Public Health Policy: COVID-19 Epidemic and SEIR Model with Asymptomatic Viral Carriers

Since the COVID-19 appeared in the late December, 2019 at Wuhan, China, it already infected more than 1,900,000 people in more than 185 countries. In Italy, there was two confirmed cases at January 31st. At same day, the government suspended all flights from China and announced a national emergency. However, these actions did not stop the spread of the virus.

Contrast with other Coronavirus, i.e. SARS and MERS, one essential different feature of COVID-19 is the asymptomatic infection. At the beginning of the spread of this virus, there were already several experts point out this. Many researchers al2020asymptomatic ; bai2020presumed ; chan2020familial ; hu2020clinical ; lai2020asymptomatic ; tang2020detection ; wang2020clinical prove the existence of asymptomatic COVID-19 carriers by comparing the nucleic acid testing of patients’ pharyngeal swab samples and chest CT scans with their typical symptoms and . Li et al. li2020early mention that more young carriers show mild symptoms than older patients. Compared with retired people, young adult have more social activities in general. Include the asymptomatic carriers into an epidemic model is crucial. About the proportion of asymptomatic patients among all viral carrier, many experts wu2020characteristics ; mizumoto2020estimating provide their own estimation based on distinct data source.

For classical contagious disease, many models can predict the number of infections, recoveries and deaths. One important type is the compartmental model. Kermack and McKendrick kermack1991contributions construct a deterministic model, called as SIR model, to quantify the susceptible population, infections and recoveries of one epidemic. Then they also updated this model in kermack1932contributions ; kermack1933contributions . Bjørnstad et al. bjornstad2002dynamics apply this model to analyze the spread of Measles. McCluskey mccluskey2010complete introduce the distributed delay and discrete delay into the SIR model. Some people update the model to a stochastic area ji2014threshold ; jiang2011asymptotic ; zhang2013stochastic . To include exposed people with latency period, Dietz dietz1975transmission construct SEIR model. Based on SEIR model, many experts update this model anderson1979population ; may1979population Many researchers use clinical and simulation data to calibrate the parameters in the epidemic model anderson1979population . Various experts apply this type of model to analyze different diseases and develop it dukic2012tracking ; lekone2006statistical ; ozalp2011fractional . Ndii and Supriatnandii2017stochastic claim that a deterministic model is more appropriate than stochastic models for an large initial susceptible population. More details of compartmental models in bailey1975mathematical ; hethcote2000mathematics .

Several containment are proposed by experts. Fraser et al. fraser2004factors recommend to isolate symptomatic patients and their contacts to control an outbreak. ferguson2006strategies apply a large scale simulation to analyze the effect of publich health policy to the spread of one virus at the beginning period. Wu et al. wu2006reducing provide recommendation of household-based public health interventions. Allred et al. allred2020regional include the effect of asymptomatic patients to the demand of the health care. Bousema et al. bousema2014asymptomatic discuss the detect tools of the asymptomatic Malaria infections with related public health. However, few paper focus on public health policy to inhibit an epidemic of aerial contagious virus with asymptomatic patients based on epidemic models.

The structure of this paper is as follow. Section 2 lists the symbols we need to construct our SEIRA model and the assumptions of to include the special characteristics of COVID-19. In Section 3, we construct the SEIRA model and several related models to include the effects of different public health policy. In Section 4, we analyze the effects of asymptomatic rate, three types of public health containment (Confirmed Cases Isolation, Personal Protections, Quarantine), and delayed policy to the spread of COVID-19. Section 5, we provide the suggestions of public health agency for a new aerial transmitted virus.

In this paper, we have the following assumptions about COVID-19.

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  No vaccine exist for COVID-19

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  People who recovered after catch COVID-19 will be insusceptible of it, all other people are susceptible.

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  The number of infectious people only include the cases with typical symptoms and get confirmation from laboratory tests

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  The natural birth rate and natural death rate are same (the population is a constant)

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  The infectious ability of symptomatic and asymptomatic patients are same.

Before we introduce our time series SEIRA model, we list necessary symbols at first.

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  $S$: Susceptible population

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  $E$: Population who are exposed to the virus and do get confirmed test

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  $I$: Population who get laboratory positive confirmation

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  $R$: Recovery (insusceptible to COVID-19)

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  $D$: Death number

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  $\alpha^{I}$: rate of people exposed to the infectious patients.

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  $\alpha^{E}$: rate of people exposed to the asymptomatic patients.

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  $\beta$: rate of exposed people catch COVID-19

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  $\beta^{I}$: rate of exposed people get COVID-19 with diagnosed confirmation

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  $\beta^{E}$: rate of exposed people without symptoms but can transmit it to others

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  $\gamma$: recover rate of infected people

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  $\mu^{I}$: death rate of symptomatic patients

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  $\mu^{E}$: death rate of asymptomatic patients

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  $e^{Q}$: Efficiency of quarantine (lock-down) policy

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  $e^{F}$: Efficiency of personal protection

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  $e^{L}$: Efficiency of Isolation for confirmed cases

Because we want to check the inhibitory effects to the spread of COVID-19 for different efficiencies of public health policy, all rate $e^{L}$, $e^{F}$, and $e^{Q}$ would run between $0\%$ to $95\%$ In our simulation, we cite the data from John Hopkins Coronavirus Resource Center (JHU CSSE) jhdata as source. Based on the data at 4:00 pm on April 5^th, 2020, the case fatality rate (CFR) for countries with Top 6 most confirmed cases are $12.32\%$ (Italy), $9.5\%$ (Spain), $8.34\%$ (France), $4.04\%$ (China) $2.85\%$ (U.S.), $1.54\%$ (Germany). In our model, we assign the constant death rate $\mu^{I}$ equal the average $6.43\%$ and $\mu^{E}$ equal 0, which means all people died of COVID-19 should be laboratory-confirmed positive.

Del Valle et al. del2007mixing conclude that the contact numbers for population are distinct from different age group, arriving peak (about 21 average number of contacts per person daily) between 20 and 50 years old. In this paper, we take a rough mean as 15. So we assume the exposure rate $\alpha^{I}=\alpha^{E}=15/S_{0}$, where $S_{0}$ is the initial susceptible population.

The reproduction number ($R_{0}$) is the average number of people catch this disease, caused by one infectious people. Liu et al. liu2020reproductive state the mean reproduction number ($R_{0}$) of COVID-19 is around 3.28, with a median of 2.79 and a interquartile range (IQR) of 1.16. Zhang et al. zhang2020estimation claim the $R_{0}$ is 2.28 with a $95\%$ confident interval as $(2.06,2.52)$. In this paper, we use the average $R_{0}=2.54$. Follow the findings from Li et al. li2020early , Lauer et al. lauer2020incubation and Anderson et al anderson2020will . In this model, we denote the incubation period as $d$ days, with assuming that the average incubation period of COVID-19 is 5.5 days. With the value of $R_{0}$ and relationship among $R_{0}$, $\beta_{I}$ and the average incubation period by colizza2007modeling ; diekmann2000mathematical , the estimation of $\beta^{I}$ is

In previous sections, we already construct an epidemic SEIRA model and use the simulation analyze the effects of confirmed case isolation, personal protection and quarantine. In this section, we want to give a guideline about the non-pharmaceutical public health interventions before and during a pandemic.

In this paper, we assume that no imported viral source to the system. In general, a ban for other countries only delay the peak, instead of prevent it. Ferguson et al. ferguson2006strategies find a 99.9% ban can postpone 6 weeks for United States. Therefore, we do not include the travel restrictions into our public health policy. We want to discuss the preparation for a government during this period.

First of all, at the start of one pandemic, the government must pay attention to the asymptomatic rate. Compared the cumulative confirmed cases and deaths with low and high asymptomatic rate in Section 4.1, the proportion of asymptomatic viral carriers (with mild symptomatic patients) among all infections is an influential parameter of any viral transmission, also supported by Fraser Fraser et al. fraser2004factors . When there is limited local casts, the researchers must pay attention to foreign researchers result about asymptomatic rate. If the pandemic start at our country, health workers need to estimate the rate as soon as possible.

Next, more tests are better. In general, at the beginning of one transmission, only symptomatic patients get tests. However, doctors cannot find asymptomatic patients in this situation, until one asymptomatic patient show typical symptoms. This test rule lead to overlook of asymptomatic infections. In fact, we should tests all contacts of every confirmed cases. Based on the results of these test, we can estimate the reproduction number $R_{0}$ and the asymptomatic rate $r$. Moreover, doctors should have the right to test potential infections with typical symptoms even though they do not have direct contacts with confirmed cases or travel history. As we all known, a dangerous sign of pandemic is community transmission without clear source, contradict to the previous rule. National Public Health Agency should permit doctors to do test for more patients without direct contacts or travel history.

For laboratory confirmed cases, the compulsory self-isolation must apply to all direct contacts compulsory immediately. In Section 4.1, we also analyzed the spread of one virus if we miss one diagnosed case initially. In Section 4.2.1, we discussed the sensitivity of the effects for the policy to the epidemic. Besides test all direct contacts, the isolation of the household immediately is also necessary.

Facing a new virus, the government should recommend general public take personal protection as soon as possible. When no information about an asymptomatic rate $r$, experts should assume there is asymptomatic patients. Compared the death and cumulative confirmed cases in Section 4.1, it is really dangerous to ignore this viral transmission. Because isolation of confirmed cases is hard to control it, a general personal protection is one of the best method. In addition, this method is also better than lock down policy. Although these two methods share similar inhibit effects, personal protection lead to fewer disruptions to the society and economic. Public health agency need to involve general public into the fight with epidemic as soon as possible, in order to avoid having to implement quarantine province-wide or nationwide.

For general personal protections, there are several types, such as social distancing, washing hands, wearing mask and others. In these three methods, the first two are much easier than the third one. Washing hand is the most useful and economical method to protect people. However, when the virus is transmitted by droplets or aerosols, only depends on washing hands could lead to dreadful result. In this case, social distancing is essential. Unfortunately, the social distance require critical condition about the location. In many place, such as the vehicle of most public transportation, school, and many workplace, no enough space exists to keep social distance. In this situation, general mask-wearing is a good substitution. Tellier tellier2006review recommend use N95 respirators as part of personal protective equipment for aerosol transmission. Considering the storage amount and price of N95, it should be saved to people work in high risk area. Li et al. li2006vivo the efficiency of surgical masks is 95%, compared with 97% for N95. As a result, an optimal choice for general public is surgical mask during an outbreak. Because of the paucity of research about homemade face masks, we cannot give any conclusion about homemade face mask in this paper. Before cumulative cases and death increase sharply, the government need take action to produce or purchase enough surgical and N95 face masks for both health workers and general population, and educate people about the correct method to wear a face mask and protect themselves on the same time. During a aerial transmitted disease, a combination of all three types of personal protections is indispensable.

Last but not least, government need lock down an outbreak zone determinedly. Although any lock-down policy brings horrible outcomes in economic and society, it is the final policy to save life. In section 4.3, we already compare isolation, personal pretection and quarantine. An effective personal protection should show similar consequence with a quarantine. But when a disease is out of control, it means the rule of isolation and general personal action are too late or too liberal. In this situation, the government has to execute more rigorous policy to inhibit the transmission, i.e. closing school, cancellation of public gatherings, working from home, limited mass transportation, and national quarantine.

In this paper, we construct two epidemic models. Firstly, the SEIRA model can be used to predict epidemic with asymptomatic viral carriers. Then to analyze the effect of different public health intervention, we use a SEIRA with policy model. By simulation, we find that asymptomatic rate $r$ is crucial parameters for the health policy. When $r\geq 10\%$, government need to include general personal protection into their public health policy, especially wearing-mask for aerial transmitted virus. When there is a new virus with transmission by droplets or aerosols, the government must

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  Test all typical symptomatic suspects and direct contacts of confirmed cases

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  Estimate the proportion of asymptomatic viral carriers

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  Prepare sufficient N95 and relative equipment for health worker

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  Prepare enough surgical face masks for general public

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  Execute compulsory isolation of confirmed case and all contacts

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  Educate people with correct personal protection methods at the beginning

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  Enforce quarantine when an epidemic is unavoidable