Collection of cases data

We obtained data of confirmed COVID-19 cases that were reported in Guangdong province, Italy and other places in the world from the Health Commission of Guangdong Province, the Ministry of Health of Italy, and the World Health Organization [2831]. The data include infected cases in the world and Europe since January 26 to March 12, 2020, the cumulative confirmed cases, newly reported cases, death cases, and recovered cases in Italy from January 30 to March 13, 2020 and in Guangdong from January 19 to February 26, 2020. The number of cumulative confirmed cases in Italy was 2 between January 30 and February 5, 2020, and was 3 from February 5 to February 19, 2020. The numbers of cumulative death and recovered cases were 0 from January 30 to February 19, 2020. We exclude the data in these periods and will focus on the Italian data between February 20 and March 13 because this is the early stage of the Italian outbreak, which allows us to compare the effectiveness of the prevention and control measures used in Italy and Guangdong by analysis of those datasets.

In Fig.1, we select February 20 as the first day when there were 4 cumulative confirmed cases in Italy, and January 19 as the first day when Guangdong had one cumulative confirmed case. On February 28 Italy closed schools in Lombardy, the most serious epidemic area, when the number of cumulative confirmed cases was already 650 (Fig.1a). This number is similar to the confirmed cases (i.e. 683) in Guangdong on February 2. In comparison, the Guangdong government took Seven Measures policy (e.g. blockade of unnecessary public places) on the third day (January 21) to prevent the spread of the epidemic and shut down all schools on January 28, when there were only 241 cumulative confirmed cases (Fig.1b). Therefore, Italy closed schools about 5 days (January 28 February 2) later than Guangdong. In Fig.1c we see that the number of cumulative cases has already exceeded 10 thousand in Italy, however, the newly confirmed cases in Guangdong began to decline since the 37th day and reached the peak of cumulative cases at only 1347 (Fig.1d). In summary, it took 14 days for Guangdongs reported cases to decline since its cumulative confirmed cases exceeded 1000.

In the early stage of the epidemic in Italy, the government did not take many prevention measures across the country. Thus, to study the epidemic of this stage, we extend the classical deterministic susceptible-exposed-infectious-removed (SEIR) epidemic model by dividing the population into susceptible (S), exposed (E), symptomatic/asymptomatic infected (I/A), confirmed (H) and recovered (R) compartments. The susceptible and exposed populations are further partitioned into quarantined susceptible (Sq) and quarantined suspected individuals (Eq). Based on the previous research [22], we adopt an autonomous model to study the early stage of the outbreak. We assume that the individuals exposed to the virus are quarantined with a proportion q by contact tracing. If the quarantined individuals are successfully infected, they will move to Eq compartment, otherwise they move to the Sq compartment. The individuals who exposed to the virus but were missed in the contact tracing with rate 1q can either move to the compartment E or still stay in compartment S, depending on whether they are infected or not. We assume that the successful transmission probability is and the contact rate is c. The infected individuals can be detected and then isolated at a rate of I or move to the compartment R at the rate of I due to recovery. The death rate of the infectious individuals with symptoms I and the isolated infected individuals H is . We also assume that the asymptomatic infectious is neither dead nor hospitalized. With these assumptions the model can be described by

$$ begin{array}{l} frac{dS}{dt}=-left(beta c+cq(1-beta)right)Sfrac{(I+theta A)}{N}+lambda S_{q},\ frac{dE}{dt}=beta c (1-q)Sfrac{(I+theta A)}{N}-sigma E,\ frac{dI}{dt}=sigma rho E-left(delta_{I}+alpha+gamma_{I}right)I,\ frac{dA}{dt}=sigma(1-rho)E-gamma_{A}A,\ frac{{dS}_{q}}{dt}=(1-beta)cqSfrac{(I+theta A)}{N}-lambda S_{q},\ frac{{dE}_{q}}{dt}=beta cqSfrac{(I+theta A)}{N}-delta_{q}E_{q},\ frac{dH}{dt}=delta_{I}I +delta_{q}E_{q}-left(alpha+gamma_{H}right)H,\ frac{dR}{dt}=gamma_{I}I+gamma_{A}A+gamma_{H}H.\ end{array} $$

(1)

The more detailed definitions of variables and parameters for model (1) are provided in Table1. As the population size is much larger than the size of the outbreak, i.e. S(t)/N1, the basic reproductive number R0 of model (1) is given by the following formula by utilizing the next generation matrix [32].

$$R_{0}=frac{betarho c(1-q)}{delta_{I}+alpha+gamma_{I}}+frac{beta(1-rho) ctheta(1-q)}{gamma_{A}}. $$

The above model will be used to study the early stage of the outbreak. However, with a series of prevention and control measures being implemented by the government, the autonomous model needs to be modified. Because of the difference before and after the implementation of control measures, piecewise functions of the contact rate and diagnosis rate are introduced to the autonomous model.

The contact rate is a constant in the autonomous model, i.e. the average number of susceptible individuals that an exposed people can contact without any control measures in a unit time. As the action of regional or national lockdown came into effect, peoples contact will gradually decrease. Thus, we assume that the contact rate is an exponential decreasing function of time t after the government has taken the control measures. The contact rate c(t) is assumed to take the following form:

$$ c(t)=left{begin{array}{l} c_{0}, tleq t^{*}+tau,\ left(c_{0}-c_{b}right)e^{-r_{1}(t-t^{*}-tau)}+c_{b}, t>t^{*}+tau.\ end{array}right. $$

(2)

Here c0 denotes the contact rate at the initial time without control measures, cb denotes the minimum contact rate under the current control strategies (cb

Similarly, because the efficiency of detection and availability of medical resources vary, we assume that the diagnosis rate is a time-dependent piecewise function rather than a constant. It is an increasing function when medical resources are adequate and a decrease function when they are not. The duration of diagnosis 1/I(t) is given by the following form:

$$ frac{1}{delta_{I}(t)}=left{begin{array}{l} frac{1}{delta_{I0}}, tleq t^{*},\ left(frac{1}{delta_{I0}}-frac{1}{delta_{If}}right)e^{-r_{2}(t-t^{*})}+frac{1}{delta_{If}}, t>t^{*},\ end{array}right. $$

(3)

where I0 is the diagnosis rate at the initial time. If the efficiency of detection is increasing with time t, then the diagnosis rate I(t) will increase. The parameter r2 measures how fast the diagnosis rate increases (i.e. the duration of diagnosis decreases) as more medical equipments or resources become available. The final diagnosis rate If is usually larger than I0. However, if the medical resource is inadequate, the diagnosis rate I(t) can decrease and the final diagnosis rate If can be less than I0.

According to the basic reproductive number R0, time-varying contact rate Eq. (2) and diagnosis rate Eq. (3), the effective reproductive number Rc(t) of time-dependent model is given by the following formula:

$$R_{c}(t)=frac{betarho c(t)(1-q)}{delta_{I}(t)+alpha+gamma_{I}}+frac{beta(1-rho) ctheta(1-q)}{gamma_{A}}. $$

Although the governments mandatory intervention plays a major role in epidemic control, peoples behavior changes such as keeping social distancing, wearing facial masks and washing hands due to media and expert suggestions cannot be ignored. Hence, the piecewise function similar to the previous contact rate and diagnosis rate is applied to the transmission rate . Considering that the impact of behavior change on the spread of the disease is not as great as the government mandatory intervention, the exponential change form is not used. If the number of reported confirmed cases increases, the public will enhance self-protection measures. Thus, we assume that the transmission rate is inversely proportional to reported confirmed cases H(t). The time-dependent transmission rate (t) takes the following form:

$$ beta(t)=left{begin{array}{l} beta_{0}, if frac{1}{klog(H(t))}>1,\ beta_{0}frac{1}{klog(H(t))} \ end{array}right. $$

(4)

where k represents the indicator measures strength of peoples awareness of self-prevention. The larger the value of k, the smaller the transmission rate.

According to the total population of Italy and the epidemic situation on February 20, 2020, we set initial values to be S(0)=60 480 000, H(0)=3 and R(0)=0. According to the WHO [33], the incubation period of COVID-19 is about 7 days. Thus, =1/7. The quarantined individuals were quarantined for 14 days, thus =1/14. We obtain other unknown parameter values by fitting data on reported number of cumulative confirmed cases, death cases and recovery cases from February 20 to March 10 in Italy. We utilized the nonlinear least-square (NLES) method in Matlab to fit model solution to the real data sets, as shown in Fig.2. The estimated parameter values are listed in Table1.

Fitting of the autonomous model to the data of COVID-19 in Italy from February 20 to March 10. (a) shows the number of cumulative confirmed cases, (b) shows the number of death cases, and (c) shows the number of recovered cases

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The effect of control measures on COVID-19 transmission in Italy: Comparison with Guangdong province in China - Infectious Diseases of Poverty -...