key: cord-1000017-rymqtww3
authors: Li, Ao; Wang, Yang; Cong, Pingping; Zou, Xingfu
title: Re-examination of the impact of some non-pharmaceutical interventions and media coverage on the COVID-19 outbreak in Wuhan()
date: 2021-07-17
journal: Infect Dis Model
DOI: 10.1016/j.idm.2021.07.001
sha: c2b21738b26e0333ebf6d2500993ffe169429160
doc_id: 1000017
cord_uid: rymqtww3

In this paper, based on the classic Kermack-McKendrick SIR model, we propose an ordinary differential equation model to re-examine the COVID-19 epidemics in Wuhan where this disease initially broke out. The focus is on the impact of all those major non-pharmaceutical interventions (NPIs) implemented by the local public healthy authorities and government during the epidemics. We use the data publicly available and the nonlinear least-squares solver lsqnonlin built in MATLAB to estimate the model parameters. Then we explore the impact of those NPIs, particularly the timings of these interventions, on the epidemics. The results can help people review the responses to the outbreak of the COVID-19 in Wuhan, while the proposed model also offers a framework for studying epidemics of COVID-19 and/or other similar diseases in other places, and accordingly helping people better prepare for possible future outbreaks of similar diseases.

impact on the global economy. 23 For this disastrous disease, efforts have been and are being made in locating the source 24 of this coronavirus, finding various treatments and developing effective vaccines. On the 25 other hand, re-examining why and how this virus has been spread to the globe is important 26 and worth doing, because such a re-examination will help human beings better prepare for 27 possible future outbreaks of other similar infectious diseases, particularly diseases caused 28 by new coronaviruses. For this purpose, it seems to be natural and reasonable to start 29 with a re-examination of controls and interventions in Wuhan, the city where SARS-CoV- to a food market called Huanan seafood wholesale market (HSWM), it was suspected 5 that the market might be the source of infection. Then on 1 January 2020, the market 6 was closed and disinfected to block the transmission. At that time, it was believed 7 that the pathogen was solely from that market and was the only source of infection, 8 and there was no human-to-human transmission. 9 (M2) Announcement of human-to-human transmission. Since the closure and hygienization 10 of HSWM, a continued and even faster increase in the number of reported COVID-19 11 cases seemed to suggest that human-to-human transmission of the coronavirus was also 12 possible. This possibility was soon confirmed and officially announced to the public 13 on 20 January 2020 through various media. This announcement has a crucial impact, 14 because before this, without realizing human-to-human transmission, many large scale 15 activities and gatherings had been going on as scheduled, and traditional pre-Chinese- 16 New-Year shoppings were as active as in all previous years, with participants and 17 shoppers having no or little personable protections. 18 (M3) Lockdown of the City. Further realizing the severity of the epidemic of COVID-19, three 19 days after announcing human-to-human transmission, the government announced lock-20 down of the whole city of Wuhan starting on 23 January 2020, and the lockdown was 21 in the most serious sense that all businesses were shut down, all public transportations 22 including buses, subways and ferry lines were suspended within Wuhan; additionally, 23 all airports and train stations were halted. Shortly after that, other cities in Hubei 24 province also announced strict restrictions similar to Wuhan. February 2020, it started admitting the first group of patients. Huoshenshan Hospital 30 was equipped with 1, 000 beds, 30 intensive care units and several quarantine wards.

A few days later, a second special emergency hospital, Leishenshan Hospital, was built 32 3 J o u r n a l P r e -p r o o f and opened on 8 February 2020. In addition, since 14 February 2020, 16 mobile cabin 1 hospitals, called Fangcang Hospitals in Chinese, were constructed which could provide 2 a total of more than 20, 000 beds. Together with the addition of these special hospitals, 3 tens of thousands of front medical workers (doctors and nurses), recruited from other 4 provinces of China, arrived in Wuhan group by group, and worked in all hospitals 24 5 hours a day and 7 days a week. These added new facilities and health workers greatly 6 improved the capability of rapid diagnosis, efficient isolation and timely treatments. and active cases reported in China maintain at a very low level since then until today.

In this paper, we focus on the coronavirus outbreak in Wuhan, where the first case ing those measures. We then use the data available to us for the outbreak in Wuhan to 19 validate some basic model parameters. Finally, we will vary some non-biological parameters 20 that represent various aspects of certain decisions and control measures by the governments 21 and public health authorities, particularly the values of those critical time points (dates) 22 when those related important measures were taken/implemented, to see if there would have 23 been significant changes in epidemics. This would help us to review Wuhan's responses to 24 the COVID-19 outbreak and seek improvement for future when facing similar situations of 25 disease outbreaks. 26 We point out that mathematical modelling can play a key role in providing evidence-27 based information for health policy decision makers and it has been recognized by the World In this paper, we focus on the coronavirus outbreak in Wuhan, where the first case The rest of the paper is organized as follows. In Section 2, we give detailed descriptions 10 on the model; we also activate the model estimating the model parameters: some are directly 11 obtained from published research works while the others are obtained using the data available 12 and the nonlinear least-squares solver lsqnonlin that is built in MATLAB. In Section 3, with 13 the activated model, we explore the impact of the timings for the four major control measures.

14 In Section 4, we summarize main results of the paper and their implications; we also discuss 15 some limitations of our model and possible improvements. 

Here the population is divided into three classes with S(t), I(t) and R(t) denoting the 20 subpopulations of these classes: susceptible, infectious and removed classes. The parameter 21 β is the transmission rate. An SIR type model assumes that recovered individuals will carry 22 immunity for the disease. Moreover, the removed class includes those died of the disease and 23 those recovered from the disease. Here in (1), for the convenience of data fitting later, we 24 distinguish these two sub-classes by incorporating two different parameters α and γ, with 25 the former denoting the recovery rate and the latter being the disease caused death rate. In

(1), the demography of the host population is omitted/ignored. This is reasonable for those to be contagious, we need to add a compartment for the population during the incubation 5 period, denoting it by E(t). We point out that E(t) here is not the population in the latent 6 period like in many other SEIR type model, and when it comes to modelling, the suscepti- 

(2)

Here the variables S, E, I, R and V have been explained above, and all the parameters are 15 non-negative with their biological meanings explained below:

(i) p 1 is the fraction of susceptible population that visit Huanan seafood wholesale market; 17 p 2 and p 3 are the fractions of the susceptible population that have a chance to be 18 exposed to I class and E class individuals respectively (see more discussion later);

(ii) β 10 , β 20 and β 30 are V (t)-to-S(t), I(t)-to-S(t) and E(t)-to-S(t) transmission rates, 20 respectively;

(iii) σ is the rate at which exposed individuals develop symptoms (incubation period ends);

(iv) α is the recovery rate; 23 (v) γ is disease-induced death rate;

Huanan seafood wholesale market; 1 (vii) d is the natural mortality rate of SARS-CoV-2 when exposed in the environment. 

Next, by the meanings of p 2 and p 3 , it is seen that p 2 S and p 3 S account for the numbers various platforms of public health organizations and media. To reflect these impacts, we 22 propose the following forms for the two fractions p 2 and p 3 :

Here, we have assumed that not only the number of infectious individuals I(t) but also 24 its increment ∆I(t) = I(t) − I(t − 1) will affect people's behaviours and activities, as the 25 increment suggests a trend of the epidemics in certain sense. The coefficients m 1 (t) and m 2 (t) point out that max{0, ∆I(t)} is a technical modification of ∆I(t) to avoid negativity and 1 will be denoted by∆I(t) = max{0, ∆I(t)} subsequently. Also, the lockdown described in 2 (M3) means that more people were forced to stay home (due to restriction rules, as well as 3 lack of public transportation and closure of commercial stores and restaurants, etc.), and 4 this would surely reduce the fractions of the I-available and E-available susceptible classes. 5 Thus, we incorporate the quantity m 3 (t) to reflect such an effect. 6 Corresponding to (M4), the main effects of the addition of those extra medical and 7 health resources are twofold. Firstly, more hospital rooms and more doctors and nurses 8 meant broader and more intensive treatments for diagnosed infected patients leading to a 9 higher recovery rate and a lower disease-induced death rate. We reflect this by assuming 10 that parameters α and γ are step functions, with α = α(t) jumping from a lower level 11 to a higher one while γ = γ(t) switching from a higher level to a lower one, after such 12 resources being available. Such an assumption is supported by the related data available J o u r n a l P r e -p r o o f a low level, while the recovery rate is very low at first and then it keeps increasing. This is because two special emergency hospitals and many mobile cabin hospitals became available 1 and many medical personnel from all other provinces were later recruited to Wuhan to help, 2 as mentioned in (M4). To make it simple, we set 16 February 2020 as a turning point and 3 use average values to represent the rates for each period. 4 Figure 1 : The daily disease-induced death rate and recovery rate. Now we rewrite (2) as the following more standard form

where β i (t) = β i0 p i (t), i = 1, 2, 3. to 16 February 2020 in Figure 1 , we obtain the recovery rate α(t) and the disease-induced The basic reproduction number R 0 represents the average number of people that one 8 person with SARS-CoV-2 is likely to infect in an otherwise susceptible population. In our 9 model, it is straightforward that

where S 0 = N is the total population in Wuhan. The basic reproduction number is of great 11 importance in measuring the risk of transmission, and its estimation ranges from 2.2 to 5.7 Table 2 . 3 Exploration of non-pharmaceutical interventions 13 We have seen in the preceding section that the disease dynamics described by the model 14 (5) with the parameters given in Tables 1 and 2 and m 2 (t) switching on at t 2 , media coverage starts affecting human's behaviours, making 12 less susceptible people available for infection. We can also explore the impact of altering 13 timing t 2 , which is illustrated in Figures 5 and 6 by the numerical results for the model. We may also look at the consequence of altering the pair of two time points t 2 and t 3 . For 2 simplicity, we let t 2 and t 3 vary synchronously in the sense of t 3 = t 2 +3, which approximately 3 accounts for a scenario that three days are needed to prepare for a lockdown after realizing 4 the severity of the disease due to massive human-to-human transmission. Figure 9 presents 5 the corresponding disease dynamics with respect to the time variable, while Figure 10 shows The market was not closed: p 1 (t) = p 10 for t t 0 implementations of some major NPIs. 22 We remark that as far as parameter estimations for ODE and PDE models for infectious 23 review and discussion in this paper. 30 We also note that in (4), the first part on the right hand side of the formulas for p 2 and p 3 ,

with q(t) = m 1 (t)I(t)+m 2 (t)∆I(t) accounting for a kind of "weighted information 32 types of decreasing functions can also be used. One example is the exponential decay function We have included the concentration of SARS-CoV-2 in the environment into the model. 10 Since most early cases were believed to be infected by the viruses discharging from wildlife 11 traded in Huanan seafood wholesale market, we have assumed that no more people catch 12 COVID-19 by exposure to coronaviruses in the environment after closure of the market.

However, this is a bit too simple and ideal, as some studies have shown that the novel coro- 

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