








key: cord-341187-jqesw4e8
authors: Yu, Xinhua
title: Modeling Return of the Epidemic: Impact of Population Structure, Asymptomatic Infection, Case Importation and Personal Contacts
date: 2020-08-27
journal: Travel Med Infect Dis
DOI: 10.1016/j.tmaid.2020.101858
sha: 
doc_id: 341187
cord_uid: jqesw4e8

BACKGROUND: Proactive interventions have halted the pandemic of coronavirus infected disease in some regions. However, without reaching herd immunity, the return of epidemic is possible. We investigate the impact of population structure, case importation, asymptomatic cases, and the number of contacts on a possible second wave of epidemic through mathematical modelling. METHODS: we built a modified Susceptible-exposed-Infectious-Removed (SEIR) model with parameters mirroring those of the COVID-19 pandemic and reported simulated characteristics of epidemics for incidence, hospitalizations and deaths under different scenarios. RESULTS: A larger percent of elderly people leads to higher number of hospitalizations, while a large percent of prior infection will effectively curb the epidemic. The number of imported cases and the speed of importation have small impact on the epidemic progression. However, a higher percent of asymptomatic cases slows the epidemic down and reduces the number of hospitalizations and deaths at the epidemic peak. Finally, reducing the number of contacts among young people alone has moderate effects on themselves, but little effects on the elderly population. However, reducing the number of contacts among elderly people alone can mitigate the epidemic significantly in both age groups, even though young people remain active within themselves. CONCLUSION: Reducing the number of contacts among high risk populations alone can mitigate the burden of epidemic in the whole society. Interventions targeting high risk groups may be more effective in containing or mitigating the epidemic.

: flowchart of the epidemic model

The above flowchart is applied to both young (age <65) and old populations (age>=65). We assume there is cross infection between younger and older age groups. The combined flowcharts can be modeled in differential equations (with index t suppressed and subscript y for young people, s for old people) as follows: Table1: Detailed parameter estimates and ranges. All rates are set as daily rates in a region with population size of 1,000,000.

Range and references f 1 Importing rate for susceptible population, proportional to the size of population 10 people per day, only for young people Arbitrary, net travel inflow of 2-20 people per day for a city of a million.

f y2 , f s2 Importing rate for exposed and infectious persons, proportional to the size of population Note:

1) The infecting rate per contact per day is based on the basic reproductive number and serial interval between case generations. Early reports suggested a R 0 of 2.2 (range 1.5-3), and a serial interval range of 3-9 days. However, several recent studies found the R 0 was greater than 3, one reported as high as 5.7. In this study, we took a conservative estimate of 2.6 and a moderate serial interval: 6 days for young people, 4 days for old people.

Based on simple SIR model, the reproductive number is k*b*t, where k is contact rate, b is infecting probability per contact, and t is serial interval. Assuming k=10 contacts per person per day for young people, and the infecting probability is b1= 0.043 for a symptomatic young people. For mild or asymptomatic cases, we assume 50% less infectious than symptomatic cases, thus b2 = 0.021 for young undiagnosed people. For older people, the parameters are set to higher than young people, with fewer contact and shorter serial interval. 2) The base rate per day is based on disease duration such as incubation period, diagnosis delay, hospitalization delay, hospitalization stay and recovery duration. We assumed an exponential distribution of duration, i.e., daily rate is 1-exp(-1/duration).

J o u r n a l P r e -p r o o f 7 We investigate the impact of population structure, case importation, asymptomatic cases, and the number of contacts on a possible second wave of epidemic through mathematical modelling.

Methods: we built a modified Susceptible-exposed-Infectious-Removed (SEIR) model with parameters mirroring those of the COVID-19 pandemic and reported simulated characteristics of epidemics for incidence, hospitalizations and deaths under different scenarios. have detected virus shedding in nasopharyngeal swap samples among asymptomatic cases [11] .

A few case reports have shown some cluster of cases initiated by asymptomatic cases [6, 8, 10] .

Researchers have postulated that asymptomatic and pre-symptomatic cases may play a significant role in sustaining the community transmission [7] .

Second, government leaders have been pressed to allow people to return to normal work and life to avoid economic recession. After social activities are restored, both international and domestic J o u r n a l P r e -p r o o f travel ban will be lifted. Social and work-related gatherings are restored. Imported symptomatic and asymptomatic cases may kindle a second wave of epidemic in the community [7] . the US, the mortality rate for age 50 or younger is below 1%, while the mortality rate increases to more than 10% among people aged 80 or above [15] . Finally, as demonstrated in the 2009 H1N1 flu pandemic [16] , a pandemic with lower hospitalization and mortality rates has less impact on the society than those with higher hospitalization and mortality rates, though it may still have heavier impact on the economy.

Epidemic model simulation has been used extensively to estimate essential epidemic parameters, In this study, we will build a modified Susceptible-Exposed-Infectious-Removed (SEIR) model [27] to simulate the COVID-19 pandemic and investigate the impact of population structure, asymptomatic cases, case importation, and the number of contacts on the epidemic progression.

We will explicitly evaluate the changes of hospitalizations and mortality under various scenarios for young and elderly people. Our analysis will provide theoretical evidence for possible strategies to prepare for a second wave of epidemic.

The COVID-19, like many other respiratory infectious diseases such as influenza, often has an incubation period during which the exposed persons cannot transmit the virus to others. After the incubation period, there is an infection period during which cases may or may not have symptoms but are able to infect other people. The infectivity may also vary at different time points of the infection period. As in the COVID-19 pandemic, the highest infectious points are 1-2 days around the symptom onset [28] . After the infection period, the patients are recovered or removed from the infectious pool. In addition, various controlling measures may be implemented J o u r n a l P r e -p r o o f during the epidemic, notably the case isolation, quarantine of high risk people through contact tracing, and also social distancing. All these measures will change the transmissibility of virus during a contact between an infectious person and a susceptible person. Therefore, the modified SEIR model as shown in Figure 1 is appropriate (also see the modeling framework section in supplemental documents for details). The SEIR model and its variants have been used in many previous studies for modeling the COVID-19 pandemic [23, 25] . Briefly, we divide the population into the susceptible population (S), self-quarantined susceptible people (Q), exposed We also assume a dynamic population in which the numbers of imported susceptible persons and assumptions that will be discussed later and also in the supplemental document.

To account for population heterogeneity, we also apply the basic framework (Figure 1 ) to both young (age < 65) and elderly (age >=65) populations. The two flowcharts are connected through cross-infection due to mutual contacts. The combined flowcharts can be translated into a set of ordinary differential equations (see supplemental document). The key equations relevant to the drive of pandemic and cross-infection between two age groups are for the change of exposed people at time t (subscript y for young, and s for elderly people, with time indicator t suppressed):

Specifically, the first equation models the exposure dynamics among young people. It includes imported exposed people (f y2 N y ), newly exposed people through contacting within the young people ( + ) and contacting between young susceptible and infected elderly people ( + ). Then some percent of exposed young people become symptomatic cases ( ), and some become asymptomatic cases ( ). A fixed percent of exposed people will die of other diseases ( ). The second equation for the exposure dynamics among elderly people can be interpreted similarly.

The model involves many parameters. Their definitions, default values, and ranges are listed in the supplemental document (supp. 

The default model is set on a region with 1 million residents, consisting of 20% elderly people and 20% of total population with past infection (or immunized). There is no existing symptomatic or asymptomatic case, and no person in self-quarantine in the region. We assume only one imported young exposed case every two days for 20 days (i.e., 10 imported cases).

Analyses are performed based on the ranges of parameter estimates. We vary one parameter at a The R package EpiModel is used for simulating the deterministic epidemic models [40] . The R codes for simulating the modified SEIR epidemic models are available (http://github.com/xinhuayu/returnepidemic/).

This study is deemed exempt from ethics approval as the research involves no human subjects and we use publicly available data. No informed consent is needed.

Under the default model setting, all epidemic measures reflect the model parameters satisfactorily (Table 2 , also refer to supplemental Table 1 ). That is, the resulting epidemic J o u r n a l P r e -p r o o f measures from the default model such as the disease incidence, epidemic peak, and duration of the epidemic are reasonable and mirror those reported in the literature. For example, starting with ten imported infectious persons and assuming 40% asymptomatic cases at the peak of epidemic, the epidemic reaches peak quickly within 73 days and lasts 172 days. It is ten days quicker among elderly people than among young people ( Table 2 ). The epidemic curves for incident cases (symptomatic and asymptomatic), hospitalizations and deaths by age groups are typical (Supplemental Figure 2 ). The modeling results in an overall hospitalization rate of 14.1%. The in-hospital mortality rate is 5.0% for young and 20.3% for elderly people, with an overall mortality rate of 1.9%, similar to those empirical measures in the COVID-19 pandemic in the early epidemic of the US. Therefore, the default model represents the current COVID-19

pandemic sufficiently well.

As summarized in Table 3 , the size of region and a small percent change of self-quarantined susceptible people do not change the epidemic progression significantly except for the total number of cases. A smaller percent of elderly slows down the epidemic, while a much higher percent of elderly does not change the epidemic curve significantly. As expected, when over 60% people have prior infections, the epidemic takes very long to reach the peak and results in substantial fewer cases. The effects are similar in both young and elderly people (supplemental 

Both the percent and infectivity of asymptomatic cases were investigated (Table 3 ). An increase of the percent of asymptomatic cases from 10% to 30% postpones the epidemic peak by 12 days J o u r n a l P r e -p r o o f due to less infectivity of asymptomatic cases, and results in significantly fewer hospitalizations and deaths. On the other hand, a higher infectivity of asymptomatic cases (e.g., 100% of symptomatic cases) results in a fast developing and narrow epidemic curve which reaches the peak within 60 days. There are more hospitalizations and deaths at the epidemic peak compared with the default model, both assumed 40% asymptomatic cases. In addition, a change of the percent of asymptomatic cases among elderly people leads to larger changes in hospitalizations and deaths than that of young people (Supplemental Table 2a & 2b) . For example, comparing 60%

with 40% asymptomatic cases, the total hospitalizations are reduced only by half among elderly people, while it is a two third decrease among young people. Furthermore, when the effects of the percent and infectivity of asymptomatic cases are combined, for example, in a low risk epidemic with 60% asymptomatic cases but with a lower (30%) infectivity, the epidemic reaches its peak slower for both young and elderly people with peak hospitalizations almost half of the default model (assuming 40% asymptomatic cases and 50% infectivity) (supplemental Figure 3) .

This epidemic model is initiated by imported infectious persons (may be asymptomatic or presymptomatic cases). The number of imported cases is in absolute sense, regardless of the size of population. A daily arrival of two infectious people speeds up the epidemic by 8 days compared with one case every two days in the default model ( Table 3 ). The magnitudes of epidemic are similar between different importation scenarios. In addition, a longer importing duration shifts the epidemic only slightly. Finally, if we assume all the imported cases are asymptomatic cases, the epidemic curves are not significantly different from that of default model (supplemental Figure 4) . respectively, all significantly lower than those of default model ( Table 2 ). The times to the epidemic peaks are also postponed in both curves. When both young and elderly people reduce contacts to 3 per day, such as under the stay-at-home rule, the epidemic curves on hospitalizations are significantly mitigated in both age groups (Figure 2d ).

Finally, we consider two extreme scenarios: 1) high risk scenario: assuming one imported case per day continuously throughout the epidemic, 30% asymptomatic cases at the epidemic peak, and the same infectivity between symptomatic and asymptomatic cases; 2) low risk scenario:

assuming one imported case every two days for twenty days, 60% asymptomatic cases, and asymptomatic cases have only 30% infectivity of symptomatic cases. In both scenarios, limiting contacts among elderly people alone still has significant impact on hospitalizations in both age J o u r n a l P r e -p r o o f groups, and a larger relative difference in the low risk scenario than high risk scenario ( Figure 3 and supplemental Figure 7 ).

We With growing availability of detection kits during the COVID-19 pandemic, more asymptomatic or mild symptomatic cases are identified. Ultimately, an optimal view is asymptomatic cases may account for 60% of infections. However, recent research and case reports have confirmed that asymptomatic or pre-symptomatic cases can shed enough quantify of virus to be infectious [6] [7] [8] [9] 11] . Furthermore, if the infectivity of asymptomatic cases is similar to that of symptomatic cases, a faster epidemic will occur. Despite more asymptomatic cases at the peak of epidemic, there are also significantly more hospitalizations and deaths, which may overwhelm the health care system. With a lower infectivity (30% infectivity) among asymptomatic cases, the epidemic reaches its peak later and results in half of hospitalizations at the peak compared with the default model (Supplemental Figure 3) . In addition, a closely related issue is case importation [25] .

Imported cases are often pre-symptomatic, asymptomatic or with mild symptoms. They seed of a second outbreak, even with just a few cases. Therefore, proactively identifying asymptomatic J o u r n a l P r e -p r o o f cases, isolating them and tracing their contacts thereafter will prevent the occurrence of an epidemic [22] .

Our study has some strengths. We have devised a modified SEIR model to incorporate both symptomatic and asymptomatic cases. We emphasize population heterogeneity such as age structure in the model. We include a self-quarantined group who will not infect other people if they are infected with the virus. Naturally, these settings can be extended to represent other high risk or special groups with revised parameters. In addition, we separate hospitalization and death from other removed compartments to explicitly estimate the impact of an epidemic on hospitalizations and deaths. From the health impact point of view, severe cases that lead to hospitalizations and deaths are more important than mild cases, as demonstrated in the 2009 H1N1 pandemic [16] . Furthermore, we explored a few key determinants of epidemic explicitly, leading to many insights on epidemic prevention strategies.

There are a few limitations in our study. As inherent in all modeling studies, simulation interpretations are heavily dependent on model assumptions and parameter estimations. Our epidemic model is a population model. Although we take account of population heterogeneity such as age in the current model, our age group is overly broad. A more detailed age grouping scheme, including children, young adults, middle age group, and elderly, may reflect the agespecific epidemic more realistically. Other factors may also be included, and additional compartments such as pre-symptomatic stage may be modeled. However, more sophisticated models require more assumptions and may not necessarily provide more insights about the epidemic process. Instead, in this study, in addition to ensuring the mathematical correctness of the models, we prioritize the epidemiological concepts and clinical relevance in setting up the models rather than model complexity. Nevertheless, our findings do not intend to provide J o u r n a l P r e -p r o o f definitive advice to design a new policy but rather gain insights of the epidemic process and provide theoretical support for a possibly more effective prevention strategy based on approaches targeting high risk populations.

In addition, we assume random mixing within and between age groups. As a population model, we cannot assess the impact of individual behaviors such as the way of reducing contacts, social distancing and travelling. Furthermore, it ignores clustering within the population such as senior group living, community gatherings (e.g., churches, community centers), worksites and schools.

These clusters are hotbeds for superspreading events which may lead to a sudden increase of new cases and overwhelm the healthcare system unexpectedly. Furthermore, the quarantine compartment in the model is not contact tracing based. Modeling contact tracing based quarantine is more relevant to public health interventions [42] . Therefore, the goal of our future research is to exploring the effect of these factors with stochastic simulations of individual behavior [24, 43] and network analysis [44] . Additionally, our model is set on a mid-size region with 1 million residents. We do not intend to model an pandemic, as all prevention strategies are ultimately local.

Our study only examines a small subset of scenarios during the epidemic. Multi-interventions and more stringent controlling measures are more effective in mitigating a pandemic but are likely less sustainable in the long run. After the initial epidemic ends, society will return to normal, and only one or two most effective interventions such as social distancing may be practiced, often partially. Thus, one parameter analysis under various scenarios is important for evaluating the probability of a second epidemic.

Finally, one critical issue in preparing for and preventing a second wave of pandemic is vaccinating the susceptible people, which is the most effective way to protect people, especially J o u r n a l P r e -p r o o f 

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The peak self-quarantined is 2% of the susceptible population but changes with the prevalence of disease. The hospitalization rate is 5% for young, and 30% for old symptomatic cases, respectively. The mortality rate is 5% for hospitalized young people and 20% for hospitalized elderly. There are no prior symptomatic or asymptomatic cases or quarantined people in the total population

The X. Yu is solely responsible for the conceptualization, data collection, analysis, report writing and reporting writing of this study.

