Data processing

Confirmed cases rarely infect others; infections are mainly caused by infectious persons outside hospitals. To clarify the spread of COVID-19 in Wuhan, we tried to estimate the number of potential infections on January 20, the day $({T_1})$ when Academician Zhong Nanshan, the leader of the high-level expert group formed by National Health Commission of the People’s Republic of China, awakened the awareness of Chinese people, and China began to take measures to prevent and control COVID-19. ^15,16

It was noticed that Wuhan Municipal Health Commission started the daily outbreak report from January 23 on its official website, and the National Health Commission of the People’s Republic of China presented the outbreak notification from Wuhan Municipal Health Commission from January 11 to January 20, followed by the national reports starting on January 21. Furthermore, clinically diagnosed confirmed cases were included on January 17, 18, and 19, then cancelled, and re-added from February 12. ^17,18 Therefore, there were missing data, which was confusing, in the first few days in Wuhan, and we decided to revise and complement these data from January 20 to 22 first. Also, since the new confirmed cases suddenly increased to 892 on January 27, it also needed to be amended. After all, these patients already existed, and we failed to confirm them in time. Based on the reported cumulative laboratory confirmed cases from January 23 to February 3, we obtained a best Gauss fitting function which is $C(t) = 132100\exp ( - {((t - 42.16)/17.3)^2})$ (Goodness of fit: SSE: $3.12e + 05$ , R-square: $0.9924$ , Adjusted R-square: $0.9907$ , RMSE: $186.2$ ). We used it for the short-term backtracking of the cumulative laboratory confirmed cases from January 20 to January 22 and got 195, 262, and 348, respectively. The revised cumulative confirmed cases from January 15 to February 3 are presented in Table 2.

Table 2. The revised cumulative laboratory confirmed cases from January 15 to February 3

* Clinically diagnosed cases are added in these data. Although the number of reported cases is more than the revised one on January 19 by 53, and more than the revised one on January 23 by 36, we think we can refer to these data because the revised data are laboratory confirmed cases and the reported number of 495 on January 23 includes some clinical diagnosis cases added before. After smoothing the reported new 892 cases on January 27, the corresponding new confirmed cases from January 23 to January 31 were 155, 198, 240, 272, 296, 319, 356, 378, and 576 respectively. See Figure 13 in Appendix A1 for the revision effect.

As clinically diagnosed cases were incorporated into the confirmed cases and the demand for clearing inventory, we see a spike in the new confirmed data on February 12. ¹⁹ We explored the trend of laboratory confirmed cases from January 23 to February 14, excluded the clinically diagnosed cases from January 12 to 14 with the purpose of maintaining the statistical consistency, and found there is a good continuity (See Figure 1). There were 20630, 21960, and 22961 laboratory confirmed cases from January 12 to 14. Since clinically diagnosed cases were added, an additional fitting from February 12 was obtained, which is provided in Appendix A2. We observed that the data of laboratory confirmed cases with clinically diagnosed cases from February 3 to February 11 were 7898, 9904, 12244, 14900, 17822, 20935, 24141, 27326, and 30375 respectively. Figure 14 in Appendix A2 illustrates the fitting effect. So far, data was revised and we figured out the evolution of cumulative confirmed cases from January 20 to February 26 in Wuhan in Figure 1.

Figure 1. The evolution of cumulative confirmed cases.

Let $\Gamma (t)$ be the revised evolution curve of cumulative confirmed cases for $t$ starting on January 20. We can see from Figure 1 that $\Gamma (t)$ changes its trend (from an accelerated rise to a decelerated rise) and appears as an inflection point on February 7. Another feature is that curve $\Gamma (t)$ starts to flatten after February 22, which means the increment is very small. In general, the epidemic inflection points are the days when new infections begin to decline and the days when there are few new cases; both cannot be accidental and should be sustainable. For a smooth $\Gamma (t)$ , they correspond to $\Gamma ''(t) = 0$ and $\Gamma '(t) \ll \varepsilon ,\varepsilon $ is small and positive. Therefore, Figure 1 shows that while the first epidemic inflection point appeared on February 7 and the second would be on February 23, it also indicates that infections declined from February 7 and new infections rarely occurred in an incubation period before February 23. That is, COVID-19 in Wuhan was almost under control by the end of February.

It can also be detected from Figure 2 that the number of new laboratory-confirmed cases declined after February 7. Figure 2 directly presents the evolution of reported new laboratory confirmed cases before February 12. Figure 3 also verifies that China had contained COVID-19 in Wuhan by the middle of February. We can see from Figure 3 that medical capacity was sufficient after February 16 and the vacancy rate of beds in hospitals remained over 20% after February 23.

Figure 2. The evolution of new laboratory confirmed cases.

Figure 3. Capacity and occupation of beds in designated hospitals in Wuhan.

It shows February 7 as an inflection point of cumulative confirmed curve $\Gamma (t)$ , and the increase in the number of confirmed patients would tend to alleviation after February 20. There are few new confirmed patients after February 23.

It only presents the number of new laboratory- confirmed cases from January 23 to February 11 because the confirmation criteria changed after February 12. An abnormal data on January 27 is excluded. It can be captured that there is a decline after February 7.

Wuhan government increased the number of designated hospitals on February 2, 3, 8, 9, 9, 10, 12, 15, 16, and 20 respectively. Data comes from Wuhan Municipal Health Commission. ²⁰ There is no running of medical resources after February 16.

Discuss the proportion of the asymptomatic $p$

There is an argument about asymptomatic infections. The infectivity of the asymptomatic is certain, ^{Reference Hu, Song and Xu21} but the proportion is not clear. On February 17, China CDC published a paper in the Chinese Journal of Epidemiology, and mentioned that there were 889 cases of asymptomatic infections in the total 72314 domestic reported cases as of February 11, accounting for about 1.2%. ²² The preprint platform medRxiv which released a study by research teams at Kyoto University in Japan, Georgia State University in the United States, and Oxford University in the United Kingdom, claimed that the asymptomatic rate was 17.9%. ^{Reference Mizumoto, Kagaya, Zarebski and Chowell23} An article published in Nature on March 20, 2020 warned that the asymptomatic or mild symptoms may account for 30-60% and the ability to spread the virus is not weak. ²⁴

Although the proportion of asymptomatic infections is not clear, there are mainly 2 ways to confirm the asymptomatic. A way is the detection of close contacts and another is the port input detection. The ratio of close contacts to confirmed cases was approximately 3.29:1 in Hubei province as of February 16, ²⁵ and this ratio was slightly higher in Wuhan. It was 6.65 in late January, 4.75 in February, and 3.62 after February 15 by data analysis directly. A follow-up survey by the China-WHO Joint Expert Group found that approximately 1 to 5% of close contacts were diagnosed with COVID-19, and many of them are asymptomatic. ²⁶ Assuming that the confirmed cases found by detecting close contacts are all asymptomatic, the proportion would not exceed 33%. As for cases found by the entry detection, academician Zhong Nanshan said that 50% of imported cases did not have fever, and only the mild symptoms such as cough and cold, were detected by nucleic acid testing after entering the country. ²⁷

There is also an important fact from children’s COVID-19 cases. Most investigations confirmed that children’s cases are mainly asymptomatic or mild, and their symptoms are generally milder than adults. Children’s Medical Center in Shanghai studied more than 700 infected children in China, and found that 56% of children had mild or asymptomatic conditions. ²⁸ A research by Lu et al. suggested that 15.8% of infected children had no symptoms, 19.3% only had upper respiratory symptoms, and 7% had only CT features without any symptoms. ^{Reference Xiaoxia, Liqiong and Hui29} Raccardo et al. also clarified that most children with COVID-19 presented with mild symptoms ^{Reference Riccardo, Martina and Licari30} ; Xu et al. stated that symptoms in children cases were nonspecific and no children required respiratory support or intensive care. ^{Reference Yi, Xufang and Bing31}

Based on these analyses, we deem the most possible value of $p$ is $p \le 0.55$ for all infections.

Determination of the disease incubation period $m$

Due to the reason that COVID-19 in Wuhan was in the waning stage in March, without loss of generality, it is supposed that $$(1 - p){N_t} \le \sum\nolimits_{it + 1} {{C_t}} $$ within 14 days before March 18. We begin investigations from January 23. It is obvious that ${N_{56}} = 0$ and ${N_{55}} = 0$ because there were no new confirmed cases after March18.

By the above presented timelines, we know the inventory was cleared and medical demands were basically met after February 15. It also can be seen from Figure 3 that available beds in hospitals continued to increase after February 16 and the vacancy rate remained above 20% after February 23. In other words, Wuhan achieved the timely confirmation and complete admission for visiting patients after February 15. Since there were no new cases after March 18, we believe there were few asymptomatic infections at that time. Also, since almost ${N_{t - 1}}/m$ persons will feel sick on day $t$ in terms of probability, then ${C_t} \le {N_{t - 1}}/m$ in March. Thus, there is $$(1 - p)m{C_{t + 1}} \le \sum\limits_{it + 1} {{C_t}} $$ within 14 days before March 18. The relationship between the number of infectious persons outside hospitals ${N_t}$ and the number of new confirmed cases per day ${C_t}$ is provided in Table 3.

Table 3. The relationship between ${C_t}$ and ${N_t}$ in March

By the direct calculation, we can get $(1 - p)m \le 2.9441,$ where 2.9441 is the median value. Thus, $m \le 2.9441/(1 - p) \le 6.667.$

That is, the incubation period of most infected persons is not more than 7 days. Therefore, we use $m = 7$ for reference.

Estimation of ${N_0}$ and ${R_0}$

${N_0}$ expresses the number of infectious persons outside hospitals on January 22, before the day of Lockdown. ${R_0}$ is the basic reproduction number of a disease, which reflects the initial infectivity of the disease.

On January 31, the third batch of 14 designated hospitals began to treat patients with COVID-19, providing nearly 10000 available beds. The first 2 batches provided a total of about 4000 beds. ³² Then, more dedicated hospitals, including the Vulcan Mountain Hospital, Thunder God Mountain Hospital (after February 4, $({T_3})$ ), and some Ark hospitals (The first 1 started operation on the evening of February 5, $({T_4})$ ) began to treat patients with COVID-19 in succession. We believe medical capacities in the first part of February increased gradually and met demands after February 15 (which also can be detected from Figure 3). Therefore, considering the inventory and the asymptomatic, the daily new confirmed cases ${C_t}$ would satisfy:

(1) $${C_t} \ge \left\{ {\matrix{ {{a \over m}(1 - p){N_{t - 1}},} \hfill \hskip 66pt{t:Jan.23\;to\;Jan.30,} \hfill \cr {{{a + 0.025(t - 8)} \over m}(1 - p){N_{t - 1}},} \hfill \hskip 26pt{t:Jan.31\;to\;Feb.5,} \hfill \cr {{{a + 0.15 + 0.06(t - 14)} \over m}(1 - p){N_{t - 1}},} \hfill \hskip 6pt{t:Feb.6\;to\;Feb.15,} \hfill \cr } } \right.$$

where $a$ is the average hospital admission rate in January. According to a report from Wuhan Municipal Health Commission, there is a 5% chance that fever patients were admitted to hospitals from January 22 to 27. ³³ We get $a = 0.063$ by report-based numbers of visiting patients and left-observing patients, which are provided in Table 5 in Appendix A3. Therefore, it is reasonable to argue that $a = 0.063$ in late January. Thus, in January there is

$${N_{t - 1}} \le {\displaystyle{{m{C_t}} \over {a(1 - p)}}} \le 247{C_t}.$$

Hence, it is possible that the number of infectious persons outside the hospitals may be 17290 on January 22. That is, infected people who were not diagnosed were already very many before the day of Lockdown. Also, a series of values of ${N_t}$ can be given. Now, to estimate the contagion of SARS-CoV-2 virus, let $\alpha $ be the daily growth rate of infectious persons outside hospitals; we define N _t as:

(2) $${N_t} = {N_{t - 1}}(1 + \alpha ) - {C_t}.$$

As the revised cases presented in Table 2 represent the definitive confirmed ones, we used these data for further investigation. Then, with Eq. (2) and these data, based on the Principle of Least Squares, using LINGO (Adobe Inc., San Jose, California, USA) for optimization, by programming and operating, beginning with 17290 on January 22, we have the optimal value of $\alpha=0.16025$ in late January after January 23, 0.3332 from January 16 to 22, and 0.2864 on January 23. Further estimating, there were 10448 infected persons on January 20 and 22242 on January 23. We also calculated the greatest number of infected patients outside hospitals ia over 72000 on February 4.

Studies have shown that the proportion of people who carried the virus for more than 14 days of latency was very small (there were only 13 cases in more than 1000 cases) ³⁴ ; it is supposed that the infected persons are no longer infectious after 14 days. Considering this natural elimination rate of virus toxicity, the infection rate may be $\alpha+1/14$ . Thus, the effective reproduction number ${R_e}$ in late January is approximately $R_{e}=m(\alpha+1/14)=7\times(0.16025+0.07143)=7\times0.2317=1.62$ and the basic reproduction number R ₀ = m(α + 1/14) = 7 × (0.3332 + 0.07143) = 7 × 0.4046 = 2.83 in the early outbreak, and $7\times(0.2864+0.07143)=2.50$ on January 23. Similarly, ${R_e} = 1.20$ in early February and about 0.33 in late February.

Functions of the removed rate $\theta (t)$ and the admission rate $\delta (t)$

On December 31, 2019, people heard that unexplained pneumonia occurred in Wuhan which was related to Wuhan’s South China Seafood Market and were not worried about it. On January 20, Zhong Nanshan confirmed the human-to-human transmission of the SARS-CoV-2 virus and urged the public not to go to Wuhan and to wear masks. ¹⁵ These words really awakened Chinese people. On the same day, Chinese Premier Li Keqiang presided over an executive meeting of the State Council to deploy the COVID-19 epidemic prevention and control, and China began the difficult control process. ¹⁶ The lives of 1.3 billion Chinese people were destroyed in the next 2 months.

On January 23, a city-wide closure notice was issued by Wuhan government. People were scared. In view of the panic, coupled with the need to buy daily necessities, seek medical treatment, and the shortage of hygienic protective articles, we assumed that the removed proportion may be increased to 30% as of January 31. Since the closed management of residential communities started on February 11 and every household was allowed 1 person to go out to buy necessities every 3 days (referring to $({T_3}),({T_4}),({T_6})$ ), we assumed that at least 60% of the people will be removed from those susceptible to the virus, taking into account family size in China. Furthermore, because the completely closed community management started on February 15, and no was allowed to go out (see $({T_7})$ ) except the required staff, we suggested that 90% of the people could avoid being infected. The removed rate $\theta $ of those susceptible to the virus is defined as $\theta (t)$ , as follows:

(3) $$\theta (t) = \left\{ {\matrix{ {0.30(t - 1)/8,} \hfill & {t:Jan.23\;to\;Jan.31,} \hfill \cr {0.30,} \hfill & {t:Feb.1\;to\;Feb.10,} \hfill \cr {0.60,} \hfill & {t:Feb.11\;to\;Feb.15,} \hfill \cr {0.90,} \hfill & {t:AfterFeb.15.} \hfill \cr } } \right.$$

According to the analysis above, the admission rate of visiting patients was 0.063 before February. Noticing that the average daily detection capacity increased 10 times in 6 days in Wuhan from January 22 to January 27, ³⁵ the Vulcan Mountain Hospital and the Thunder God Mountain Hospital had been in operation since February 4 and February 8, and Ark hospitals started operations on February 5, it means the admission rate slowly increased before February 5 and it was growing faster after the 5th. This happened even though Wuhan had the capability of timely detection and admission of all patients after February 15. Hence, the admission rate of visiting patients $\delta $ is defined as the following, $\delta (t)$ :

(4) $${\delta (t) = \left\{ {\matrix{ {0.063,} \hfill & {t:Jan.23\;to\;Jan.30,} \hfill \cr {0.063 + 0.025(t - 8),} \hfill & {t:Jan.31\;to\;Feb.5,} \hfill \cr {0.063 + 0.15 + 0.06(t - 14),} \hfill & {t:Feb.6\;to\;Feb.15,} \hfill \cr {1,} \hfill & {t:Feb.16\;to\;Mar.17.} \hfill \cr } } \right.}$$

The model

Susceptible people may be infected in contact with latency, the asymptomatic, and the infectious patients outside hospitals. Close contacts may be quarantined, including infected and uninfected. Infected close contacts would be confirmed by PCR-testing, and uninfected ones will return to being susceptible after isolation. Infected people may show symptoms after the incubation period or may be asymptomatic. Of course, asymptomatic infections have no medical demands. Typical patients will visit doctors, some are admitted to hospitals, while others are not admitted. Hospitalized patients will not infect others (disregarding infections of medical staff). Due to information disclosure, social-distancing, and stay-at-home, some people can avoid being infected and the contact rate and infection rate will also be reduced.

We divided individuals in Wuhan into the susceptible $S$ , the latent $E$ , the asymptomatic ${I_a}$ , the not admitted patients ${I_m}$ , the hospitalized patients ${I_h}$ , the awareness protective population ${S_p}$ , the quarantined contacts ${S_q}$ , and the recovered people $R$ . Based on intervention, control process, and individual status, the COVID-19 transmission mechanism in Wuhan is presented in Figure 4, and the corresponding epidemic model is as follows:

(5) $$\left\{ {\matrix{ {{{{\rm{d}}S} \over {{\rm{d}}t}} = } & {A - \beta (1 - q)S(\mu E + {I_a} + {I_m}) - \theta S - (1 - b)r(\delta {I_m} + bc{S_q}) + (1 - b)c{S_q} - dS,} \cr {{{{\rm{d}}E} \over {{\rm{d}}t}} = } & {\beta (1 - q)S(\mu E + {I_a} + {I_m}) - br(\delta {I_m} + bc{S_q}) - \varepsilon E - dE,} \cr {{{{\rm{d}}{I_a}} \over {{\rm{d}}t}} = } & {\varepsilon pE - {\gamma _1}{I_a} - d{I_a},} \cr {{{{\rm{d}}{I_m}} \over {{\rm{d}}t}} = } & {\varepsilon (1 - p)E - \delta {I_m} - {\gamma _2}{I_m} - d{I_m},} \cr {{{{\rm{d}}{I_h}} \over {{\rm{d}}t}} = } & {\delta {I_m} + bc{S_q} - {\gamma _3}{I_h} - \rho {I_h} - d{I_h},} \cr {{{{\rm{d}}{S_p}} \over {{\rm{d}}t}} = } & {\theta S - d{S_p},} \cr {{{{\rm{d}}{S_q}} \over {{\rm{d}}t}} = } & {r(\delta {I_m} + bc{S_q}) - c{S_q} - d{S_q},} \cr {{{{\rm{d}}R} \over {{\rm{d}}t}} = } & {{\gamma _1}{I_a} + {\gamma _2}{I_m} + {\gamma _3}{I_h} - dR,} \cr } } \right.$$

Figure 4. Spread mechanism of COVID-19 in Wuhan.

with the initial conditions S(0)>0,E(0)>0,I_a(0)>0,I_m(0)>0,I_h(0)>0,S_p(0)0,S_q(0)0,R(0)>0. It is obvious that all solutions initiating in $\mathbb{R}_ + ^8$ exist for all $t 0$ in $\mathbb{R}_ + ^8$ . Where: $\mathbb{R}_ + ^8 = \{ (x,y,z,u,v,w,s,t) \in {\mathbb{R}^8}:x 0,y 0,z 0,u 0,v 0,w 0,s 0,t 0\} .$

Noticing that Wuhan had achieved the complete admission of visiting patients after February 15, then the disease incidence item $\beta (1 - q)S(\mu E + {I_a} + {I_m})$ in model (5) should be changed to $\beta (1 - q)S(\mu E + {I_a})$ . ${I_m}$ can be regarded as the mild patients in Ark hospitals, $\delta $ is the conversion rate from mild to severe, and the quarantine item $r(\delta {I_m} + bc{S_q})$ , will be replaced with $r(\varepsilon (1 - p)E + bc{S_q})$ . See said model as (5+) presented in Appendix A4.

The effective reproduction number ${R_e}$

It is straightforward to get the disease-free equilibrium (DFE) point ${M_0}$ . System (5) can be written in a vector form as

$$\displaystyle{{dX} \over {dt}} = F(X)$$

with $X = (S,E,{I_a},{I_m},{I_h},{S_p},{S_q},R{)^T}$

Computing the effective reproduction number ${R_e}$ using the method of van den Driessche and Watmough, ^{Reference van den Driessche and Watmough36} a subsystem which consisted of the infectious compartments $E,{I_a}$ and ${I_m}$ of system (5) is presented as:

$$\displaystyle{{d{x_i}} \over {dt}} = F({x_i}) - V({x_i}),i = 1,2,3.$$

Hence,

$${{\cal F} = \left( {\matrix{ {\beta (1 - q)S(\mu E + {I_a} + {I_m})} \cr 0 {} {} \cr 0 {} \cr } } \right),{\cal V} = \left( {\matrix{ {\varepsilon E + dE} {} \cr {{\gamma _1}{I_a} + d{I_a} - \varepsilon pE} {} {} \cr {\delta {I_m} + {\gamma _2}{I_m} + d{I_m} \!- \!\varepsilon (1 - p)E} {} {} \cr } } \right)}.$$

The associated next-generation matrices are

$${F = \displaystyle\left({{\partial {\cal F}} \over {\partial {x_i}}}({M_0})\right) = \left( {\matrix{ {\mu \beta (1 - q){S_0}} &{\beta (1 - q){S_0}} &{\beta (1 - q){S_0}} \cr 0 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right)},$$

$$V = \displaystyle\left ({{\partial {\cal V}} \over {\partial {x_i}}}({M_0})\right) = \left( {\matrix{ {\varepsilon + d} & 0 & 0 \cr { - \varepsilon p} & {{\gamma _1} + d} & 0 \cr { - \varepsilon (1 - p)} & 0 & {\delta + {\gamma _2} + d} \cr } } \right).$$

Giving out

$${{V^{ - 1}} = {1 \over \scriptstyle{{(\varepsilon + d)({\gamma _1} + d)(\delta + {\gamma _2} + d)}}}\left( {\matrix{ {({\gamma _1} \!+\! d)(\delta \!+\! {\gamma _2} \!+\! d)} & 0 & 0 \cr {\varepsilon p(\delta \!+\! {\gamma _2} + d)} & {\varepsilon \!+\! d} & 0 \cr {\varepsilon (1 \!-\! p)({\gamma _1} \!+\! d)} & 0 & {(\varepsilon \!+\! d)({\gamma _1} \!+\! d)\!} \cr } } \right).}$$

The effective reproduction number ${R_e}$ is determined as the spectral radius $\rho (F{V^{ - 1}})$ , thus we obtain:

$${{R_e} = \rho (F{V^{ - 1}}) = \beta (1 - q){S_0}\left({\mu \over {\varepsilon + d}} + {{\varepsilon p} \over {(\varepsilon + d)({\gamma _1} + d)}} + {{\varepsilon (1 - p)} \over {(\varepsilon + d)(\delta + {\gamma _2} + d)}}\right)}.$$

Let the transmission rate of latency be $\mu = 2/m$ because the infected persons have been contagious 2 days before onset, ³⁷ and the conversion rate $\varepsilon = 1/m$ as $m$ is the incubation period. Therefore,

(6) $$\displaystyle{R_e} = {{m\beta (1 - q){S_0}} \over {1 + dm}}\left({2 \over m} + {p \over {m({\gamma _1} + d)}} + {{1 - p} \over {m(\delta + {\gamma _2} + d)}}\right).$$

It shows that the awareness-protective rate $q$ will affect ${R_e}$ greatly, and the admission rate $\delta $ of mild patients will also contribute to decrease ${R_e}$ . Based on the estimated parameters in the next subsection, surface ${R_e}(q,\delta )$ , and curve ${R_e}(q,\delta ) = 1$ are depicted in Figure 5. We can see that the maximum value of ${R_e}$ may exceed 4, it is possible to make ${R_e} \lt 1$ if the awareness-protective rate $q$ is over 60% for high medical admission rate $\delta $ . It should be 75% for low medical admission rate.

Figure 5. The control reproduction rate ${R_e}$ .

${R_e}$ will exceed 4 for the low $q$ and $\delta $ . It will decrease with the increasing $q$ and $\delta $ . ${R_e} \lt 1$ only when $q \gt 0.6$ . It should be $q \gt 0.75$ for the low $\delta $ .

With respect to the dynamics of infectious diseases, at the beginning of the outbreak, Eq. (6) expresses the basic reproduction number ${R_0}$ , reflects the ability of an infectious disease to invade a population, and is ‘the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual.’ ^{Reference Diekmann, Heesterbeek and Metz38} ${S_0}$ is generally taken as the total population. On day $t$ , we can hold Eq. (6) and replace ${S_0}$ with ‘ $(1 - \theta ){S_0}$ ’ to claim the control reproduction number ${R_e}$ by Eq. (6) because some susceptible people can avoid being infected due to protective measures and staying at home. In this paper, $q$ is defined as the awareness-protective rate of the susceptible population, then it also is the removed rate of the susceptible population in Eq. (6) when we use Eq. (6) for referring to the control reproduction number ${R_e}$ . Therefore, the representation in Figure 5 tells us that it is necessary to Stay-at-Home in the case of limited medical resources, and protective measures such as social distancing, wearing masks, and washing hands are also necessary. Using $q$ in Eq. (6) can also be interpreted as the group immunity rate because the reduction in susceptible population is mainly due to the immunization of the recovered persons if there is no implementation of Stay-at-Home. Therefore, we conclude that COVID-19 would be controlled only when the group immunity rate is over 75% of total population if most mild patients are allowed to heal themselves at home.

The awareness-protective rate $q$ can be referred to the decline of infection rate from early January to late January mentioned above; we took $q = 0.3$ for reference. It could also be detected from Figure 8 for the sound fitting of data in January. For the following estimated $\beta $ , ${\gamma _{1,2}}$ , and Eq. (3,4,6), ${R_e}$ is figured out roughly in Figure 6. The depiction is consistent with the above-mentioned estimation of ${R_e}$ by the method of Least Squares. We can see that the strict Stay-at-home orders starting on February 11 quickly made ${R_e} \lt 1$ so that COVID-19 in Wuhan could finally be contained within 1 month. Figure 6 (b) clearly shows that the increasing removal rate of the susceptible is extremely effective in reducing infections. Hence, it is very important to stay at home.

Figure 6. Evolution of the reproduction number ${R_e}$ and the necessary of θ. (a) Curve ${R_e}(t)$ continues falling with the recruitment of designated hospitals and temporary hospitals. The sudden drop comes from the strict implementation of the Stay-at-home order which was issued on February 11 in Wuhan. (b) Here, evolutions of $E(t) + {I_a}(t) + {I_m}(t) + {I_h}(t)$ are depicted to illustrate infections. Depictions show that infections will obviously decrease with the removed rate.

Figure 7 presents the evolution of COVID-19 without active interventions (corresponding to $\theta = 0,q = 0,\delta = 0.063$ ). From Figure 7(a), we can further observe that the number of infections will reach 75% after 37 days, and the latency at its peak will take up 60% of the total population. Visiting patients will get to the maximum after 2 months with 15.8% day visit and the maximum number of daily hospitalized patients will be 8%, as no country or region has the capability to support such huge medical demands. Peak of death will come 100 days after the outbreak, with daily deaths of 2500 which would fall below 100 after 7 months.

Figure 7. Evolution of the epidemic on the low admission rate.

The low admission rate corresponds to $\delta = 0.063$ . Graphics show the severeness of the corresponding infection if COVID-19 is treated as a cold or flu and there is no appropriate response ( $\theta = 0,q = 0$ ). (a) ${2.8*10^6}{/11.081*10^6} = 0.25$ , $6.65/11.081 = 0.6$ , $1.75/11.081 = 0.158$ , ${8.53*10^5}{/11.081*10^6} = 0.08.$ (b) Proportion of infections will be over $10/11 = 0.909$ . Top 3 curves include the recovered population. So, it also verifies the high self-healing rate and cure rate of COVID-19.

Figure 7 also hints that group immunity would be all people’s infections. In theory, ${R_e} \lt 1$ means that if an infected individual can produce less than 1 new infected individual in his infectious period, then the infection cannot grow. As for group immunity, it seems there will not have new infections after 75% of people have been infected conceptually. Simulations tell us that the proportion needs to be 3 out of 4 and infection continues. The total deaths will be 220000 by curve fitting and integral calculation. It is the result of group immunity for a city with a total population of 11081000, such as Wuhan. In fact, there were no COVID-19 patients and the cumulative deaths were 3869 as of May 1 with the comprehensive and strong interventions in Wuhan. Figure 7 (b) also shows that COVID-19 patients have higher self-healing rate and cure rate, and the increasing admission rate can significantly reduce infections.

Of course, ${R_e} = \displaystyle{{{m\beta (1 - q){S_0}} \over {1 + dm}}\left({2 \over m} + {p \over {m({\gamma _1} + d)}}\right)}$ if all infected people with symptoms are admitted, corresponding to model (5+). It means the reproduction number ${R_e}$ will significantly be reduced. Also, ${R_e}$ will be further reduced if the mild infected person can wear a mask consciously or even self-isolate to avoid spreading the virus to others. Therefore, for the existence of an uncertain large proportion of the asymptomatic, and the easy spread of the new coronavirus SARS-CoV-2, it is very important to call on all people to take measures to avoid being infected or spreading COVID-19.

Parameter determination and estimation

Since SARS-CoV-2 virus is susceptible to all people (the youngest patient in Wuhan was just 17 days old), daily births need to be considered. At the end of 2018, the population of Wuhan was 11.081 million and the number of births was 119400. ³⁹ We refer to $S(0) = 11.081$ million and $A = 375$ . The average life expectancy is 81 years old in Wuhan, so $d = 1/81*365 \approx 0.0000338$ .

For the 14 days quarantine, the daily release rate of close contacts is $c = 1/14$ . A study showed that the infection ratio of close contacts to confirmed cases is 6.3%. ⁴⁰ Liang Wannian, the leader of the expert group for the treatment of COVID-19 in China, introduced that approximately 1 to 5 percent of close contacts are confirmed with SARS-CoV-2 virus infections. ²⁸ Then, we take the confirmation rate of close contacts as $b = 0.05$ .

For the above estimated incubation period $m = 7$ , the incidence of latency will be $\varepsilon = 1/7$ . The transmission rate of latency is $\mu = 2/m = 2/7$ . Because the average length of hospital stay for discharged patients is 20 days in Wuhan and 9 days in other provinces outside Hubei, ⁴¹ we suppose the curative rate of mild patients is ${\gamma _2} = 1/9$ and the curative rate of patients in designated hospitals is ${\gamma _3} = 1/20$ . As mentioned above, there are few people who carried the virus for more than 14 days, ³⁴ considering the incubation period, hence we can assume that the self-healing rate of the asymptomatic is ${\gamma _1} = 1/7$ .

The median time from onset of symptoms to death was 18.5 days and the average time for patients from onset to seeing doctors was 5 days, ^{Reference Zhou, Yu and Du42,43} so the mean hospitalized days of the deaths would be 14. Case fatality in Wuhan was about $4.05$ %, ⁴⁴ and the death rate of patients in designated hospitals is $\rho = 0.0405/14 = 0.00289$ . Let the ratio of close contacts to confirmed cases be 4.54 as introduced above.

Beginning with January 23, we take ${I_h}(0) = 440$ because the cumulative confirmed cases in Wuhan as of January 23 was 495 (excluding the 31 discharged and the 24 deaths). $R(0) = 31$ . Let ${S_q}(0) = 2776 \times 495/549 = 2503$ because there were 2776 close contacts under observation on January 23 and 549 cumulative confirmed cases in Hubei by January 23. We take the number of infected persons outside hospitals on January 23 as: ${N_1} = 22242$ as $E(0)$ , ${I_a}(0)$ , and ${I_m}(0)$ arose from ${N_0} = 17290$ on January 22, then ${I_a}(0) = p \cdot {N_0}/7 = 0.55 \times 17290/7 = 1358$ , ${I_m}(0) = (1 - p){N_0}/7 = 1112$ .

According to the estimated infection rate $\alpha $ , the coefficient of incidence rate may be $\beta = 0.4046/K = 3.65 \times {10^{ - 8}},K$ is the total population in Wuhan. On this basis, fitting model (5) to new confirmed cases in January, we determine that the appropriate coefficient of incidence rate would be $\beta = 3.33 \times {10^{ - 8}}$ at the beginning of the outbreak in Wuhan, and will reduce to $2.15 \times {10^{ - 8}}$ at the end of January with human interventions. The sound fitting effects can be observed from Figure 15 in Appendix A5 or Figure 8.

Sensitivity analysis

In this section, we will investigate the effects of some important control measures including Staying-at-home, Social-distancing, Handwashing, Mask-wearing, and Temporary hospitals corresponding to parameters $\theta ,q$ and $\delta $ . We can observe from Figure 8 and Figure 9 that the highest daily new confirmed cases will exceed 25000 and the expected removed susceptible population should be at least 50%, so an administrative ban is needed. It also can be detected from Figure 9 that the total number of infections will exceed 120000 (Estimated by integral) for the removal rate of the 60%, therefore Wuhan implemented the strict community management of Stay-at-home after February 11 to avoid more infections. Details are also presented in Figure 10. Figure 10 shows that the awareness-protective rate $q$ can reduce infections effectively, so the awareness protection measures should be encouraged. Publicity, social-distancing, and hand hygiene are required.

Figure 8. Control of COVID-19 in Wuhan.

Figure 9. Comparison of removal effect and actual data.

Figure 10. Importance of θ and q.

From Figure 11(a), we can see that the increasing admission rate $\delta $ accelerated the findings of cases so that the peak of new confirmed cases will come in advance with a larger value, but greatly shortened the control period and reduced the total infections. Figure 11(b) indicates that personal protection measures other than Stay-at-home can really reduce infections. It can be observed that a certain percentage of people staying at home (only 30%) with well-implemented protective behaviors can sufficiently control COVID-19. We can see new cases will not exceed 2000 with $\theta = 0.3$ and $q = 0.3$ . That is, it is necessary for the public to maintain physical distance, to wash hands frequently, and to wear masks.

Figure 11. The admission rate and the awareness effect on the disease control. (a) The increasing admission rate δ raises the peak and speeds up the arrival of the peak but reduces infections and accelerates control process. (b) Certain values of θ and q can effectively reduce the incidence of infections. 30% of people’s staying at home and 30% of the effect of protection measures can reduce the maximum number of daily new cases to 2000.

Figure 12(a) investigates the role of temporary hospitals. It is clear the recruitment of temporary hospitals greatly reduces infections. Figure 12(b) illustrates that calling for protective behaviors such as social distancing can continue to reduce the number of infections by nearly 50% in the premise of complete admission of patients. However, the control results that can be achieved by Stay-at-home are much more significant. A complete admission in combination with a removed rate of 20% is enough to control COVID-19.

Figure 12. The admission rate and the awareness effect on the disease control.

The most effective control strategy for COVID-19 is Stay-at-home, but at least 60% of removed rate should be required to achieve a control effect like that of Wuhan. The removed rate only needs to be 30% if social-distancing and personal hygiene can be maintained successfully. That is, if some non-required people need to stay at home. It means it is possible for us to minimize social disruption and economic loss while controlling COVID-19. To end COVID-19 and restart the economy as soon as possible, it is necessary for the admission of patients. However, admission alone cannot be very effective because of the extremely fast spread of the SARS-CoV-2 virus and the presence of asymptomatic infections.

Social effects and individual awareness caused by the epidemic information can play a certain role. Fitting based on the data in January hints that the awareness-protective rate $q$ may decrease 30% of infection rate. The removal rate of susceptible population should be at least 50% in pursuit of controlling COVID-19.

The removal rate of susceptible population should be at least 50 percent; it requires the guarantee of some administrative bans.

The strategy of Stay-at-home really guarantees the removal of most susceptible people and contributes to disease control. At least 60% of removed rate $\theta $ can effectively control COVID-19. $q$ expresses the awareness-protection rate, it can be aroused by behavior changes. $\theta $ in combination with $q$ may be more effective.

Model (5+) corresponds to the case of complete admission. (a) Infections can be greatly reduced with the introduction of temporary hospitals. (b) It is obvious that strategy of stay-at-home can control COVID-19 more effectively.