \(\Delta l\) is the newly confirmed cases per day and \(\lambda_{i}\) is the functions of model parameters \(l,\beta,w_{l},\rho,a,S(0)\) 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.