What the reproductive number R0 can and cannot tell us about COVID-19 dynamics

The importance of quarantine: modelling the COVID-19 testing process

We incorporate the disease state and testing state into the formulation of a COVID-19 epidemic model. For this model, the basic reproduction number is identified and its dependence on model parameters related to the testing process and isolation efficacy is discussed. The relations between the basic reproduction number, the final epidemic and peak sizes, and the model parameters are further explored numerically. We find that fast test reporting does not always benefit the control of the COVID-19 epidemic if good quarantine while awaiting test results is implemented. Moreover, the final epidemic and peak sizes do not always increase along with the basic reproduction number. Under some circumstances, lowering the basic reproduction number increases the final epidemic and peak sizes. Our findings suggest that properly implementing isolation for individuals who are waiting for their testing results would lower the basic reproduction number as well as the final epidemic and peak sizes.

COVID-19 epidemic and public health interventions in Shanghai, China: Statistical analysis of transmission, correlation and conversion

Background

The Shanghai COVID-19 epidemic is an important example of a local outbreak and of the implementation of normalized prevention and disease control strategies. The precise impact of public health interventions on epidemic prevention and control is unknown.

Methods

We collected information on COVID-19 patients reported in Shanghai, China, from January 30 to May 31, 2022. These newly added cases were classified as local confirmed cases, local asymptomatic infections, imported confirmed cases and imported asymptomatic infections. We used polynomial fitting correlation analysis and illustrated the time lag plot in the correlation analysis of local and imported cases. Analyzing the conversion of asymptomatic infections to confirmed cases, we proposed a new measure of the conversion rate (C _r ). In the evolution of epidemic transmission and the analysis of intervention effects, we calculated the effective reproduction number (R _t ). Additionally, we used simulated predictions of public health interventions in transmission, correlation, and conversion analyses.

Results

(1) The overall level of R _t in the first three stages was higher than the epidemic threshold. After the implementation of public health intervention measures in the third stage, R _t decreased rapidly, and the overall R _t level in the last three stages was lower than the epidemic threshold. The longer the public health interventions were delayed, the more cases that were expected and the later the epidemic was expected to end. (2) In the correlation analysis, the outbreak in Shanghai was characterized by double peaks. (3) In the conversion analysis, when the incubation period was short (3 or 7 days), the conversion rate fluctuated smoothly and did not reflect the effect of the intervention. When the incubation period was extended (10 and 14 days), the conversion rate fluctuated in each period, being higher in the first five stages and lower in the sixth stage.

Conclusion

Effective public health interventions helped slow the spread of COVID-19 in Shanghai, shorten the outbreak duration, and protect the healthcare system from stress. Our research can serve as a positive guideline for addressing infectious disease prevention and control in China and other countries and regions.

A comparative study on the three calculation methods for reproduction numbers of COVID-19

Objective

This study uses four COVID-19 outbreaks as examples to calculate and compare merits and demerits, as well as applicational scenarios, of three methods for calculating reproduction numbers.

Method

The epidemiological characteristics of the COVID-19 outbreaks are described. Through the definition method, the next-generation matrix-based method, and the epidemic curve and serial interval (SI)-based method, corresponding reproduction numbers were obtained and compared.

Results

Reproduction numbers (R eff ), obtained by the definition method of the four regions, are 1.20, 1.14, 1.66, and 1.12. Through the next generation matrix method, in region H R eff = 4.30, 0.44; region P R eff = 6.5, 1.39, 0; region X R eff = 6.82, 1.39, 0; and region Z R eff = 2.99, 0.65. Time-varying reproduction numbers (R t ), which are attained by SI of onset dates, are decreasing with time. Region H reached its highest R t = 2.8 on July 29 and decreased to R t < 1 after August 4; region P reached its highest R t = 5.8 on September 9 and dropped to R t < 1 by September 14; region X had a fluctuation in the R t and R t < 1 after September 22; R t in region Z reached a maximum of 1.8 on September 15 and decreased continuously to R t < 1 on September 19.

Conclusion

The reproduction number obtained by the definition method is optimal in the early stage of epidemics with a small number of cases that have clear transmission chains to predict the trend of epidemics accurately. The effective reproduction number R eff , calculated by the next generation matrix, could assess the scale of the epidemic and be used to evaluate the effectiveness of prevention and control measures used in epidemics with a large number of cases. Time-varying reproduction number R t , obtained via epidemic curve and SI, can give a clear picture of the change in transmissibility over time, but the conditions of use are more rigorous, requiring a greater sample size and clear transmission chains to perform the calculation. The rational use of the three methods for reproduction numbers plays a role in the further study of the transmissibility of COVID-19.

Testing and Isolation Efficacy: Insights from a Simple Epidemic Model

Testing individuals for pathogens can affect the spread of epidemics. Understanding how individual-level processes of sampling and reporting test results can affect community- or population-level spread is a dynamical modeling question. The effect of testing processes on epidemic dynamics depends on factors underlying implementation, particularly testing intensity and on whom testing is focused. Here, we use a simple model to explore how the individual-level effects of testing might directly impact population-level spread. Our model development was motivated by the COVID-19 epidemic, but has generic epidemiological and testing structures. To the classic SIR framework we have added a per capita testing intensity, and compartment-specific testing weights, which can be adjusted to reflect different testing emphases-surveillance, diagnosis, or control. We derive an analytic expression for the relative reduction in the basic reproductive number due to testing, test-reporting and related isolation behaviours. Intensive testing and fast test reporting are expected to be beneficial at the community level because they can provide a rapid assessment of the situation, identify hot spots, and may enable rapid contact-tracing. Direct effects of fast testing at the individual level are less clear, and may depend on how individuals' behaviour is affected by testing information. Our simple model shows that under some circumstances both increased testing intensity and faster test reporting can reduce the effectiveness of control, and allows us to explore the conditions under which this occurs. Conversely, we find that focusing testing on infected individuals always acts to increase effectiveness of control.

Deathdaily: A Python Package Index for predicting the number of daily COVID-19 deaths

This paper presents a new open-source program called deathdaily for predicting the number of daily COVID-19 deaths in the next 7 days. The predictions can be used by policymakers to determine whether current policies should be strengthened/mitigated or new policies should be challenged to mitigate the COVID-19 pandemic. Although vaccines have been mitigating the pandemic initially, the recent resurgence with new variants has been observed in many vaccinated countries. This paper shows how to use deathdaily to detect symptoms of resurgence. The proposed deathdaily is available in public and can be installed by a Python package manager PyPI. The deathdaily has been downloaded by 15,964 users worldwide, according to https://pepy.tech/project/deathdaily. The fact shows that the applicability, practicality, and usefulness of the proposed program have been duly evaluated.

Modeling the evolution of SARS-CoV-2 under non-pharmaceutical interventions and testing

Background and objectives

Social and behavioral non-pharmaceutical interventions (NPIs), such as mask-wearing, social distancing and travel restrictions, as well as diagnostic tests, have been broadly implemented in response to the COVID-19 pandemic. Epidemiological models and data analysis affirm that wide adoption of NPIs helps to control the pandemic. However, SARS-CoV-2 has extensively demonstrated its ability to evolve. Therefore, it is crucial to examine how NPIs may affect the evolution of the virus. Such evolution could have important effects on the spread and impact of the pandemic.

Methodology

We used evo-epidemiological models to examine the effect of NPIs and testing on two evolutionary trajectories for SARS-CoV-2: attenuation and test evasion.

Results

Our results show that when stronger measures are taken, selection may act to reduce disease severity. Additionally, the timely application of NPIs could significantly affect the competition between viral strains, favoring the milder strain. Furthermore, a higher testing rate can select for a test-evasive viral strain, even if that strain is less infectious than the detectable competing strain. Importantly, if a less detectable strain evolves, epidemiological metrics such as confirmed daily cases may distort our assessment of the pandemic.

Conclusions and implications

Our results highlight the important implications NPIs can have on the evolution of SARS-CoV-2.

Lay Summary

We used evo-epidemiological models to examine the effect of non-pharmaceutical interventions and testing on two evolutionary trajectories for SARS-CoV-2: attenuation and test evasion. Our results show that when stronger measures are taken, selection may act to reduce disease severity.

Lessons from a pandemic

Several interventions have been used around the world trying to contain the SARS-CoV-2 pandemic, such as quarantine, prohibition of mass demonstrations, isolation of sick people, tracing of virus carriers, semi-containment, promotion of barrier gestures, development of rapid self-tests and vaccines among others. We propose a simple model to evaluate the potential impact of such interventions. A model for the reproduction number of an infectious disease including three main contexts of infection (indoor mass events, public indoor activities and household) and seven parameters is considered. We illustrate how these parameters could be obtained from the literature or from expert assumptions, and we apply the model to describe 20 scenarios that can typically occur during the different phases of a pandemic. This model provides a useful framework for better understanding and communicating the effects of different (combinations of) possible interventions, while encouraging constant updating of expert assumptions to better match reality. This simple approach will bring more transparency and public support to help governments to think, decide, evaluate and adjust what to do during a pandemic.

Estimating the impact of interventions against COVID-19: From lockdown to vaccination

Coronavirus disease 2019 (COVID-19) is an infectious disease of humans caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Since the first case was identified in China in December 2019 the disease has spread worldwide, leading to an ongoing pandemic. In this article, we present an agent-based model of COVID-19 in Luxembourg, and use it to estimate the impact, on cases and deaths, of interventions including testing, contact tracing, lockdown, curfew and vaccination. Our model is based on collation, with agents performing activities and moving between locations accordingly. The model is highly heterogeneous, featuring spatial clustering, over 2000 behavioural types and a 10 minute time resolution. The model is validated against COVID-19 clinical monitoring data collected in Luxembourg in 2020. Our model predicts far fewer cases and deaths than the equivalent equation-based SEIR model. In particular, with R0 = 2.45, the SEIR model infects 87% of the resident population while our agent-based model infects only around 23% of the resident population. Our simulations suggest that testing and contract tracing reduce cases substantially, but are less effective at reducing deaths. Lockdowns are very effective although costly, while the impact of an 11pm-6am curfew is relatively small. When vaccinating against a future outbreak, our results suggest that herd immunity can be achieved at relatively low coverage, with substantial levels of protection achieved with only 30% of the population fully immune. When vaccinating in the midst of an outbreak, the challenge is more difficult. In this context, we investigate the impact of vaccine efficacy, capacity, hesitancy and strategy. We conclude that, short of a permanent lockdown, vaccination is by far the most effective way to suppress and ultimately control the spread of COVID-19.

Modelling Heterogeneity and Super Spreaders of the COVID-19 Spread through Malaysian Networks

Malaysia is multi-ethnic and diverse country. Heterogeneity, in terms of population interactions, is ingrained in the foundation of the country. Malaysian policies and social distancing measures are based on daily infections and R0 (average number of infections per infected person), estimated from the data. Models of the Malaysian COVID-19 spread are mostly based on the established SIR compartmental model and its variants. These models usually assume homogeneity and symmetrical full mixing in the population; thus, they are unable to capture super-spreading events which naturally occur due to heterogeneity. Moreover, studies have shown that when heterogeneity is present, R0 may be very different and even possibly misleading. The underlying spreading network is a crucial element, as it introduces heterogeneity for a more representative and realistic model of the spread through specific populations. Heterogeneity introduces more complexities in the modelling due to its asymmetrical nature of infection compared to the relatively symmetrical SIR compartmental model. This leads to a different way of calculating R0 and defining super-spreaders. Quantifying a super-spreader individual is related to the idea of importance in a network. The definition of a super-spreading individual depends on how super-spreading is defined. Even when the spreading is defined, it may not be clear that a single centrality always correlates with super-spreading, since centralities are network dependent. We proposed using a measure of super-spreading directly related to R0 and that will give a measure of ‘spreading’ regardless of the underlying network. We captured the vulnerability for varying degrees of heterogeneity and initial conditions by defining a measure to quantify the chances of epidemic spread in the simulations. We simulated the SIR spread on a real Malaysian network to illustrate the effects of this measure and heterogeneity on the number of infections. We also simulated super-spreading events (based on our definition) within the bounds of heterogeneity to demonstrate the effectiveness of the newly defined measure. We found that heterogeneity serves as a natural curve-flattening mechanism; therefore, the number of infections and R0 may be lower than expected. This may lead to a false sense of security, especially since heterogeneity makes the population vulnerable to super-spreading events.

A comprehensive estimation of country-level basic reproduction numbers R0 for COVID-19: Regime regression can automatically estimate the end of the exponential phase in epidemic data

In a compartmental epidemic model, the initial exponential phase reflects a fixed interaction between an infectious agent and a susceptible population in steady state, so it determines the basic reproduction number R₀ on its own. After the exponential phase, dynamic complexities like societal responses muddy the practical interpretation of many estimated parameters. The computer program ARRP, already available from sequence alignment applications, automatically estimated the end of the exponential phase in COVID-19 and extracted the exponential growth rate r for 160 countries. By positing a gamma-distributed generation time, the exponential growth method then yielded R₀ estimates for COVID-19 in 160 countries. The use of ARRP ensured that the R₀ estimates were largely freed from any dependency outside the exponential phase. The Prem matrices quantify rates of effective contact for infectious disease. Without using any age-stratified COVID-19 data, but under strong assumptions about the homogeneity of susceptibility, infectiousness, etc., across different age-groups, the Prem contact matrices also yielded theoretical R₀ estimates for COVID-19 in 152 countries, generally in quantitative conflict with the R₀ estimates derived from the exponential growth method. An exploratory analysis manipulating only the Prem contact matrices reduced the conflict, suggesting that age-groups under 20 years did not promote the initial exponential growth of COVID-19 as much as other age-groups. The analysis therefore supports tentatively and tardily, but independently of age-stratified COVID-19 data, the low priority given to vaccinating younger age groups. It also supports the judicious reopening of schools. The exploratory analysis also supports the possibility of suspecting differences in epidemic spread among different age-groups, even before substantial amounts of age-stratified data become available.

Nonlinear Dynamics of the Introduction of a New SARS-CoV-2 Variant with Different Infectiousness

Several variants of the SARS-CoV-2 virus have been detected during the COVID-19 pandemic. Some of these new variants have been of health public concern due to their higher infectiousness. We propose a theoretical mathematical model based on differential equations to study the effect of introducing a new, more transmissible SARS-CoV-2 variant in a population. The mathematical model is formulated in such a way that it takes into account the higher transmission rate of the new SARS-CoV-2 strain and the subpopulation of asymptomatic carriers. We find the basic reproduction number R0 using the method of the next generation matrix. This threshold parameter is crucial since it indicates what parameters play an important role in the outcome of the COVID-19 pandemic. We study the local stability of the infection-free and endemic equilibrium states, which are potential outcomes of a pandemic. Moreover, by using a suitable Lyapunov functional and the LaSalle invariant principle, it is proved that if the basic reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable. Our study shows that the new more transmissible SARS-CoV-2 variant will prevail and the prevalence of the preexistent variant would decrease and eventually disappear. We perform numerical simulations to support the analytic results and to show some effects of a new more transmissible SARS-CoV-2 variant in a population.

A Review of Matrix SIR Arino Epidemic Models

Many of the models used nowadays in mathematical epidemiology, in particular in COVID-19 research, belong to a certain subclass of compartmental models whose classes may be divided into three “(x,y,z)” groups, which we will call respectively “susceptible/entrance, diseased, and output” (in the classic SIR case, there is only one class of each type). Roughly, the ODE dynamics of these models contains only linear terms, with the exception of products between x and y terms. It has long been noticed that the reproduction number R has a very simple Formula in terms of the matrices which define the model, and an explicit first integral Formula is also available. These results can be traced back at least to Arino, Brauer, van den Driessche, Watmough, and Wu (2007) and to Feng (2007), respectively, and may be viewed as the “basic laws of SIR-type epidemics”. However, many papers continue to reprove them in particular instances. This motivated us to redraw attention to these basic laws and provide a self-contained reference of related formulas for (x,y,z) models. For the case of one susceptible class, we propose to use the name SIR-PH, due to a simple probabilistic interpretation as SIR models where the exponential infection time has been replaced by a PH-type distribution. Note that to each SIR-PH model, one may associate a scalar quantity Y(t) which satisfies “classic SIR relations”,which may be useful to obtain approximate control policies.

Estimating the impact of interventions against COVID-19: From lockdown to vaccination

Coronavirus disease 2019 (COVID-19) is an infectious disease of humans caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Since the first case was identified in China in December 2019 the disease has spread worldwide, leading to an ongoing pandemic. In this article, we present an agent-based model of COVID-19 in Luxembourg, and use it to estimate the impact, on cases and deaths, of interventions including testing, contact tracing, lockdown, curfew and vaccination. Our model is based on collation, with agents performing activities and moving between locations accordingly. The model is highly heterogeneous, featuring spatial clustering, over 2000 behavioural types and a 10 minute time resolution. The model is validated against COVID-19 clinical monitoring data collected in Luxembourg in 2020. Our model predicts far fewer cases and deaths than the equivalent equation-based SEIR model. In particular, with R0 = 2.45, the SEIR model infects 87% of the resident population while our agent-based model infects only around 23% of the resident population. Our simulations suggest that testing and contract tracing reduce cases substantially, but are less effective at reducing deaths. Lockdowns are very effective although costly, while the impact of an 11pm-6am curfew is relatively small. When vaccinating against a future outbreak, our results suggest that herd immunity can be achieved at relatively low coverage, with substantial levels of protection achieved with only 30% of the population fully immune. When vaccinating in the midst of an outbreak, the challenge is more difficult. In this context, we investigate the impact of vaccine efficacy, capacity, hesitancy and strategy. We conclude that, short of a permanent lockdown, vaccination is by far the most effective way to suppress and ultimately control the spread of COVID-19.