If you would like a quick refresher before we get started, check out our previous posts in this series (especially the first 2):

1. Epidemic Modeling 101: Or why your CoVID19 exponential fits are wrong

2. All CoVID-19 models are wrong, but some are useful

3. Adding confidence intervals and stochastic effects to your CoVID-19 Models

4. Impact of Seasonal effects on CoVID-19

5. Competing CoVID-19 Strains

6. Network Structure, Super-spreaders and Contact Tracing

7. Network Models, the effect of degree correlations

8. How to model the effects of vaccination

9. The Impact of Age Structure on Epidemic Spreading

10. Meta-population Models

In all the models we have considered so far, we have kept the total population constant by making sure that all transitions between compartments balanced out. For e…

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Features

• Homogeneously mixed Model: Easily define compartmental models by just specifying the transitions and rates

• Seasonality: Allow parameters to evolve in time

• Age Structure: Convert any compartmental model to an age-structured model by simply adding an age contact matrix and a the population of each age group

• Network Model: Easily define epidemic models running over an arbitrary network structure

• Meta-Population Models: Define metapopulation model connecting subpopulations through a travel matrix

Installation

You can install the latest version of Epidemik using pip:

pip install epidemik

Alternatively, you can download the source code from the GitHub repository and install it manually.

Documentation and Support

Checkout the documentation o…

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Welcome to the 10th post of the Epidemic Modeling series. In this post, we go beyond single populations and start modeling the spread of a disease through multiple coupled populations. Our fundamental assumptions are that the disease structure is the same everywhere and that individuals can travel between any two populations with a given fixed probability.

If you would like a quick refresher before we get started, check out our previous posts in this series (especially the first 2):

1. Epidemic Modeling 101: Or why your CoVID19 exponential fits are wrong

2. All CoVID-19 models are wrong, but some are useful

3. Adding confidence intervals and stochastic effects to your CoVID-19 Models

4. Impact of Seasonal effects on CoVID-19

5. Competing CoVID-19 Strains

6. Network Structure, Super-spreaders and Contact Tracing

7. Network Models, the effect of degree correlations

8. How to model the effects of vaccination

9. The Impact of Age Structure on Epidemic Spreading

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Welcome to the latest post on our long-running epidemic modeling series. In this post, we’ll cover how to introduce age structure into our compartmental models and its impact on how an epidemic spreads.

If you need a quick refresher, check out our previous posts in this series (especially the first 2):

1. Epidemic Modeling 101: Or why your CoVID19 exponential fits are wrong

2. All CoVID-19 models are wrong, but some are useful

3. Adding confidence intervals and stochastic effects to your CoVID-19 Models

4. Impact of Seasonal effects on CoVID-19

5. Competing CoVID-19 Strains

6. Network Structure, Super-spreaders and Contact Tracing

7. Network Models, the effect of degree correlations

8. How to model the effects of vaccination

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Welcome to the latest post of the “Epidemic Modeling” series where I aim to guide you through the mathematical and conceptual details underlying epidemic modeling with a focus on what’s most relevant to the current CoVID-19 pandemic.

In this post, we explore some simple network models in order to better understand the role that degree correlations play in disease spreading. I aim to make each post self contained, but you might want to check out our previous posts to brush up on the basics of epidemic modeling:

• Epidemic Modeling 101: Or why your CoVID-19 exponential fits are wrong

• Epidemic Modeling 102: All CoVID-19 models are wrong, but some are useful

• Epidemic Modeling 103: Adding confidence intervals and stochastic effects to your CoVID-19 Models

• Epidemic Modeling 104: Impact of Seasonal effects on CoVID-19

• Epidemic Modeling 105: Competing CoVID-19 Strains

• Epidemiology 201: Network Structure, Super-spreaders and Contact Tracing

• Epidemiology 301: How to model the effects of vaccination

Each p…

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The largest vaccination campaign in history continues around the world with millions of new doses distributed everyday. In the US, the number of doses peaked at over 4 million per day in mid April and has been steadily declining since:

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Welcome to the first 2021 post of the “Epidemic Modeling” series where I aim to guide you through the mathematical and conceptual details underlying epidemic modeling with a focus on what’s most relevant to the current CoVID-19 pandemic.

While I strive to make each post self contained, this post does build on previous ones that you might want to go back to the first one before getting started:

• Epidemic Modeling 101: Or why your CoVID-19 exponential fits are wrong

You can find the complete list of posts in this series at the bottom.

Each post also comes with a Jupyter notebook detailing all the analysis and calculations. You can find them in the companion GitHub repository

Epidemiology 101 GitHub

These are simply my personal views and opinions, as an individual with some background real world experience in this fascinating field and shouldn’t reflect on any group or institution I might be affiliated with.

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On December 1st, 2019, the first few patients in Wuhan province, China started displaying symptoms. By Dec 18 there had been 9 total cases detected, 2 of which were connected to the Wuhan Seafood market. Less than a month later, on Jan 9, 2020 the discovery of a new Coronavirus was announced.

In the months since, the pandemic went through several phases in its spread around the world and it now reaches over 170 countries and regions:

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This is the latest of a series of blog posts analyzing CoVID-19 from both the data and model perspective. Check out Part I, II, III, IV, V and VI and the GitHub repository with all the underlying Python code:

Epidemiology 101 GitHub

You can also sign up for my newsletter to be kept up to date on future posts.

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As the number of people affected by the epidemic started increasing so did the public’s interest, resulting in a large number of public datasets, visualization, hackathons, forecasting models, etc. We’ve already covered in detail, the structure of simple epidemic models here, here, here and here, as well as background information specifically to CoVID-10 here.

In this post, we take a dive into the public data and on how we may visualize it to gain a better understanding of how it spread around the world. Among all the publicly available data sources, the a GitHub repository curated by the Center for Systems Science and Engineering (CSSE) at Jonhs Hopkins University has steadily become the gold standard data source and the one we will use for this post.

Geographical Spread

After starting in Wuhan, China, CoVID-19 has gradually spread around the world, and is currently (May 31, 2020) present in 6 continents (sorry, Antarctica).

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Viruses are fascinating things: Tiny, micron-sized, capsules containing RNA or DNA code whose sole purpose is to hijack the internal mechanisms of living cells to create thousands of copies of itself. They are the simplest of machines, not even counting as being alive according to most definitions. Their simplicity is their most powerful weapon. They do a single thing and do it exceedingly well: create more copies of themselves. To achieve this, each virus specializes in invading and highjacking the internal machinery of a specific type of cells in a specific animal (or plant, or bacteria, etc) species.

When a cell becomes infected, it diverts all of its internal processes to produce thousands of copies of the virus before “exploding” releasing the viruses to go on to infect other cells. As infected cells typically die resulting in sometimes significant damage and even the death of the host organism. Some viruses are so successful that they eventually become an intrinsic part of the ge…

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This is the latest post of the “Epidemic Modeling” series where I aim to guide you through the mathematical and conceptual details underlying epidemic modeling with a focus on what’s most relevant to the current CoVID-19 pandemic. While I aim to make each post self contained, but you might want to check out our previous posts:

• Epidemic Modeling 101: Or why your CoVID-19 exponential fits are wrong

• Epidemic Modeling 102: All CoVID-19 models are wrong, but some are useful

• Epidemic Modeling 103: Adding confidence intervals and stochastic effects to your CoVID-19 Models

Each post comes with a corresponding Jupyter notebook that implements the models and generates the figures. You can find them over at the GitHub repository dedicated to this series:

Epidemiology 101 GitHub

What follows are my personal views and opinions, as an individual with some background real world experience in this fascinating field and shouldn’t reflect on any group or institution I might be affiliated with.

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This is the third post of the “Epidemic Modeling” series. We will be building up on our discussion from the first two posts:

• Epidemic Modeling 101: Or why your CoVID-19 exponential fits are wrong

• Epidemic Modeling 102: All CoVID-19 models are wrong, but some are useful

so you might want to start reading there.

The next post in the series has already been published, and you might want to check it out:

• Epidemic Modeling 104: Impact of Seasonal effects on CoVID-19

The jupyter notebooks I wrote to implement the models and generate the figures can be found over at the GitHub repository dedicated to this series:

Epidemiology 101 GitHub

What follows is my personal perspective, as an individual with some real world experience in epidemic modeling during previous pandemics and shouldn’t reflect on any group or institution I might be affiliated with.

So, without further ado…

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This is the second post of the “Epidemic Modeling” series. We will be building up on our discussion from the first post, “Epidemic Modeling 101: Or why your CoVID-19 exponential fits are wrong”, so you might want to start reading there. You might also find a follow up posts interesting

• Epidemic Modeling 103: Adding confidence intervals and stochastic effects to your CoVID-19 Models

• Epidemic Modeling 104: Impact of Seasonal effects on CoVID-19

The notebooks I wrote to implement the models and generate the figures over at the GitHub repository I made specifically for this series:

Epidemiology 101 GitHub

What follows is my personal perspective, as an individual with some real world experience in epidemic modeling during previous pandemics and shouldn’t reflect on any group or institution I might be affiliated with.

So, without further ado…

Models vs the real world

As George E. P. Box, a statistician, famously said “all models are wrong, but some are useful”. This is perhaps never more true than during a crisis. Information is limited, often wrong, but decisions must be made and implemented based on what is known at the time.

It is also during an ongoing crisis that models play their most fundamental role, that of allowing us to explore scenarios and work through the consequences of our decisions:

However, care must be taken to avoid mistaking the model for the reality. After all, “the map is not the territory”. The development of a model, regardless of the domain of application, typically follows a common pattern:

A simplified version of the world is created, to which a specific modeling approach can be applied, resulting in a working model. The simplifications made can be due to a variety of factors, such as the lack of specific data, excessive complexity, and intractability, among others. The modeling approach chosen is both influenced by and helps drive the assumptions that are made, often resulting in the stereotypical overabundance of Physicists concerned about Spherical Cows or trying to apply Ising Spins to every possible problem.

Once a working model is obtained, we can use it to explore scenarios, the consequences of specific decisions, etc. Finally, it is by studying the scenarios that are outputted by our models that decisions are taken in the Real World. Graphically, we have:

Naturally, this is a simplified and schematic view (and a model in and of itself) to help illustrate the various points at which our modeling efforts can go awry, leading the results of our models to differ from what we actually observe in the real world.

While in many cases, mismatches between the model and reality can be traced back to errors made during the process, they can also be due to the fact that our model was successful and it resulted in appropriate measures being taken to prevent the undesirable scenarios it predicted. This is specially true in the case of highly visible models that are used to guide government decisions and interventions such as in the case of an ongoing pandemic like the one we’re living through now:

“The most important function of epidemiological models is as a simulation, a way to see our potential futures ahead of time, and how that interacts with the choices we make today. With COVID-19 models, we have one simple, urgent goal: to ignore all the optimistic branches and that thick trunk in the middle representing the most likely outcomes. Instead, we need to focus on the branches representing the worst outcomes, and prune them with all our might. Social isolation reduces transmission, and slows the spread of the disease. In doing so, it chops off branches that represent some of the worst futures. Contact tracing catches people before they infect others, pruning more branches that represent unchecked catastrophes.” — Zeynep Tufecki, The Atlantic

It is this kind of misunderstanding that leads to public distrust in scientific models in particular and Science in general.

My hope is that this (and many other posts out there) can help the general public to understand the underlying assumptions, power, and limitations of scientific models and how they can be put to good use.

Susceptible-Infectious-Recovered (SIR) Model

Now that we have established both the advantages and limitations of using models to understand the world, we can start exploring how to improve the simple models we introduced in the previous post.

The SIR model is one of the simplest and best known epidemic models. Its popularity is due, in no small part, to its ability to establish a perfect balance between simplicity and usefulness. It is still relatively amenable to mathematical and analytical exploration while at the same time it is able to capture the fundamental features of the epidemic process: healthy (Susceptible) people become infected when coming in contact with Infectious individuals only to eventually Recover after a certain period of time. The process is illustrated schematically in the figure at the top of this section.

This model can be written mathematically using a simple set of partial differential equations:

Which can be numerically integrated to obtain the values of each compartment as a function of time, just as done in the previous blog post:

While this kind of equations can be useful to explore analytical results for simple models like the SIR model, they quickly become unwieldy for more complex models. However, it is easy to note how they have a one-to-one correspondence with the illustration above:

• Interactions correspond to terms involving two compartments and the total number of individuals in the population:

• While spontaneous transitions correspond to terms involving just a single compartment:

• The sign of each term is determined by whether the equation we are considering corresponds to the “source” or “target” compartments. Notably, “agent” compartments are not affected unless they are also “targets”.

This one-to-one correspondence between transitions and terms allows us to simply “draw up” arbitrarily complex models that can be trivially implemented using generic code (like the one in EpiModel.py) without having to write out and debug all the rules “by hand”.

In the rest of the discussion we will focus on discussing the assumptions and details of the various models while avoiding as much as possible the use of complex mathematical expressions.

Incubation Period

One of the main limitations of the SIR model is the fact that the infection develops instantaneously without any incubation period what so ever. You’ll recall from recent news that this is not a very realistic scenario and the incubation or latent period is one of the most important factors that must be understood in order to contain an epidemic: For how long must a suspected case be kept under watch until we can be certain that the person will not become infectious?

We can address this limitation by adding one extra step (compartment) to our epidemic model: The Exposed (or Latent) compartment. When a Susceptible person comes in contact with an infectious one s/he moves to the Exposed from which s/he transitions to the Infectious compartment at a fixed rate ε. While in the Exposed compartment the person is said to be “incubating” the disease, possibly even starting to develop symptoms, but is not yet able to infect other individuals. The resulting model is known as the Susceptible-Exposed-Infectious-Recovered (SEIR) model:

Here we have 4 distinct compartments connected by one interacting transition and two spontaneous ones:

And the evolution of the various compartments is simply:

Here we highlight that the addition of the extra compartment didn’t change the total number of people who become infected, but it does have a strong impact on the temporal evolution of the epidemic, significantly delaying and widening the peak of infectious cases. It effectively “flattens” the curve:

It should be clear how this has a direct impact on the likelihood of the healthcare system being overwhelmed and the necessary duration of any quarantine measures imposed: a lower peak reduces the stress in the healthcare system, while a longer duration implies that longer period of social distancing is necessary.

Temporary Immunity

Another fundamental assumption underlying the SIR model is the idea that Recovered persons are permanently immune from the disease. While this is the case with many common diseases, there have been some reports of CoVID-19 patients being re-infected after recovery.

Re-infection in such a short period of time is unlikely (even temporary immunity typically lasts for a few months or years) and these cases are more likely to be due to faulty tests, but it is certainly a possibility that should be considered.

By simply adding a spontaneous transition from the Recovered compartment back to the Susceptible compartment, we obtain the SEIRS (can you guess what the letters stand for? 😀):

This seemingly innocuous addition to the model has a very important effect. By allowing Recovered individuals to once more become Susceptible, we replenish the group of people that can once again be infected. The end result is that the epidemic never burns itself out (its fuel is never exhausted) and the disease becomes endemic, with a constant fraction of the population remaining infected!

The rate ρ at which immunity is lost has a determinant effect in the progress of the epidemic and the rise of endemicity. If ρ is sufficiently small (immunity is longer lasting) we can even have several epidemic peaks before the steady state of a fixed fraction of the population is reached.

The appearance of the peak is due to the fact that the temporary immunity afforded by the disease is sufficiently long to allow the epidemic to follow most of its course before the number of Susceptibles starts to increase again, adding fuel to the fire.

Asymptomatic Cases

In many diseases, a significant fraction of infected individuals remain asymptomatic throughout the course of the disease. In the case of seasonal Influenza, this number is typically around 33%, while for CoVID19 the number is thought to be 40% or higher, thus significantly skewing the total number of cases.

Asymptomatic individuals are often less infectious than those displaying symptoms by some fraction rᵦ. We can model their effect by splitting the Infectious compartment in two: a Symptomatic, Is, and an Asymptomatic, Ia. A fraction pₐ of all of those Exposed become asymptomatic while the remaining (1-pₐ) develop symptoms. Our model is then:

As we now have two Infectious compartments we must also redo our Rₒ calculation. Fortunately, the modification is simple: since we have split the original Infectious compartment in two, our value of β is simply the weighted average of the original and the reduced β.

We can easily verify that if rᵦ is 1 we recover the original SIR value, while if rᵦ is 0 (the asymptomatic and completely non-infectious) we reduce the original Rₒ by a factor of (1-pₐ) as we effectively have that much smaller Infectious population.

In order to maintain the same value of Rₒ we simply calculate the value of β as:

This approach makes it easier to compare the results from both models since they both have the same value of Rₒ.

As we add more and more compartments to our models, the smaller the population of each individual compartment becomes.

We can easily verify that the value of Rₒ remains the same as before by looking at the Recovered and Susceptible curves at the end of the epidemic. On the other hand, we now have 3 distinct infected compartments, 2 of which are Infectious and peak at the same time and a few days after the Exposed population:

Here we should note that we explicitly decided to keep the recovery rate, μ for both Symptomatic and Asymptomatic individuals. Had we chosen them to be different then the peaks would occur at different times and the expression for Rₒ would have to be revised even further.

Mortality rate

Finally, we look at the effect of explicitly considering mortality. We assume that only symptomatic cases die from the disease or, similarly, that any asymptomatic individuals that do die from the disease are not counted as such. If we assume that a fraction pd of symptomatic cases end up dying, our model becomes:

So we now have 6 compartments and a total of 7 transitions and 6 parameters, denoting how the more details we include the more complex the model becomes and the more parameters must be specified. In the early days of an epidemic most, if not all, of these parameters are partially or completely unknown. As the epidemic progresses, more and more information is gathered and more detailed models can be used. This constant refinement also helps improve the reliability of the scenarios we are able to analyze and the decisions made.

If we assume that 10% of the symptomatic cases eventually die, we have:

It should be noted that 10% mortality rate is huge and unrealistic for the kind of diseases we are considering. The reason we choose such a large number is to make the effects obvious when plotting.

By including the possibility of Death, the number of Recovered individuals is naturally reduced, despite the fact that none of the disease parameters have been changed. If we focus on just the relationship between the most significant compartments we have:

The total number of dead can be easily estimated. We know that for our set of parameters, 80% of the population eventually becomes infected. Of those, 60% are symptomatic and of those, 10% eventually die, so we expect that the total number of fatal cases to be 4.8% as shown in the plot above.

This value is significantly smaller that the actual mortality rate for the symptomatic cases. This is due to the fact that the number of recovered is inflated by the milder asymptomatic cases.

Resources

Congratulations on making it through two loooong posts on Epidemic modeling. You already know a lot more about epidemic modeling than most mortals, but the journey has just begun.

You can move on to the next posts of this series

• Epidemic Modeling 103: Adding confidence intervals and stochastic effects to your CoVID-19 Models

• Epidemic Modeling 104: Impact of Seasonal effects on CoVID-19

All the code necessary to implement the models described above and to generate the figures used can be found in this posts GitHub repository:

Epidemiology 101 GitHub

Parts II, III and IV of this blog series is already published:

• Epidemic Modeling 102: All CoVID-19 models are wrong, but some are useful

• Epidemic Modeling 103: Adding confidence intervals and stochastic effects to your CoVID-19 Models

• Epidemic Modeling 104: Impact of Seasonal effects on CoVID-19

Over the past few weeks, a terrible affliction has been spreading across the world. Otherwise healthy and productive members of society have been infected with this devastating illness that causes them to fire up Excel, Python or R and start extrapolating the latest numbers of confirmed CoVID-19 cases in their town, state, country or even the entire world!

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