Hossein Gorji^a^*, Markus Arnoldini^b^*, David F. Jenny^c, Wolf-Dietrich Hardt^d^†, Patrick Jenny^e^†

^a Laboratory of Multiscale Studies in Building Physics, Empa, Swiss Federal Laboratories for Materials Science and Technology, Dübendorf, Switzerland

^b Department of Health Sciences and Technology, Swiss Federal Institute of Technology, Zurich, Switzerland

^c Department of Mathematics, Swiss Federal Institute of Technology, Zurich, Switzerland

^d Institute of Microbiology, D-BIOL, Swiss Federal Institute of Technology, Zurich, Switzerland

^e Department of Mechanical and Process Engineering, Swiss Federal Institute of Technology, IFD, Zurich, Switzerland

Summary

Relevant pandemic-spread scenario simulations can provide guiding principles for containment and mitigation policies. We devised a compartmental model to predict the effectiveness of different mitigation strategies with a main focus on mass testing. The model consists of a set of simple differential equations considering the population size, reported and unreported infections, reported and unreported recoveries, and the number of COVID-19-inflicted deaths. We assumed that COVID-19 survivors are immune (e.g., mutations are not considered) and that the virus is primarily passed on by asymptomatic and pre-symptomatic individuals. Moreover, the current version of the model does not account for age-dependent differences in the death rates, but considers higher mortality rates due to temporary shortage of intensive care units. The model parameters have been chosen in a plausible range based on information found in the literature, but it is easily adaptable, i.e., these values can be replaced by updated information any time. We compared infection rates, the total number of people getting infected and the number of deaths in different scenarios. Social distancing or mass testing can contain or drastically reduce the infections and the predicted number of deaths when compared with a situation without mitigation. We found that mass testing alone and subsequent isolation of detected cases can be an effective mitigation strategy, alone and in combination with social distancing. It is of high practical relevance that a relationship between testing frequency and the effective reproduction number of the virus can be provided. However, unless one assumes that the virus can be globally defeated by reducing the number of infected persons to zero, testing must be upheld, albeit at reduced intensity, to prevent subsequent waves of infection. The model suggests that testing strategies can be equally effective as social distancing, though at much lower economic costs. We discuss how our mathematical model may help to devise an optimal mix of mitigation strategies against the COVID-19 pandemic. Moreover, we quantify the theoretical limit of contact tracing and by how much the effect of testing is enhanced, if applied to sub-populations with increased exposure risk or prevalence.

Introduction

The recent outbreak of COVID-19 in Wuhan, China has led to a pandemic with significant impacts on public health and economies across the globe. As the number of infected people increases in a community, public health policies move away from containment of the outbreak to mitigation strategies such as social distancing and isolation, with considerable detrimental effects on public life and the economy. Although less restrictive mitigation strategies would be desirable, alternative choices are limited owing to a lack of resources and technologies. To better understand the potential effects of a particular mitigation strategy, we must assess the underlying factors that impact the spread of the outbreak, and for this mathematical models integrating the relevant underlying mechanisms are a good tool.

Various biomathematical approaches have been proposed and pursued for epidemic-spread modelling. At the highest level, one can categorise them into agent based [1], network [2, 3] and compartmental models [4]. The chosen model can then be closed using empirical/machine-learning, statistical or deterministic approaches [5]. Agent/network based models may provide highly refined scenario analysis tools, but their black-box nature may not be appropriate for large-scale mitigation scenario assessments. Compartmental models, on the other hand, can provide us with insightful and explicit solutions more relevant for drawing fundamental conclusions. These type of models may employ deterministic or stochastic methodologies to tackle the evolution of the epidemic within a susceptible population. The former category belongs to deterministic descriptions, which include susceptible-infectious-removed (SIR), susceptible-infectious-susceptible (SIS) and susceptible-exposed-infectious-removed (SEIR) models [6]. A more complicated class of models incorporates the stochastic nature of the epidemic spread via the framework of, for example, Ito- or Levy-type processes [7–9]. Both deterministic and stochastic descriptions, at their fundamental level, rely on reaction mechanisms that characterise infections, recoveries and deaths within different sub-groups of the population.

Although recently there has been a massive effort in pandemic-spread investigations of COVID-19, for example using SEIR models [10–13], network models [14] and agent-based simulations [15, 16], no studies that investigated the effect of mass testing are found in the literature. As recent efforts in, for example, China are channelled towards mass testing in relatively large cities [17], it is necessary to also provide theoretical foundations for mass testing-based mitigation strategies. To achieve this, we separated the category of detected cases from infected ones who remain undetected (e.g., by the sheer lack of test kits) and predicted their coupled dynamics. Note that detected here refers to persons being isolated, which comprises not only those who tested positive, but also those who have strong symptoms and thus stay in self-quarantine. It is also important to note that in the case of this specific virus, the asymptomatic and pre-symptomatic infected people contribute significantly to the spread of the pandemic [18]. Therefore, early detection and containment of infected but asymptomatic individuals can be extremely relevant for the dynamic behaviour. Hence, we devised a set of reaction equations focusing on both detected and undetected categories. Moreover, the impact of the shortage of intensive care units during peaks of the pandemic were integrated in the outcome of the scenarios. The model coefficients were calibrated on the basis of existing data, and the model was employed to investigate two main mitigation approaches, one relying on social distancing and one on more frequent infection testing. Also, a combination of the two is studied, and we quantified by how much the testing efficiency is enhanced, if available resources are focused on subpopulations with increased prevalence. Finally, theoretical limitations of contact tracing, which relies on isolating contacts of symptomatic individuals, was studied. We argue that as contact tracing provides little means to prevent virus transmission from asymptomatic cases, it cannot lead to the pandemic containment as a stand-alone approach.

Materials and methods

Results

Discussion

As more and more affected populations are focusing on risk mitigation plans for COVID-19, it is important to identify strategies to mitigate and eventually suppress the pandemic. For this reason, it is crucial to understand the mechanisms underlying this pandemic-spread. Outcomes of social distancing and mass testing are investigated in this paper. It was found that the latter can significantly reduce the percentage of people getting infected and the death toll. It is important to emphasise here that repetitive testing of individuals without symptoms (if sufficiently large numbers of tests are available and can be applied) has a much stronger effect on the reproduction number than testing people with symptoms, since in most cases the latter are contained. As testing capacities improve, our approach may help to decide by how much social distancing measures can be relaxed. Whereas social distancing is currently essential, it is of utmost importance that testing capabilities are upgraded such that they cover large portions of affected populations in the near future.

From our analysis we conclude that testing every individual without symptoms every few days (with our assumptions roughly every week) would reduce the reproduction number of COVID-19 to 1 and thereby stabilise the pandemic, which is very promising. After a while, fewer and fewer infected people (who spread the virus) will be detected. In this way, continued large scale testing can verify the success of the mitigation strategy. Mass testing should be continued beyond this point, though at a reduced frequency. This would allow determination of whether the fraction of infected persons tends to increase. If this were the case, testing frequencies should again be ramped up. In any case, unless the virus can be defeated completely and globally by reducing the number of infected individuals to zero, there is a risk of COVID-19 re-emergence after such mitigation measures are abandoned.

In this context, the estimates for mass testing that are provided by our analysis should be regarded as estimates for the upper boundary of the tests needed. This upper boundary test number can be used as a guideline for development of mass testing technology and logistics. For further improvements of the predictions a more reliable data base for parameter tuning would be necessary. The model itself can be refined by accounting for different age groups and latency, which would involve additional parameters. In the future it would be of utmost interest to further investigate combined strategies such as social distancing for old and endangered persons and mass testing for the remaining population, including the work force. Ideally, contact tracing and repetitive testing should be combined with some sort of social distancing to successfully suppress the virus spread and to keep the death toll low.

Appendix Supplementary material

Acknowledgement

The authors are very thankful to Alexandre Duc for his contribution to the former version of the manuscript with a bluetooth app-based implementation of targeted testing, Dario Ackermann for the website corona-lab.ch, containing the simulation tool based on the model presented in this paper, and Emma Slack, Erik Bakkeren and Noemi Santamaria for helpful comments on the manuscript. Hossein Gorji acknowledges the funding provided by Swiss National Science Foundation under the grant number 174060.

Notes

Disclosure statement

No potential conflict of interest relevant to this article was reported.

References

2 Ball F , Sirl D , Trapman P . Analysis of a stochastic SIR epidemic on a random network incorporating household structure. Math Biosci. 2010;224(2):53–73. doi:.https://doi.org/10.1016/j.mbs.2009.12.003

3Kiss IZ, Miller JC, Simon PL. Mathematics of epidemics on networks. Vol. 46. Springer; 2017.

4Allen LJS, Bauch CT, Castillo-Chavez C, Earn DJD, Feng Z, Lewis MA, et al. Mathematical Epidemiology. Brauer F, van den Driessche P, Wu J, editors. 2008. 415 p.

5 Siettos CI , Russo L . Mathematical modeling of infectious disease dynamics. Virulence. 2013;4(4):295–306. doi:.https://doi.org/10.4161/viru.24041

6Daley DJ, Gani J. Epidemic modeling: an introduction. Cambridge: Cambridge University Press; 1999. 15.

8 Allen LJS . A primer on stochastic epidemic models: Formulation, numerical simulation, and analysis. Infect Dis Model. 2017;2(2):128–42. doi:.https://doi.org/10.1016/j.idm.2017.03.001

9 Zhou Y , Yuan S , Zhao D. Threshold behavior of a stochastic SIS model with Lévy jumps. Appl Math Comput. 2016;275:255–67. doi:https://doi.org/10.1016/j.amc.2015.11.077

10 Wu JT , Leung K , Leung GM . Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study. Lancet. 2020;395(10225):689–97. doi:.https://doi.org/10.1016/S0140-6736(20)30260-9

11 Hou C , Chen J , Zhou Y , Hua L , Yuan J , He S , et al. The effectiveness of quarantine of Wuhan city against the Corona Virus Disease 2019 (COVID-19): A well-mixed SEIR model analysis. J Med Virol. 2020;92(7):841–8. doi:.https://doi.org/10.1002/jmv.25827

12 Rǎdulescu A , Williams C , Cavanagh K . Management strategies in a SEIR-type model of COVID 19 community spread. Sci Rep. 2020;10(1):21256. doi:.https://doi.org/10.1038/s41598-020-77628-4

14 Block P , Hoffman M , Raabe IJ , Dowd JB , Rahal C , Kashyap R , et al. Social network-based distancing strategies to flatten the COVID-19 curve in a post-lockdown world. Nat Hum Behav. 2020;4(6):588–96. doi:.https://doi.org/10.1038/s41562-020-0898-6

15 Rockett RJ , Arnott A , Lam C , Sadsad R , Timms V , Gray K-A , et al. Revealing COVID-19 transmission in Australia by SARS-CoV-2 genome sequencing and agent-based modeling. Nat Med. 2020;26(9):1398–404. doi:.https://doi.org/10.1038/s41591-020-1000-7

16 Silva PCL , Batista PVC , Lima HS , Alves MA , Guimarães FG , Silva RCP . COVID-ABS: An agent-based model of COVID-19 epidemic to simulate health and economic effects of social distancing interventions. Chaos Solitons Fractals. 2020;139(10223):110088. doi:.https://doi.org/10.1016/j.chaos.2020.110088

18 Li R , Pei S , Chen B , Song Y , Zhang T , Yang W , et al. Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2). Science. 2020;368(6490):489–93. doi:.https://doi.org/10.1126/science.abb3221

19Ferguson N, Laydon D, Nedjati Gilani G, Imai N, Ainslie K, Baguelin M, et al. Impact of non-pharmaceutical interventions (NPIs) to reduce COVID19 mortality and healthcare demand. 2020 Mar 16. Available from: http://spiral.imperial.ac.uk/handle/10044/1/77482

20 Diekmann O , Heesterbeek JA , Metz JA . On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J Math Biol. 1990;28(4):365–82. doi:.https://doi.org/10.1007/BF00178324

21 Virlogeux V , Fang VJ , Wu JT , Ho L-M , Peiris JSM , Leung GM , et al. Brief Report: Incubation Period Duration and Severity of Clinical Disease Following Severe Acute Respiratory Syndrome Coronavirus Infection. Epidemiology. 2015;26(5):666–9. doi:.https://doi.org/10.1097/EDE.0000000000000339

22 Men K , Wang X , Yihao L , Zhang G , Hu J , Gao Y , et al. Estimate the incubation period of coronavirus 2019 (COVID-19). medRxiv. 2020;24(2):219.

24 Ferretti L , Wymant C , Kendall M , Zhao L , Nurtay A , Abeler-Dörner L , et al. Quantifying SARS-CoV-2 transmission suggests epidemic control with digital contact tracing. Science. 2020;368(6491):eabb6936–9. doi:.https://doi.org/10.1126/science.abb6936

25 Anastassopoulou C , Russo L , Tsakris A , Siettos C . Data-based analysis, modelling and forecasting of the COVID-19 outbreak. PLoS One. 2020;15(3):e0230405–21. doi:.https://doi.org/10.1371/journal.pone.0230405

26 Liu Y , Gayle AA , Wilder-Smith A , Rocklöv J . The reproductive number of COVID-19 is higher compared to SARS coronavirus. J Travel Med. 2020;27(2): taaa021. doi:.https://doi.org/10.1093/jtm/taaa021

27 Liu Z , Bing X , Zhi XZ ; Epidemiology Working Group for NCIP Epidemic Response, Chinese Center for Disease Control and Prevention. [The epidemiological characteristics of an outbreak of 2019 novel coronavirus diseases (COVID-19) in China]. Zhonghua Liu Xing Bing Xue Za Zhi. 2020;41(2):145–51. Article in Chinese. doi:10.3760/cma.j.issn.0254-6450.2020.02.003.

28 Karagiannidis C , Mostert C , Hentschker C , Voshaar T , Malzahn J , Schillinger G , et al. Case characteristics, resource use, and outcomes of 10 021 patients with COVID-19 admitted to 920 German hospitals: an observational study. Lancet Respir Med. 2020;8(9):853–62. doi:.https://doi.org/10.1016/S2213-2600(20)30316-7

29 Baud D , Qi X , Nielsen-Saines K , Musso D , Pomar L , Favre G . Real estimates of mortality following COVID-19 infection. Lancet Infect Dis. 2020;20(7):773. doi:.https://doi.org/10.1016/S1473-3099(20)30195-X

30 Mizumoto K , Chowell G . Estimating Risk for Death from Coronavirus Disease, China, January-February 2020. Emerg Infect Dis. 2020;26(6):1251–6. doi:.https://doi.org/10.3201/eid2606.200233

32 Corman VM , Landt O , Kaiser M , Molenkamp R , Meijer A , Chu DK , et al. Detection of 2019 novel coronavirus (2019-nCoV) by real-time RT-PCR. Euro Surveill. 2020;25(3):1–8. doi:.https://doi.org/10.2807/1560-7917.ES.2020.25.3.2000045

34 Verity R , Okell LC , Dorigatti I , Winskill P , Whittaker C , Imai N , et al. Estimates of the severity of coronavirus disease 2019: a model-based analysis. Lancet Infect Dis. 2020;20(6):669–77. doi:.https://doi.org/10.1016/S1473-3099(20)30243-7

35 Hinch R , Probert WJM , Nurtay A , Kendall M , Wymatt C , Hall M , et al. OpenABM-Covid19 - an agent-based model for non-pharmaceutical interventions against COVID-19 including contact tracing. medRxiv. 2021 Jan 30:1–23.

36 Fraser C , Riley S , Anderson RM , Ferguson NM . Factors that make an infectious disease outbreak controllable. Proc Natl Acad Sci USA. 2004;101(16):6146–51. doi:.https://doi.org/10.1073/pnas.0307506101

37 Lavezzo E , Franchin E , Ciavarella C , Cuomo-Dannenburg G , Barzon L , Del Vecchio C , et al. Suppression of COVID-19 outbreak in the municipality of Vo. Nature. 2020;584(7821):425–9. doi:.https://doi.org/10.1038/s41586-020-2488-1

Notes

Equally contributing first authors

†

Equally contributing last authors

Disclosure statement

No potential conflict of interest relevant to this article was reported.

Dynamic modelling to identify mitigation strategies for the COVID-19 pandemic

Summary

Introduction

Materials and methods

Results

Discussion

Appendix Supplementary material

Acknowledgement

Notes

References

Notes