Fighting Coronavirus with Mathematics
Dan Coombs, UBC, February 7, 2020
The beginning of 2020 has seen the emergence of a new public health threat: a transmissible respiratory infection caused by a novel coronavirus, or nCoV. The purpose of this post is to briefly describe how scientists around the world are using mathematical modelling to understand the disease spread and make predictions about the future course of the epidemic. Specific information about the virus reflects the general understanding of the situation up to the first week of February 2020. However, this is a fast moving epidemic and the situation may change rapidly!
Facts of the nCoV Epidemic
The first cases of infection are believed to have occurred in very early December, 2019, in Wuhan, China. Genetic sequencing of nCoV sampled from patients has since revealed that it is closely related to known coronaviruses that circulate in bat populations. Coronaviruses have affected humans before. There are four widely circulating coronaviruses that are known to cause colds, although these infections are usually mild. A more serious example that is well known in Canada is Sudden Acute Respiratory Syndrome (SARS), also caused by a coronavirus. SARS caused thousands of infections in China and hundreds of infections in Canada during 20022003. The case fatality rate (in this case, proportion of confirmed cases that caused death) of SARS exceeded 10% in Canada. For comparison, the case fatality rate of seasonal influenza is usually estimated to be less than 0.01%. Finally, Middle East Respiratory Syndrome (MERS) is also caused by a coronavirus that is occasionally transmitted from dromedary camels to humans. MERS does not seem to spread well among humans, which is fortunate because the case fatality rate exceeds 20%. Due to the high case fatality rates of SARS and MERS, there is great concern among infectious disease epidemiologists about new coronaviruses spreading into the human population.
As of today’s reports, there have been over 31,000 confirmed cases in mainland China, and over 600 deaths attributed to the virus. Approximately twothirds of cases, and the vast majority of deaths, have been reported in Hubei province (where the city of Wuhan is located) but spread to other provinces has also been apparent since approximately January 20. There have also been over 300 cases reported outside mainland China. Most of these cases have been travellers who were infected in Wuhan although there have been a few persontoperson infection events reported outside China.
Mathematical Modelling
In the early stages of a fastspreading epidemic, mathematical modelling has an important role to play in estimating certain key parameters which can then influence public health decisions. Interest in realtime modelling of epidemics has grown, primarily since the SARS epidemic, and driven also by pandemic influenza and the Ebolavirus epidemic in West Africa.
The most important epidemiological parameter is the basic reproductive number, or R_{0}. This is defined as the average number of new cases caused by an infected individual entering a fully susceptible population. R_{0} is a threshold quantity. If R_{0}<1, then the disease will go extinct with probability one, perhaps after a few transmission events. On the other hand, if R_{0}>1, the disease has a positive probability of spreading and (at least initially) growing exponentially in the population. It is helpful to think of R_{0} as a product of three factors:
 the contact rate between infectious and susceptible individuals,
 the probability of infection, per contact event, and
 the duration of the infectious period of the disease.
This formulation allows epidemiologists to estimate the strength of intervention that must occur to prevent an epidemic – for example, by vaccination of the population or quarantining of potentially infectious people (to reduce the contact rate with susceptible individuals), by encouraging people to wash hands frequently (to reduce the probability of infection per contact), or by treating known infections to hasten viral clearance. In the case of measles, the estimated R_{0} is 18. This means that (11/18) ~ 95% of the population must be vaccinated to prevent epidemics. On the other hand, influenza epidemics typically show an R_{0} of less than 2 – so a vaccination rate of less than 50% would be predicted to prevent an epidemic. For nCoV, vaccination is not available, but the same calculation could indicate what level of quarantine will suffice to prevent an epidemic, for example. R_{0} also determines the final size of an idealized epidemic in a mathematical model.
We can break down R_{0} further by considering the rate of new infections β(τ) caused by an individual, structured over time since infection τ, and then integrating this quantity:
R_{0}= ∫_{0}^{∞} β(τ) dτ.
This equality indicates the importance of understanding the time dynamics of transmission. Epidemiologists talk about this in terms of the serial interval, or the average time between successive infections. One such model that has been applied to nCoV is the SusceptibleExposedInfectiousRemoved (SEIR) model. In this model, individuals transition through the states in this order: Exposed individuals are understood to be infected, but not yet transmitting the virus. After transitioning to the infectious class, individuals are fully infectious until they move to the removed class (by recovering, being isolated or dying). We can then parameterize each stage of the disease process, for example by estimating the average length of time a typical patient spends in each stage.
Let the incidence of the disease be the rate of appearance of new cases. Denoting this by i(t), we can then write the recursion
i(t) = ∫_{0}^{∞} β(τ) i(tτ) dτ.
In the early stages of the nCoV epidemic, the incidence was observed to be approximately exponential, i(t) ~ e^{rt}. If we estimate r from the observed data, then can we determine R_{0}? Plugging in, we obtain the simple formula
1 = ∫_{0}^{∞} β(τ) e^{rτ }dτ = L[β] (r) ,
where L[β](r) is the Laplace transform of the transmission rate function evaluated at r. We can use this equation to help us fit the parameters in the functional form of β and thus obtain R_{0}. However, this is a single equation and there are usually several parameters describing the course of disease, that must all be known before R_{0} can be estimated. Researchers studying nCoV have generally supposed that the disease course of nCoV is similar to that of SARS, with an incubation/exposed period of around 5 days followed by an infectious period of around 3 days. One study has shown, using limited data from nCoV patients outside China, that these assumed parameters are probably quite close to reality (Backer et al).
Mathematics Applied to the nCoV Epidemic
Using essentially this approach, and confirmedcase data from the early phase of the epidemic in Wuhan (before the imposition of travel restrictions), two independent groups generated estimates of the R_{0} for nCoV as 2.9 (Liu et al, 95% confidence interval of 2.33.6) and 3.3 (Zhao et al, 95% confidence interval of 2.74). These numbers are very much in line with those obtained by a number of other studies that applied different simulationdriven approaches, fit to Wuhan, China and international data (Imai, Riou, Li, Wu, Read, Kucharski). The bottom line, drawn from all these early estimates, is that a successful quarantine policy will probably need to prevent at least 60% of all potentially transmissive contacts in order to reduce the transmission rate below one and send the epidemic into decline.
Modelling Challenges
Questions remain around the validity of these estimates around transmission. They are based on the reporting of confirmed or suspected cases. If many people suffer only a mild or asymptomatic illness, they may not seek medical attention and thus not be counted in the data. Simultaneously, the epidemic has placed extreme stress on the health care system, hindering identification, confirmation and reporting of cases. For both reasons, it is possible that the estimated R0 is an underestimate. Jonathan Dushoff (McMaster) and colleagues have also emphasized the importance of properly dealing with uncertainties in parameter estimation when seeking to analyze ongoing epidemics (Park et al).
A second important quantity that we need to estimate is the case fatality rate. This is the proportion of deaths experienced by “cases” and it is important to distinguish between, for example, cases of people experiencing serious illness, cases of people hospitalized, or just all people who are infected. It is difficult to do this at present, because the typical period between infection and death is uncertain, but probably more than ten days, which is a significant fraction of the length of the epidemic to date. Other challenges include questions around the true number of cases, as mentioned previously. Christian Althaus at the University of Bern has been updating an estimate based only on international confirmed cases and deaths. So far, it appears that the case fatality rate is lower than SARS, but higher than seasonal influenza, but this is based on a single death outside China, and must therefore be treated with suspicion.
Outlook
Looking forward, I think that the main questions are around the effectiveness of quarantine (of known or suspected infectious people) and travel restrictions (applied within China, as well as internationally). As described by Joseph Wu and colleagues (Hong Kong University), sustained and responsive quarantine restrictions directly reduce transmissibility and reduce the expected size of the epidemic. Within Wuhan, we would hope to see declining numbers of cases reported quite soon, given that quarantining has now been applied for over two weeks, and the serial interval probably does not exceed that duration. Of course, if capacity to report cases was previously a bottleneck, but now resources are available, this might not occur even if quarantine restrictions were succeeding. Similar considerations apply to other cities in China and elsewhere when significant numbers of infections begin to be reported.
Wu et al also point out that imperfect travel restrictions can delay the onset of the epidemic, but do not reduce the expected size of an epidemic that spreads to a new location anyway. At present, the policy of detecting and quarantining newly detected cases in most locations outside China appears to be succeeding, but continued vigilance will be required as new infections continue to appear.
Daniel Coombs
Department of Mathematics
Institute of Applied Mathematics
University of British Columbia
References:
General descriptions of the disease and epidemic:
 From The Lancet: https://www.thelancet.com/coronavirus
 Wikipedia: Timeline of the 201920 Wuhan coronavirus outbreak. https://en.wikipedia.org/wiki/Timeline_of_the_2019–20_Wuhan_coronavirus_outbreak
 From Imperial College (includes Imai et al simulationbased modelling report): https://www.imperial.ac.uk/mrcglobalinfectiousdiseaseanalysis/news–wuhancoronavirus/
Two early, regressionbased estimates of R_{0}:
 Liu et al. Transmission dynamics of 2019 novel coronavirus (2019nCoV). bioRxiv (2020). https://www.biorxiv.org/content/10.1101/2020.01.25.919787v1
 Zhao et al. Preliminary estimation of the basic reproduction number of novel coronavirus (2019nCoV) in China, from 2019 to 2020: A datadriven analysis in the early phase of the outbreak. bioRxiv (2020). https://www.biorxiv.org/content/10.1101/2020.01.23.916395v1
Simulation and differentialequation models:

 Riou and Althaus. Pattern of early humantohuman transmission of Wuhan 2019 novel coronavirus (2019nCoV), December 2019 to January 2020. Eurosurveillance (2020). https://www.eurosurveillance.org/content/10.2807/15607917.ES.2020.25.4.2000058
 Li et al. Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus–Infected Pneumonia. New England Journal of Medicine (2020). https://www.nejm.org/doi/full/10.1056/NEJMoa2001316
 Kucharski and coworkers. Analysis and projections of transmission dynamics of nCoV in Wuhan. https://cmmid.github.io/ncov/wuhan_early_dynamics/index.html
 Read and coworkers. Novel coronavirus 2019nCoV: early estimation of epidemiological parameters and epidemic predictions. medRxiv (2020). https://www.medrxiv.org/content/10.1101/2020.01.23.20018549v2
 Joseph T Wu, Kathy Leung, Gabriel M Leung. Nowcasting and forecasting the potential domestic and international spread of the 2019nCoV outbreak originating in Wuhan, China: a modelling study. The Lancet (2020).
https://www.thelancet.com/journals/lancet/article/PIIS01406736(20)302609/fulltext
Estimating disease course parameters, and sensitivity or findings to parameter estimation:
 Backer, Klinkenberg and Wallinga. The incubation period of 2019nCoV infections among travellers from Wuhan, China. medRxiv (2020). https://www.medrxiv.org/content/10.1101/2020.01.27.20018986v2
 Christian Althaus fatality rate analysis from international cases: https://github.com/calthaus/ncovcfr
 Park et al. Reconciling earlyoutbreak preliminary estimates of the basic reproductive number and its uncertainty. (2020) https://github.com/parksw3/nCoV_framework/blob/master/v2/ncov.pdf
Visualizing genetic changes in the virus over time (recommended!)