Analyzing Wuhan’s Strategies for Controlling Covid-19 Epidemic

Because of the high infectiousness of COVID-19, the paper divided infected people into two groups, the confirmed cases to be treated and isolated in hospitals and unconfirmed carriers in free environment. First, a COVID-19 transmission model was built based on the classification of infected population, and a sensitivity analysis algorithm was constructed to optimize unknown parameters in the model, such as the probabilities of transmission and the diagnosis rate. Second, the transmission model and an optimization procedure were used to simulate the COVID-19 epidemic in Wuhan and Diamond Princess Cruise, and the simulation results were compared with actual data reported by governments. Finally, Wuhan’s strategies for controlling COVID-19 epidemic at different stages were analyzed through the COVID-19 transmission model. The results showed: only isolating and treating the confirmed patients suffering severe symptoms could not effectively inhibit the rapid spread of COVID-19; isolating all the confirmed patients could reduce the infected population over 30 times; besides isolating all the confirmed patients, the city-wide lockdowns and fast test methods could dramatically contain the spread of the epidemic, including decreasing the cumulative infected population and shortening the period of epidemic; compared with Wuhan’s control strategies, the protection and isolation measures of Diamond Princess Cruise could not effectively inhibit the spread of COVID-19.

promoted to control rubella [8]. Zhang Cai-jun et al. analyzed the control strategies of hand-foot-andmouth disease, and advocated health education for children under 5 years old [9]. Kimberlyn Roosa et al. predicted the spread of COVID-19 in Guangdong and Zhejiang, China, by applying phenomenological models and actual data reported by National Health Commission of China, and pointed out that the predictions could reflect the effectiveness of control strategies [10]. Fang Yaqing et al. employed the SEIR model to simulate the COVID-19 epidemics in Hubei, China, and highlighted the importance of developing medicine for COVID-19 and optimizing therapeutic plans [11]. This paper built a COVID-19 transmission model to assess Wuhan's control strategies at different stages, such as building mobile cabin hospitals, isolating all the confirmed patients, implementing citywide lockdowns and adopting fast COVID-19 diagnostic methods. First, due to the high infectiousness of COVID-19, the infected people were divided into the confirmed patients in hospitals and unconfirmed carriers in free environment. Second, based on the classification of infected population, a COVID-19 transmission model was built by modifying the traditional SIR model. Then, the transmission model was verified through using it to simulate the COVID-19 epidemic in Wuhan and Diamond Princess Cruise and comparing the simulation results with the actual data reported by governments. Finally, Wuhan's strategies for controlling COVID-19 were assessed.

Building and verifying the COVID-19 transmission model
Compared with the past epidemics, COVID-19 is a highly contagious disease, and a COVID-19 carrier can even affect others prior to the onset of the illness. According to official statistics, the average incubation period of COVID-19 is 4.95 days. Based on the probabilities of transmission 0.3365 (1/day), an evaluated value in Table 4, in the initial epidemic, it could be estimated that an infected person could affect 5 people prior to the onset of the illness and the confirmed population would not exceed 18.91% of the infected people with no symptoms. Epidemic transmission models mainly include SI、SIR、SIRS and SEIR on the basis of different population classification methods. In SIR model, the whole population includes susceptible people, infected patients and recovered individuals with lifelong immunity [12,13]. In this section, a COVID-19 transmission model is built by modifying the traditional SIR model. Due to the high infectiousness of COVID-19, the whole population is divided into three groups, the susceptible people S , the confirmed patients I1 , who are treated in isolated environment, and the unconfirmed carriers I2 , who are in free environment and transmit the virus to the susceptible people.
where, N 0 = S ( ) + I ( ). If ( ) is used to denote the probabilities of transmission between a carrier of COVID-19 and a susceptible individual, ( ) the diagnosis rate describing a percentage of newly confirmed cases per unit time relative to the unconfirmed infected population, the COVID-19 transmission model can be expressed as Here, the parameter α is set to 0 or 1 to evaluate the influence of control strategies on the COVID-19 epidemic. If the confirmed cases are isolated, α = 0, otherwise α = 1.
The COVID-19 transmission model (3) is the first-order ordinary differential equations, and can be solved by means of Runge-Kutta method. However, some undetermined parameters in equation (3), such as the probabilities of transmission ( ), the diagnosis rate ( ), initial values of the confirmed and the unconfirmed infected population, N I1 (t 0 ) and N I2 (t 0 ), should be given first. In order to solve the undetermined parameters, an objective function is constructed where, I1 * ( ) is the actually cumulative confirmed population reported by governments; t 0 and t f are the initial and end times among the actual data; ρ( ) is the weight function and can be set as required. In the following numerical examples, set ρ( ) = 1 . The probabilities of transmission ( ) , the diagnosis rate ( ), initial values N I1 (t 0 ) and N I2 (t 0 ) can be obtained by minimizing the objective function (4). Some efficient optimization algorithms like BFGS can be adopted to minimize the objective function (4). During the solving procedure, the derivative or sensitivity of the objective function with respect to its undetermined parameters should be known. To obtain the derivative or sensitivity information, an iterative algorithm is constructed as followed.
Step ①: in the i th iteration, solve equation (3) in the internal [t 0 , t f ] by means of the current values of ( )， ( )，N I1 (t 0 ) and N I2 (t 0 ) to obtain the current values I1 ( ) and I2 ( ), and calculate the value of the objective function (4) through the numerical integration; Step ②: based on the results in Step ①, solve the conjugate equation (5) in the internal [t 0 , t f ] by means of the boundary condition M 1 (t f ) = 0 and M 2 (t f ) = 0, and calculate the current values 1 ( ) and 2 ( ); Step ③: calculate the sensitivity of the objective function (4) with respect to ( ), ( ), N I1 (t 0 ) and Assuming ( ) and ( ) are piecewise functions in the internal [t 0 , t f ], k and k are the values of the two functions in the i th internal [t k , t k+1 ], the last two expressions in equation (6) can be rewritten as Please refer to Appendix for the derivation procedure of the formulas.

Simulating the COVID-19 epidemic on Diamond Princess Cruise
The COVID-19 transmission model ( (3), the COVID-19 epidemic on Diamond Princess Cruise was simulated, as shown in figure 1. The horizontal and vertical axes were date and cumulative confirmed population, respectively. The blue curve stood for official data, and the red one represented simulation results of the transmission model (3).

Simulating the COVID-19 epidemic in Wuhan
The COVID-19 transmission model (3) was utilized to simulate the COVID-19 epidemic in Wuhan, and the undetermined parameters in the model were obtained by means of actual data reported by Chinese National Health Commission. Wuhan has a population of 14 million. In order to stop COVID-19 virus from spreading across China, Wuhan locked down the city on Jan. 23, 2020. In the lockeddown Wuhan, there were about 9 million residents. So set N 0 = 9,000,000 . Table 3

Analyzing Wuhan's strategies for controlling COVID-19 at different stages
From Jan. 20 to Mar. 11, 2020, Wuhan adjusted control strategies 2 times, which could be reflected by the parameter variation in However, because the COVID-19 diagnosis was a time consuming process and the accuracy of the test method was not high, the diagnosis rate remained much low, i.e. (t) = 0.0899 (1/day). On Feb. 11, 2020, Wuhan adjusted the control strategies again, including implementing the city-wide lockdowns and adopting the fast test methods like CT imaging to confirm COVID-19 cases for the purpose of the goal that all the infected people in Wuhan could be treated and isolated. These new and strict control strategies greatly increased the confirmed population and effectively controlled the epidemic spread. On Feb. 12, 2020, the increased confirmed cases was 13,436 and after this date, the diagnosis rate (t) was much higher than the probabilities of transmission (t). Based on the COVID-19 transmission model (3) and the parameters in table 4, the COVID-19 epidemic in Wuhan was simulated as shown in figure 3, where the green and red curves stood for the cumulative infected cases and the cumulative confirmed population, respectively. It could be found that almost all the infected people were isolated and treated by the end of March, and the cumulative confirmed population was about 50,000. The simulation showed that the COVID-19 epidemic would be finally over in late March, 2020. In order to assess the influence of Wuhan's control strategies at different stages on the COVID-19 epidemic, the COVID-19 transmission model (3) was used to simulate the epidemic under the different strategies. Case 1: Assuming that Wuhan had been implementing the control strategies ---mainly isolating and treating confirmed cases suffering severe symptoms in hospitals. The COVID-19 epidemic was shown in figure 4. It could be found that the cumulative infected population would reach up to 8,483,400 by the mid-May, 2020, about 170 times of the population in figure 3. Case 2: Assuming that Wuhan had been implementing the control strategies ---building mobile cabin hospitals, isolating and treating all the confirmed patients in designated hospitals and mobile cabin hospitals. The COVID-19 epidemic was shown in figure 5. Compared with figure 4, the cumulative infected population dropped dramatically to 247,620 although the period of epidemic was longer, over 2 years. Compared with figure3, the cumulative infected population was 4.95 times of the population in figure  3, and the period of epidemic was 13 times of that in figure 3. The differences between figure 3 and figure 5 demonstrated that the control strategies adjusted by Wuhan on Feb. 11, 2020, namely the citywide lockdowns and fast test methods like CT imaging to confirm COVID-19 cases, could dramatically contain the spread of the epidemic, including decreasing the cumulative infected population and shortening the period of epidemic. Case 3: Assuming that Wuhan had been implementing the control strategies ---building mobile cabin hospitals, isolating all the confirmed patients, adopting the fast test methods like CT imaging to confirm COVID-19 cases, and implementing the city-wide lockdowns, such as suspending public transportation, temporarily closing factories and schools, and asking people to stay at home. The COVID-19 epidemic was simulated under these strict control strategies, as shown in figure 6. It could

Conclusions
This paper built a COVID-19 transmission model to analyze the control strategies of Wuhan and Diamond Princess Cruise. Some valuable conclusions were drawn: (1) the COVID-19 transmission model could be used to predict the spread of COVID-19 in various regions by means of actual data reported by governments; (2) Wuhan's control strategies prior to Feb. 05, 2020, i.e. mainly isolating and treating the confirmed patients suffering severe symptoms in hospitals, could not effectively inhibit the rapid spread of COVID-19; (3) Wuhan's control strategies implemented from Feb. 05, 2020, i.e. building mobile cabin hospitals and isolating all the confirmed patients in the designated hospitals and the mobile cabin hospitals, could reduce the infected population over 30 times; (4) Wuhan's control strategies implemented from Feb. 11, 2020, i.e. city-wide lockdowns and fast test methods like CT imaging to confirm COVID-19 cases for the purpose of the goal that all the infected people could be treated and isolated, could dramatically contain the spread of the epidemic, including decreasing the cumulative infected population and shortening the period of epidemic; (5) the protection and isolation measures of Diamond Princess Cruise could not effectively inhibit the spread of COVID-19; (6) assuming that Wuhan had been implementing the control strategies announced on Feb. 05 and Feb. 11, 2020, i.e. building mobile cabin hospitals, isolating all the confirmed patients, adopting the fast test methods like CT imaging to confirm COVID-19 cases, and implementing the city-wide lockdowns, the COVID-19 epidemic would be over in 2 months and the cumulative confirmed population was not more than 1,700.

3,367,800
In the year of 2020