Modelling a pandemic: Asymptomatic Transmission
Part 4 of a layman's guide to modelling a pandemic.
Introduction.
This is the fourth in a series of articles introducing the Susceptible-Infectious-Recovered (S-I-R) compartmental model. This class of models was used by various expert groups to explore transmission dynamics and help decision-makers during the early stages of the Covid pandemic. For newcomers to my Substack, the first article of the series is available here, the second article here, and the third here.
The next article in this series looks at modelling policies to manage a pandemic through the isolation of symptomatic individuals and can be accessed here.
The aim is to offer a guide for the layman, focusing on simplicity over technical complexity, hence any mathematics will be kept to a bare minimum. As in previous articles, I provide links to on-line models for you to explore. Interacting with these models is a very good way to understand them, and I would encourage you to experiment with the simple models through the links provided.
This article completes the S-I-R model for infectious diseases transmitted through close contact by incorporating the asymptomatic infectious population. The updated S-I-R model is then used to demonstrate how the different biological factors of a variety of respiratory diseases, including Covid, affect their pandemic potential.
Recap.
The first article in this series on modelling a pandemic, focussed on the Susceptible-Infectious (S-I) component of the S-I-R model using a System Dynamics ‘Stock and Flow’ model. The model showed how the progress of an epidemic infection depends on the balance between a reinforcing contagion feedback loop and a balancing depletion loop.
The second article, called ‘Time to Recover’, completed the simple S-I-R model by adding the recovered population who after infection retained full immunity. Making this addition changed the dynamics of the simple S-I-R model, leading to the important finding that population immunity, also know as 'herd' immunity, can halt an outbreak before it infects all susceptible individuals.
The third article posted, called ‘Infection Matters’ highlighted the importance of the timeline of infectiousness and the onset of symptoms which plays a critical role in disease outbreaks and the policies required to bring them under control. The simple S-I-R model was enhanced to include an incubation period where individuals were infectious but did not show symptoms to better represent this important factor for some diseases.
The following image shows the stock and flow diagram for this enhanced S-I-R model which includes a new stock representing the Incubating Infectious population.
In the following sections we will complete the S-I-R transmission model, but first a cautionary tale to set the scene.
Typhoid Mary.
During the summer of 1906, an isolated outbreak of typhoid flared up in the household of wealthy family of New York banker Charles Henry Warren, at a rented summer home on the northern coast of Long Island. By late August, six of the eleven people in the house were sick with typhoid fever. The owner of the summer house called in public health experts to investigate the outbreak who first blamed a bad batch of clams. However, some of the outbreak's victims hadn't eaten the clams, leading the investigators to look for a human carrier. Eventually, they narrowed the potential culprits down to the cook, Mary Mallon, who had left the family's service shortly after the outbreak.
Mallon was a difficult woman to find, because she moved around so much. During the search, several of her previous employers reported that she had left a series of household typhoid outbreaks in her wake. When she was eventually found, despite being in perfect health, she tested positive for typhoid. The newspapers quickly picked up on the story and gave her the graphic headline name of ‘Typhoid Mary’ which has now become synonymous for any asymptomatic carrier of a disease.
Modelling asymptomatic transmission.
The story of ‘Typhoid Mary’ shows that the important role of asymptomatic transmission in driving disease outbreaks has been known for a long time. Although unaffected by the pathogen and showing no symptoms, some carriers can transmit it to susceptible individuals. Asymptomatic carriers play a critical role in the transmission of common infectious diseases such as typhoid, cholera, tuberculosis, and Covid.
In the absence of vaccines and mass testing, the onset of symptoms to identify individuals for isolation plays a vital role to suppress an outbreak. Clearly any disease which has asymptomatic carriers will require different interventions and, consequently, should be included in the S-I-R model.
The following image shows the stock and flow diagram for the S-I-R model which has been enhanced to include asymptomatic transmission. Although the picture has more elements and looks busier the basic structure of the S-I-R model remains unchanged.
The diagram shows that there are now two flows rates going from the Incubating Infectious stock. The Asymptomatic Cases per Day flow rate moves individuals to the Asymptomatic Infectious stock and the Symptomatic Cases per Day flow moves individuals to the Symptomatic Infectious stock. In both cases individuals spend the same average time incubating the infection as determined by the Duration of Incubation variable. For simplicity, the recovery times for both asymptomatic and symptomatic is assumed to be the same and is the Duration of Recovery variable.
A new Percent Asymptomatic variable is used to establish the mix of asymptomatic and symptomatic individuals flowing into their respective stocks. When Percent Asymptomatic is 0% then all incubating individuals are symptomatic, whereas setting this variable to 20% gives a mix of 80% symptomatic and 20% asymptomatic.
Finally, infectivity for incubating or asymptomatic individuals often differ from those showing symptoms. This is modelled by applying the new Incubating infectious Factor and Asymptomatic Infection Factor as multipliers to the Total Asymptomatic Infectious population. When these factors are set to 1.0 then individuals have the same infectivity as symptomatic individuals, but if set to 0.8 then they are 80% as infectious.
In general, asymptomatic individuals are less infectious than those showing symptoms and, consequently, the factor is set to less than 1.0. However, certain infectious diseases are more contagious during the incubation stage, and the Incubating Infection Factor is adjusted to be greater than 1.0.
We have now completed the core S-I-R model, which will be extended in subsequent articles to examine the impact on health systems and assess the effectiveness of policies designed to control pandemics.
The S-I-R Model: Limits and Usefulness.
“All models are wrong, but some are useful” – is a quote from the statistician, George Box, that should be the guiding principle for any model builder. Box used the phrase to refer to the limitations of models, arguing that while no model is ever completely accurate, simpler models can still provide valuable insights if applied judiciously. The Susceptible-Infected-Recovered (S-I-R) model, a cornerstone of epidemiology, illustrates this principle. But how ‘wrong’ is it and how ‘useful’ can it be?
While the S-I-R model has limitations, such as assuming everyone has the same number of daily contacts and not considering the latent period between exposure to a disease and the onset of infection, its important role in epidemiology is well-established. This is because it has proved to be reliable due to:
Simplified Representation: The model divides a population into three compartments: susceptible (S), infected (I), and recovered (R). This simplification allows for a clear analysis of disease dynamics without unnecessary complexity.
Mathematical Rigor: The model is grounded in equations that describe the rates at which individuals transition between compartments. This mathematical framework provides predictions about disease progression.
Historical Validation: Since its introduction by Kermack and McKendrick in 1927, the S-I-R model has been applied to various infectious diseases, demonstrating its ability to capture essential aspects of disease outbreaks.
In addition, the S-I-R model has also been most ‘useful’ through its:
Predictive Accuracy: Despite its simplicity, the SIR model has been effective in forecasting disease trends, including during the Covid pandemic.
Impact on Policy: The model has been used to inform vaccination strategies (e.g. measles), plan healthcare requirements (e.g. during Flu outbreaks), and design disease eradication strategies (e.g. polio).
Adaptability: The model can be extended to include additional compartments or factors, such as latency periods or varying contact rates, enhancing its applicability to a wide range of diseases.
In future articles, we will examine the predictive accuracy of the S-I-R model and its impact on policy. However, the remaining sections of this article, will explore the adaptability of the S-I-R model.
One model - many infectious diseases.
An important consideration when building any model is to make it as flexible as possible. For example, the models used to help inform policy decisions at the start of the Covid pandemic were not built from scratch. Rather existing models of disease transmission were rapidly adapted to model Covid. Our completed S-I-R model demonstrates how this can be achieved by distinguishing the critical social and biological variables that govern a disease outbreak from the basic logic that models the transmission mechanism.
So far, in these articles, we have run the model using the default input variables defined in the base line scenario. While these settings represent a typical infectious disease, we are now able to adjust the values of these input variables to run the model to show the transmission behaviour of different infectious diseases.
The following image shows the interface screen for the completed S-I-R model.
As before, the main panel shows the logic of the model with the disease input variables depicted as light brown circles or ellipses (e.g. Duration of Incubation). These input variables can be changed using the sliders in the scrollable panel on the right hand side of the screen. Alternatively, you can select one of the pre-configured scenarios from the pull down menu that represent different infectious diseases. The screen shot shows that the ‘Covid: Original Strain’ scenario has been selected.
The model has five pre-set scenarios available representing the Covid Original Wuhan strain, Seasonal Flu, the 2009 H1N1 Swine Flu, SARS, and MERS. The table below gives the values assigned to each variable for the scenarios.
These estimates provide a general guide, but the factors can vary widely depending on the environment (e.g. housing density), demographics (e.g. population health and age), and quality of healthcare.
For context, the estimated infection hospitalization and infection fatality rates are included to show the severity of each virus, along with an assessment of their relative risk profiles. The table illustrates that it is the combination of transmissibility and severity that results in the highest level of risk.
You can now run the S-I-R model with the pre-set scenarios by clicking on the button below to see the model results for these recent pandemic viruses.
A full description of how to run the model is provided in Appendix 2.
What can we learn from the model?
The following panel chart illustrates the models results for the number of daily infections for Covid, SARS, Swine Flu, and Seasonal Flu. MERS has been excluded because it has an R0 less than one, indicating that it is unable to sustain growth in the general population.
The charts indicates that the trend for Covid daily infections differs significantly from other diseases, showing the fastest growth and the earliest and highest peak, thereby making it particularly challenging to control. Remember that the model does not yet include any control measures, so these numbers represent an uncontrolled pandemic.
Interestingly, the modelled results for Covid and SARS show very different trends despite having similar levels of infectivity. However, individuals with Covid are infectious during the incubation period, whereas individuals with SARS do not become infectious until after the incubation period. In addition, asymptomatic transmission is slightly higher for Covid. This factors when combined with the different durations of incubation and (a)symptomatic infection helps explain the difference.
The two flu illnesses were also infectious during the incubation and asymptomatic recover period. However, while Swine Flu has a comparable level of infectivity to Covid, individuals were infectious for a shorter period. In contrast, Seasonal Flu infectivity was lower than for Covid as well as individuals being infectious for a shorter duration. These factors help explain why the two flu illnesses progressed more slowly and had lower peaks.
In addition to examining the infection rate when evaluating the impact of a pandemic, it’s important to consider the number of individuals infected who subsequently recovered . The following chart illustrates the models output for the cumulative number of people infected by day for Covid, SARS, Swine Flu, and Seasonal Flu. Remember that this model is for an uncontrolled pandemic.
This chart demonstrates that although Covid and SARS, both coronaviruses, follow very different paths, they infect a similar number individuals. In contrast, the influenza viruses Swine Flu and Seasonal Flu infect fewer people.
In summary the model demonstrates that even small differences in the biological factors of a virus can significantly impact its pandemic potential.
In conclusion.
In this article we completed the S-I-R model by adding asymptomatic transmission creating a generic model that can be readily configured to model different pandemic viruses.
The model was used to demonstrate the varying pandemic potentials of the original strain of Covid, SARS, Swine Flu, and a typical Seasonal Flu in the absence of any control policies. Covid daily infections were found to accelerate the fastest, peaked the highest, and occurred the earliest, presenting the greatest control challenge.
Having now completed the S-I-R transmission model we can now move on to consider pandemic control policies. In the next article the S-I-R model will be enhanced to include policies to quarantine symptomatic individuals and isolate incubating and asymptomatic individuals.
Finally, although this article is longer than I intended I trust that you have found it of interest. As always, please ask any questions or leave your comments below.
Appendix 1. Sources
The following are links to the main source material used in this article.
Kiona N. Smith, Forbes (Sep 2017)
An Introduction to Deterministic Infectious Disease Models
World Bank, 2015
Modelling Epidemics With Compartmental Models
Juliana Tolles and ThaiBinh Luong, JAMA Guide to Statistics and Methods, 2020
Neil M Ferguson et al, Imperial College Report, 2020
J Eisenberg, University of Michigan, The Conversation, 2020
Zhonglan Wu et al, Reviews in Medical Virology, 2020
Jessica E. Biddle et al, medRxiv (pre-print), 2024
The role of asymptomatic infections in influenza transmission: what do we really know
Martha P Montgomery et al, The Lancet, 2024
Appendix 2. Overview of the InsightMaker Interface
This section describes in more detail the InsightMaker screen interface using the simple S-I model as an example. The following image shows the landing page for the model. Click on the image and select full width from the … menu to view the detail.
The screen features three sections outlined in red. The main portion of the screen presents a diagram depicting the model's logic as a stock and flow diagram. The two boxes are the susceptible and infectious population stocks, linked by a flow, indicated by a blue arrow, labelled 'Infections per Day'. The circles represent the variables of the model that determine the 'Infections per Day' rate.
The right-hand panel has a dual function. The default setting, shown in the image above, displays information about the model and allows the user to adjust the Contact Rate or Infectivity using sliders. It also allows you to choose from a set of scenarios that will automatically adjust the levels of the default settings.
The second function is to allow the user to examine the model's logic in more detail. By selecting any of the stocks, flow rates, or variables, the panel shifts to present comprehensive information about the chosen building block. To revert to the default display, click anywhere on the main portion of the screen that is not a building block.
Finally, the area along the top of the screen allows the user to change the length of the simulation run and, most importantly, run the model by pressing the blue ‘SIMULATE’ button.
Thanks for the effort, but it does not account for the evolving COVID-19 variant strains, many of which are swirling around in particularly more dense populations, including the willfully unvaxxed, and the now well known sequestration capacity of COVID. Thus, many of us unknowingly harbor virus and this may account for LONG COVID.