key: cord-0576167-2vptkcas
authors: Rabajante, Jomar F.
title: Insights from early mathematical models of 2019-nCoV acute respiratory disease (COVID-19) dynamics
date: 2020-02-13
journal: nan
DOI: nan
sha: 73c10497ee3598dd2688be6988354c7ac21b0574
doc_id: 576167
cord_uid: 2vptkcas

In December 2019, a novel coronavirus (SARS-CoV-2) has been identified to cause acute respiratory disease in humans. An outbreak of this disease has been reported in mainland China with the city of Wuhan as the recognized epicenter. The disease has also been exported to other countries, including the Philippines, but the level of spread is still under control (as of 08 February 2020). To describe and predict the dynamics of the disease, several preliminary mathematical models are formulated by various international study groups. Here, the insights that can be drawn from these models are discussed, especially as inputs for designing strategies to control the epidemics. Proposed model-based strategies on how to prevent the spread of the disease in local setting, such as during large social gatherings, are also presented. The model shows that the exposure time is a significant factor in spreading the disease. With a basic reproduction number equal to 2, and 14-day infectious period, an infected person staying more than 9 hours in the event could infect other people. Assuming the exposure time is 18 hours, the model recommends that attendees of the social gathering should have a protection with more than 70 percent effectiveness.

A new coronavirus has been identified to cause respiratory illness, such as an atypical pneumonia, in humans (European Centre for Disease Prevention and Control 2020) . This disease, with the interim name '2019-nCoV acute respiratory disease (ARD)' [official name: , is first detected in the winter month of December 2019 in a city of 11 million people -Wuhan in Hubei Province, China (World Health Organization 2020a, Tweeten et al. 2020) . The 2019-nCoV ARD is believed to be zoonotic in origin, from bats to intermediate host to humans (Zhou P. et al. 2020) ; and its initiation is geographically associated, but with uncertainty, with the Huanan Seafood Market in Wuhan (Cohen 2020a) . Human-to-human transmission of 2019-nCoV has been established, such as through respiratory droplets (Chan et al. 2020) , and there is also a suspicion of asymptomatic infection (Chan et al. 2020 , Kupferschmidt 2020 . To contain the epidemics, the government of China has ordered cancellation of huge events for the Chinese New Year celebration, and the lockdown of Wuhan and other cities (Tweeten et al. 2020) .

symptoms of the secondary case infected by the primary. The results of the study are (Zhao et al. 2020a ):

a. The estimated mean " ranges from 2.24 to 5.71 based on the reporting rate of cases.

b. If there is no change in the reporting rate (as of 10-24 January 2020), the estimated " is 5.71, which is estimated to be between MERS-CoV-like (5.31) and SARS-like (6.11) respiratory syndrome.

c. If the reporting rate will increase 2-fold, the estimated " is 3.58, which is estimated to be between MERS-CoV-like (3.38) and SARS-like (3.77) respiratory syndrome.

d. If the reporting rate will increase 8-fold, the estimated " is 2.24, which is estimated to be between MERS-CoV-like (2.16) and SARS-like (2.32) respiratory syndrome.

A comparison of the pathogenicity and transmissibility of 2019-nCoV with other viruses can be found in the paper by Chen (2020) . Based on clinical studies, 2019-nCoV is less pathogenic than MERS-CoV and SARS, but there is still debate on the transmissibility of 2019-nCoV with estimated " ranging 1.4-5.5 (SARS is 2-5 and MERS-CoV is <1) (Chen 2020) . The estimated case fatality rate for 2019-nCoV is 3 percent (10% for SARS and 40% for MERS-CoV) (Chen 2020) . Case fatality rate is the number of deaths due to the disease divided by the total number of cases having the disease. It should be noted that case fatality rate is not equal to an infected's chance of death due to the disease (although many use this as approximation). Probability of death or survival is person or situation-dependent. Also, a case fatality rate computed at global or national scale could be different from the case fatality rate computed at local scale. If 2019-nCoV ARD outbreak happens in a community with inferior health system, then case fatality rate in that community may be higher. The case fatality rate currently being computed is still an initial estimate since the epidemics is still ongoing, the number of deaths and total number of cases could still change. Shen et al. (2020) predict epidemic of 2019-nCoV in China totalling to 8,042 infecteds with case fatality rate of 11.02 percent. Their prediction used data prior to 25 January 2020, and is not anymore accurate as of 8 February 2020. As of 08 February 2020, total confirmed cases in mainland China is 34,611, total deaths is 723 and total recovered is 2,370 (Johns Hopkins CSSE 2020). Shen et al. (2020) also estimated a basic reproduction number " = 4.71 at the start of the epidemic on 12 December 2019 and the effective reproduction number + decreasing to 2.08 as of 22 January 2020. + is the average number of new cases caused by an infected when only a fraction of the population is susceptible. Shen et al. (2020) predict that if effective intervention continues, the epidemics is expected to peak in March 2020. They suggested that every one day reduction in the average duration from disease onset to isolation of infecteds will reduce the peak population size by 72-84 percent, and both the cumulative infected cases and deaths by 68-80 percent. Also, every 10 percent reduction in transmission rate could reduce the peak population size by 20-47 percent, and both the cumulative January 2020 which is way larger than the number of reported cases, and the epidemic doubling time is 6.4 days. They also predict that, if transmissibility in cities in mainland China is similar to Wuhan, then localized outbreaks are being sustained and the epidemics is already growing exponentially with a lag time of 1-2 weeks behind the Wuhan outbreak. Wu et al. (2020) predict that large cities with close transportation links with China may become another epicenter of the epidemics unless effective control strategies are placed. Without reduction in mobility and transmissibility, daily 2019-nCoV ARD incidence will peak in the last week of March and early week of April 2020 with around 35 cases per 1,000 population. Reducing transmissibility by 25 percent will reduce the daily incidence to around 20 cases per 1,000 population and delay the peak of the outbreak to early May 2020. Julien Riou and Christian L. Althaus of the Institute of Social and Preventive Medicine, University of Bern predict that " is around 2.2 (90% high density interval 1.4-3.8) (Riou and Althaus 2020) . Riou and Althaus (2020) also estimated the median dispersion parameter k = 0.54 that measure the Early models of 2019-nCoV ARD dynamics chance of super-spreading (k is a parameter of the negative-binomial offspring distribution where value near zero denotes overdispersion). They indicated that transmission characteristics of 2019-nCoV is similar to SARS and to the 1918 pandemic influenza. They suggested heightened screening, surveillance and control at airports and other travel hubs to prevent further global outbreak. Read et al. (2020) Assumption ii. delay from symptom onset to isolation follows an Erlang distribution (rate=2) with mean of 2.9 days;

Early models of 2019-nCoV ARD dynamics Assumption iii. delay from symptom onset to reporting follows an exponential distribution with mean 6.1 days;

Assumption iv. proportion of reported cases = 16 percent;

Assumption v. probability of disease being exported from Wuhan to a certain location depends on the number of cases in Wuhan, the air travel connections and volume between the two places (3,300 passengers per day) before the lockdown on 23 January 2020, and the probability of reporting an (sensitivity analysis is also done by changing to 10 initial cases); Assumption ix. super-spreading is modeled using a branching process with negativebinomial offspring distribution. The probability that an outbreak will occur after n introductions is computed as

where + is the probability that a single case will initiate an outbreak; Assumption x. latent period is equal to the incubation period (which means asymptomatics are non-infectious); and Early models of 2019-nCoV ARD dynamics Assumption xi. the model has two sets of compartments: one for the population in Wuhan, and one for international travelers. The compartment for the population in Wuhan has SEIR sub-compartments; while EIR subcompartments for the international travelers. Early models of 2019-nCoV ARD dynamics Other models have also considered importation of 2019-nCoV ARD from Wuhan to other countries using transportation data (Gardner 2020 , Chinazzi et al. 2020 . Chinazzi et al. (2020) ranked cities based on associated risk of disease importation, with Shanghai (risk value = 5.7%) and Beijing (risk value = 5.1%) as the top two in mainland China; and Hong Kong (risk value = 6.6%), Bangkok (risk value = 1.4%), Seoul (risk value = 0.6%), Taipei (risk value = 0.6%) and Tokyo (risk value = 0.5%) as the top 5 cities outside mainland China. Assuming that 20 million passengers is the catchment size of Wuhan International Airport, the estimated median outbreak size is 31,200 cases (95% credible interval 23,400-40,400). Nishiura et al. (2020) It is important to know the proportion of reported cases vs total number of cases (Nishiura et al. 2020 , Zhao et al. 2020b . Nishiura et al. (2020) estimated the ascertainment rate of infection in Wuhan at 9.2 percent (95% confidence interval 5.0-20.0). This rate informs us that 90 percent of the cases is potentially undiagnosed or unreported. The infection fatality risk is estimated at 0.3-0.6 percent, comparable to 1957-1958 Asian influenza pandemic (Nishiura et al. 2020 , Jung et al. 2020 ).

In the model by Nishiura et al. (2020) , they used the data from the 565 Japanese passengers evacuated from Wuhan who were screened for symptoms using portable thermo-scanners. There were 63 who showed symptoms. The passengers were also tested for the presence of 2019-nCoV using reverse transcription polymerase chain reaction (RT-PCR). Eight were positive with five of Early models of 2019-nCoV ARD dynamics them are asymptomatic. Assuming the population size in Wuhan is 11 million, the balance equation for the risk of infection used by Nishiura et al. (2020) is

where ( ) is the cumulative number of reported cases in Wuhan as of time (as of 29 January 2020, there were 1,905 confirmed cases), is the window of virus detection based on the serial interval (e.g., = 7.5 days), and is the ascertainment rate. The cumulative incidence can be b. If public health interventions are implemented with 70 percent efficacy, the forecast can be reduced (26,498 total cases as of 31 January 2020). The implication of this is that continued large-scale anti-transmission controls should be executed across all members of the population (e.g., closure of schools and suspension of public transport).

Early models of 2019-nCoV ARD dynamics An outbreak may disrupt the activities of a community. From the models discussed above, it can be observed how mathematical modelers can help in crafting decisions or policies to prevent or control infectious diseases. In the next model to be presented, the author formulated a model that can be used to recommend auxiliary strategies to prevent the spread of 2019-nCoV ARD in large social gatherings, especially during the early period of a possible outbreak. The following model can be used to inform the public on possible scenarios that may happen during social events, and the quantitative insights drawn from the model results can be translated to actual strategies. G. A model for preventing disease spread in large social gatherings Figure 1 . Susceptible-Exposed-Infected model framework. An infected may shed virus during the social gathering and a susceptible may be exposed. An exposed individual can be protected from being infected by having a protection (e.g., by washing hands or using face mask)

In this model, a simple Susceptible-Exposed-Infected (SEI) framework ( Figure 1 ) is used to propose measures to prevent epidemics during large events, e.g., during parties or concerts with huge crowds. The aim of the model is prevention so the Recovered compartment in SEIR framework is not considered, hence the simulation runs do not predict the whole disease outbreak in a community. The focus of the model is on disease transmission during an event that is short in duration (max 24 hours) in a population that is formed randomly (well-mixed).

Let us suppose is the level of protection of exposed persons. For simplicity, the net arrivaldeparture rate of attendees is denoted by integer . The in the force of infection is derived based on the known " (e.g., " = 2) and population size of the susceptibles ( " ) as follows

The is a timescale and tuning factor to adjust the parameters to the unit of time used (i.e., 'hours').

The assumed default value for is 14 × 24 = 336 , assuming " is computed for a 14-day infectious period with 1 day = 24 hours. If the assumed infectious period associated with " is decreased, the disease transmission rate in the model will increase ( where = + + (0) + 1 (the constant 1 assures presence of at least one person in the venue of the event) and is less than or equal the maximum capacity of the event venue. The initial condition for is assumed to be one infected. In the Philippines, there are few confirmed cases but more than 200 persons under investigation as of 07 February 2020 (Modesto 2020) . The probability of having this one initial case could be low in a country where the disease is possibly contained; but to account for the uncertainty, it is assumed that there is one possible case.

Insights that can be drawn from this model are as follow:

Insight A. The exposure time (number of hours an infectious person is staying in the event venue)

is a significant factor in spreading the disease (Figure 2 ). Under the assumptions used, an infected person staying more than 9 hours could infect other people.

Early models of 2019-nCoV ARD dynamics Insight B. The level of risk asymptotically increases when population size of the susceptible increases but the exposure time dictates if disease transmission will occur (Figure 3 and 4).

Insight C. Assuming the exposure time is 18 hours, an exposed person could be protected from being infected if the level of protection is >70 percent ( Figure 5 ). Exposure time = 18 hours 95% protection

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