Covid-19: Simulation models for epidemics

Ivar Sønbø Kristiansen; Emily Annika Burger; Birgitte Freiesleben de Blasio

doi:10.4045/tidsskr.20.0225

Perspectives

Covid-19: Simulation models for epidemics

Norwegian

Ivar Sønbø Kristiansen, Emily Annika Burger, Birgitte Freiesleben de Blasio

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Ivar Sønbø Kristiansen

E-mail: i.s.kristiansen@medisin.uio.no

Ivar Sønbø Kristiansen, professor emeritus at the University of Oslo and adjunct professor at the Department of Public Health, University of Southern Denmark, Odense, with research interests in economic evaluation and modelling.

The author has completed the ICMJE form and declares no conflicts of interest.

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Emily Annika Burger

Emily Annika Burger, researcher specialising in the modelling of screening and vaccination programmes for human papillomavirus-related disease at the Department of Health Management and Health Economics, University of Oslo and the Harvard T.H. Chan School of Public Health.

The author has completed the ICMJE form and declares no conflicts of interest.

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Birgitte Freiesleben De Blasio

Birgitte Freiesleben De Blasio, Director of the Department of Infectious Diseases Epidemiology and Modelling at the Norwegian Institute of Public Health, and professor in the Department of Biostatistics, University of Oslo. She leads the Institute of Public Health's research group on infection modelling.

The author has completed the ICMJE form and declares the following conflicts of interest: She leads the research group at the Norwegian Institute of Public Health responsible for modelling the coronavirus pandemic; this modelling forms the basis for the decisions made by the authorities.

Article

No human brain has the capacity to think through all possible outcomes of an epidemic. A simulation model can keep track of many individuals and factors that influence the course of the epidemic; however, simulation models can never fully replicate reality.

The healthcare service needs answers to a number of questions when epidemics threaten. How many will be infected? How many will require intensive care treatment? How many will die? Should we close schools? Should all those who can, stay home from work? Who should be tested for the infection? Who should be quarantined? How might a vaccine affect the course of the epidemic? These types of questions cannot be answered by searching for randomised trials or registry studies if what we are facing is, as now, a new virus with unknown characteristics. Increasingly, statistical modelling – so-called infectious disease models – are being used nationally and internationally to understand and manage epidemics and the challenges related to infection outbreaks (1).

Modelling infections

Epidemic models are a type of infectious disease models and are based on theories of infectious diseases and knowledge of previous epidemics. Such models simulate the spread of disease from person to person with the aid of computer models or mathematical systems of equations (Figure 1). The starting point is that a single person can infect one or more others. This is quantified as the basic reproduction number, R₀. R₀ is defined as the average number of new cases generated by one infected individual in a fully susceptible population. R₀ increases with the number of persons the infected individual comes into contact with, the transmission probability of each contact, and the length of time the infected individual is infectious. During the swine flu epidemic (H1N1pm09), R₀ was calculated to be 1.35 (2). R₀ is believed to be 5–7 for chickenpox and 16–18 for measles (3). Early in an epidemic when the proportion of susceptible individuals in the population is high, the number of infected individuals will increase exponentially (Figure 2).

Figure 1 Schematic overview of a simple infectious disease model. Individuals start out as susceptible with a certain risk… — **Figure 1** Schematic overview of a simple infectious disease model. Individuals start out as susceptible with a certain risk of being infected based on the effective reproduction number (Reff). Infected individuals can be asymptomatic or symptomatic, after which each individual either becomes immune or dies. The costs to society can be divided into direct costs and indirect costs (lost productivity).

Figure 2 Number of infected individuals with and without societal interventions during an epidemic. — **Figure 2** Number of infected individuals with and without societal interventions during an epidemic.

Eventually, as the proportion of susceptible individuals in the population decreases, the effective reproduction number R_eff falls, where R_eff is R₀ multiplied by the proportion of susceptible individuals (Figure 2). An epidemic dies out when R_eff becomes lower than 1. The total proportion of infected individuals in a population ('attack rate') will be higher when the basic reproduction number is higher. When R₀ is above 3, more than 90 % will become infected, unless effective interventions reduce the transmission chain of infection. Information on the basic reproduction number is important early on in the initial phase of an epidemic, which helps informs policy makers about how extensive the measures needed to control the epidemic must be.

The simplest epidemic models, so-called SIR models, are based on the premise that a population can be divided into three groups: susceptible, infected, and immune or recovered (dead). Based on the basic reproduction number and a number of other parameters, one can calculate the number of asymptomatic and clinical cases, and the number of hospitalised and deceased individuals. A key element is data on the extent to which the groups 'schoolchildren', 'workers' and 'older people' interact with each other and with other members of their respective groups. Such data are necessary to simulate the spread of infection and can be obtained from social contact studies (contact tracing). National contingency plans should, as far as possible, be based on local data because differences in these contact patterns can give rise to substantial variation in the development of epidemics.

Simpler models, such as cancer models, assume that the risk factors remain constant by time and place, and that the disease in one individual does not affect the risk of disease in others. However, with infections, the likelihood of events change over time and from one place to another such that one individual or group of individuals can affect the risk of disease in others. Infectious disease models are thus said to be dynamic, which makes developing this type of model far more demanding.

For all simulation models the same rule applies: the results are no more reliable than the input data supplied

Sensitivity analyses are important in all modelling studies. Sensitivity analysis involves running numerous simulations while changing various parameters in the model, such as R₀ or the risk of death in infected individuals. This provides information on the extent to which the uncertainty in the data entered into the model affects the results of the simulations. It may also be appropriate to examine whether changes in the logical structure of the model affect the results.

Once an infectious disease model has been developed, it must be validated to verify that it contains no miscalculations and that its predictions are in line with reality. As new data become available, the models must be updated and re-validated to ensure that they are consistent with the status quo, to the extent that this is known at any given time. During an epidemic in which new measures are being introduced on an ongoing basis, developments in the spread of the infection and the effective reproduction number must be continuously monitored to evaluate the effectiveness of the measures taken and the need for new interventions.

Predicting and evaluating effects of interventions

An infectious disease model can have many applications. During an epidemic, knowledge of the likely consequences of the outbreak is required. The healthcare authorities need to know how the number of infected people is likely to develop over time, how many may require hospitalisation or respiratory support, and how many may die. In the course of a typical epidemic, there is a period of exponential growth in the number of infected individuals, after which the proportion of susceptible individuals in the population falls (R_eff decreases) and the epidemic dies out (Figure 2). It is also possible to simulate the effects of interventions such as quarantine, school closures, drug treatment and vaccines where appropriate (Figure 2).

Linking epidemic models with health economic models can provide a framework that may assist in making health policy decisions. It can provide, for example, some indication of the need for hospital beds, medicines and intensive care units, and of the cost-effectiveness of interventions.

School closures and sick leave

Yiting Xue and colleagues studied the costs and benefits of school closures during influenza pandemics in Norway (4). They assumed that individuals infected with influenza could either remain asymptomatic, develop mild to moderate disease, develop serious disease requiring hospitalisation, or die as a result of the influenza. The outcomes were 'translated' into loss of quality-adjusted life years. Reductions in morbidity or mortality as a result of interventions could thus be measured in terms of quality-adjusted life years. The model captured the costs avoided by the healthcare service as a result of school closures, including lost productivity to society due to individuals caring for children at home. It also captured the value of lost teaching, but interestingly enough, there are almost no data on what pupils lose out on as a result of short-term absence from school. The results indicated that it is profitable to close schools for students who will not require supervision at home during the closure. For younger children, the results depended on the severity of the epidemic and on whether lost productivity to society due a child's guardian requiring time off work was taken into account.

A related issue is the guidelines for sick leave in cases of suspected influenza (5). Edwards and colleagues estimated the associated costs and benefits based on a study of sick leave in cases of influenza-like symptoms (6). The analysis varied the proportion of individuals who stayed away from work due to symptoms, and the duration of their absence. The study indicated that it would be socio-economically profitable for a high proportion of employees to promptly take sick leave, especially for an infection with high morbidity and mortality. Somewhat surprisingly, sick leave was most cost-effective in epidemics with a low basic reproduction number.

Interventions such as travel bans, isolation of infected individuals and school closures are intended to reduce the transmission of infection in an epidemic so that the effective reproduction number falls below the basic reproduction number. The aforementioned influenza analyses illustrate an important aspect of such interventions: they delay the epidemic and reduce the maximum number of cases at any single point in time (Figure 2). The most effective measures seem able to delay the peak of an influenza epidemic by 50–60 days, providing healthcare services with more time to prepare for the epidemic, while simultaneously reducing its peak burden.

Covid-19 modelling

Norway has a number of research communities with expertise in the modelling of infectious diseases, at institutions including the University of Oslo and the Norwegian Institute of Public Health. Researchers from these organisations have published analyses on methicillin-resistant Staphylococcus aureus (MRSA), herpes zoster, rotavirus, hepatitis C, human papillomavirus and influenza.

In 2017, the Norwegian Institute of Public Health conducted a study in which 4 300 randomly selected Norwegians were asked to complete a diary of all their contacts over the course of a single day (7). This provided data on social networks and the possibilities for spread of infection in Norwegian society.

In recent years, the Norwegian Institute of Public Health has also collaborated with Telenor and the University of Oslo on the use of mobile phone data as inputs into infectious disease models. Since February 2020, they have been working with the Norwegian Computing Center to further develop a model that can predict the spread of Covid-19 at the municipal level in Norway over the next week and the next month. The model uses Telenor's mobile phone data from Norwegian subscribers as well as daily updated epidemiological data. Many simulations are run to estimate uncertainty, and the model is adjusted to fit new data as they become available. Real-time data thus have a part to play in epidemic preparedness, but unfortunately, it is still difficult to quickly retrieve and link registry data on infections (8).

Figures from the Norwegian Institute of Public Health on March 9^th suggested that Norway could end up with approximately 22 000 hospital admissions because of Covid-19 infection, of which 5 500 would be to intensive care units (9). According to the calculations, at the epidemic's peak, hospitals would face 1 700 admissions simultaneously, including 600 individuals requiring intensive care. Such figures are linked to scenario planning, in which consequences are assessed under a given set of assumptions. The assumptions must be interpreted with caution because knowledge about the virus and its spread was, and still is, limited. The Norwegian Institute of Public Health's models are updated as new information comes in. The model also makes it possible to evaluate the effectiveness of currently implemented measures.

Even though the Covid-19 epidemic has been ongoing since December 2019, at the time of writing (17 March 2020) the basic reproduction number (R₀) is still uncertain, although it is believed to be higher than for seasonal influenza. In a recent study, R₀ was estimated at 2.35 in Wuhan on 16 January, a week before restrictions were introduced. The effective reproduction number (R_eff) was 1.05 after measures were implemented (10). The proportion of infected individuals who die is also uncertain. The number of infected individuals is easily underestimated because some of those infected have few or no symptoms and because of insufficient contact tracing or testing capacity. Criteria for testing may vary between countries, and some countries lack the necessary equipment. Overall, this leads to the number of infected individuals being underestimated, and accordingly to the proportion of deaths due to Covid-19 being overestimated.

Discussion

For all simulation models, the same rule applies: the results are no more reliable than the input data supplied ('garbage in – garbage out'). A key limitation of infectious disease modelling is precisely the uncertainty associated with the underlying data. However, despite this uncertainty, models can provide us with useful insights, including which type of data it is important to gather. When panic over HIV prevailed in Norway in the 1980s, Hein Stigum and colleagues established a set of differential equations based on assumptions about sexual behaviour and the characteristics of HIV (11, 12). They concluded that Norway had no reason to fear an extensive HIV epidemic. Some regarded these results as counterintuitive back then, but time would show that their assumptions and presuppositions were essentially correct.

An important insight from the modelling of infections is that no human brain has the capacity to think through all possible outcomes in the complicated chain of events seen in an epidemic. A system of equations or a computer program is able to simultaneously keep track of many individuals and factors that affect the course of the epidemic. Nevertheless, no simulation model can fully replicate reality – it is, and will remain, a model. We therefore join the statistician George Box in arguing that 'all models are wrong, but some are useful' (13).

Literature

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Heesterbeek H, Anderson RM, Andreasen V et al. Modeling infectious disease dynamics in the complex landscape of global health. Science 2015; 347: aaa4339. [PubMed][CrossRef]
2.
Blasio BF, Iversen BG, Tomba GS. Effect of vaccines and antivirals during the major 2009 A(H1N1) pandemic wave in Norway–and the influence of vaccination timing. PLoS One 2012; 7: e30018. [PubMed][CrossRef]
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Centers for Disease Control and Prevention. Smallpox: Disease, Prevention, and Intervention. Slides 16–17. https://stacks.cdc.gov/view/cdc/27929 Accessed 7.3.2020.
4.
Xue Y, Kristiansen IS, de Blasio BF. Dynamic modelling of costs and health consequences of school closure during an influenza pandemic. BMC Public Health 2012; 12: 962. [PubMed][CrossRef]
5.
Edwards CH, Tomba GS, Sonbo Kristiansen I et al. Evaluating costs and health consequences of sick leave strategies against pandemic and seasonal influenza in Norway using a dynamic model. BMJ Open 2019; 9: e027832. [PubMed][CrossRef]
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de Blasio BF, Xue Y, Iversen B et al. Estimating influenza-related sick leave in Norway: was work absenteeism higher during the 2009 A(H1N1) pandemic compared to seasonal epidemics? Euro Surveill 2012; 17: 20246. [PubMed]
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Folkehelseinstituttet. Vaksinasjonsdekning og spredningspotensiale for smittsomme sykdommer i Norge – prosjektbeskrivelse. https://www.fhi.no/prosjekter/vaksinasjonsdekning-og-spredningspotensiale-prosjektbeskrivelse/ Accessed 10.3.2020.
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Time JK. Vitenskap i en unntakstilstand. Morgenbladet 4.3.2020. https://morgenbladet.no/aktuelt/2020/03/vitenskap-i-enunntakstilstand Accessed 5.3.2020.
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Kalveland J. Anslår 1 700 korona-innleggelser. Dagens Medisin 9.3.2020. https://www.dagensmedisin.no/artikler/2020/03/09/anslar-1.700-korona-innleggelser-pa-epidemitoppen/ Accessed 16.3.2020.
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Kucharski AJ, Russell TW, Diamond C et al. Early dynamics of transmission and control of COVID-19: a mathematical modelling study. Lancet Infect Dis 2020 doi: 10.1016/S1473-3099(20)30144-4. [PubMed][CrossRef]
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Magnus P, Stigum H, Grønnesby JK et al. Spådommer for omfanget av heteroseksuelt betinget HIV-infeksjon i Norge i 1990-årene. Tidsskr Nor Lægeforen 1990; 110: 3225–8. [PubMed]
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Stigum H, Magnus P, Grønnesby JK et al. Nytten av simuleringsmodeller i forståelsen av HIV-epidemien. Tidsskr Nor Lægeforen 1988; 108: 115–9. [PubMed]
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Box GE. Science and statistics. J Am Stat Assoc 1976; 71: 791–9. [CrossRef]

Comments ( 3 )

Dette kommentarfeltet modereres, men kommentarer blir ikke redaksjonelt behandlet ut over å sikre at de følger retningslinjer for vårt kommentarfelt.

30.03.2020:

Å spå om fremtiden med dårlige data og dårlige modeller er vanskelig.

England bommet på Tyskland sin produksjon av stridsvogner med en faktor på 4 i 1942/3. I sin etterretning bommet USA på militæret i Sovjetunionens andel av BNP med en faktor på 2,5. Bedre er det ikke i medisin når man modellerer influensa (1). Etter rundt 50 år med modellering og overvåking (med gode data) varierer dødeligheten med en faktor på minst 2,5. Enda verre kan det være med Covid-19. Her vet man nesten ingenting om prevalens og dødelighet ligger kanskje i områder 0,05 til 1 % (1). Det blir rundt 40 millioner dødsfall (som ved Spanskesyken) hvis 60 % blir syke med 1 % dødelighet. Men disse er meget gamle. Hvis forventet gjenstående levealder er 5 % av ved spanskesyken, så blir effekten 1 % av hva man så ved spanskesyken. En prosent! I beste fall blir det som ved mild influensa.

Litteratur:

1. Ioannidis, J. A fiasco in the making? As the coronavirus pandemic takes hold, we are making decisions without reliable data. Stat 17.03.2020. https://www.statnews.com/2020/03/17/a-fiasco-in-the-making-as-the-coronavirus-pandemic-takes-hold-we-are-making-decisions-without-reliable-data/ Lest 30.03.2020.

30.03.2020:

John PA Ioannidis etterspør mer og bedre data for å forstå Covid-19 epidemien og advarer mot at tiltakene som nå gjøres ikke er basert på evidence (1). Foreløpig mangler data om prevalens (leger må prioritere testing av syke fremfor å samle inn data til forskning fra asymptomatiske) og letalitet kan ikke studeres før alle syke enten er friske eller har dødd. Myndighetene må gjøre tiltak basert på de data man har.

Kunnskap om tidligere epidemier kan og bør også brukes. Forskere siterer statistiske modeller som estimerer infeksjonsrater, antall døde, og epidemiens varighet, men ingen konsensus finnes (2). Forskere er ikke enige. Hvem skal vi tro på? Forskjell mellom forskerne skyldes ikke data, men hvordan forskere modellerer data. Usikkerheten i prognoser kan ikke gies med en modell. Usikkerheten er representert ved hele variasjonsbredden til hva eksperter mener, inkludert hver ekspert sin usikkerhet (2). Antall nye Covid-19 syke i USA om 14 dager varierer med en faktor på 50 (fra 10 000 til 500 000 syke) når man spurte dem sist uke. Hvis man sammenlikner alle ekspertenes mest sannsynlig estimater, varierte de med en faktor på 7 (fra 10 000 til 70 000). Covid-19 sin ukjente natur er en selvfølgelig forklaringen, men også mangel på kunnskap om hvordan folk vil beskytte seg mot infeksjon. Når de viktigste parametrene i modellene er ukjente, blir det vanskelig å planlegge ut fra modeller, tenker jeg.

Men man kan ta antall syke i dag (2383) og si at om en uke vil man trenge rundt 2383/20=120 respiratorer (5% blir alvorlig syke).

Og istedenfor å telle letalitet i prosent av alle syke, burde man oppgi letalitet etter f eks 65 år og sammenlikne med influensa. Nesten ingen under 65 år dør av Corona. Ti prosent letalitet hos de eldste er høyt , og 100 ganger høyere enn hos unge (3). Kanskje burde man isolert alle på sykehjem og informert eldre i Norge bedre?

Litteratur:

1. Ioanidis JPA. A fiasco in the making as the coronavirus pandemic takes hold we are making decisions without reliable data. Statsnews 17.03.2020. https://www.statnews.com/2020/03/17/a-fiasco-in-the-making-as-the-coronavirus-pandemic-takes-hold-we-are-making-decisions-without-reliable-data/

2 Boice J, Wiederkehr A. Infectious Disease Experts Don’t Know How Bad The Coronavirus Is Going To Get, Either. 538.com.

3 Begley S. New analysis breaks down age-group risk for coronavirus — and shows millennials are not invincible. Statnews 18.03.2020. www.statnews.com/2020/03/18/coronavirus-new-age-analysis-of-risk-confirms-young-adults-not-invincible/

15.04.2020:

Takk for god lesing. Jeg finner ikke kildekoden til modellen. Kunne dere tydeliggjøre link hvis den er ute, eller legge ut koden på github slik Imperial College har gjort hvis dere ikke allerede har gjort det?

This article was published more than 12 months ago and we have therefore closed it for new comments.

Published: 18 March 2020

Tidsskr Nor Legeforen 18 March 2020 Vol. 140.

doi:

10.4045/tidsskr.20.0225

Published: 18 March 2020

Tidsskr Nor Legeforen 2020 Vol. 140.

doi: 10.4045/tidsskr.20.0225

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Covid-19: Simulation models for epidemics

Modelling infections

Predicting and evaluating effects of interventions

School closures and sick leave

Covid-19 modelling

Discussion

Å spå om fremtiden

Hva hvis man ikke vet hvilken modell man skal bruke?

Kildekode

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