Modelling the Spread of COVID-19. Part 1: Introduction

Infectious Diseases | Modeling Complex Systems | COVID-19 | Epidemics | Sanitary Crisis | Pandemic Responses | Sanitary Policies

Disclaimer

This series of articles was written in early 2020 and therefore most of the views and assumptions are outdated given our current state of knowledge.

Why Understanding Virus Spread Is Important

Early 2020, the world was gripped with fear and panic as the novel corona virus became a global pandemic. The virus was new and the world was unprepared; nothing on that scale has been experienced since the Spanish Flu in 1918. Authorities frantically wavered on a number of crucial topics such as the usage of masks, infectiousness of the disease, virus transmission channels, possible treatment (novel vs existing medication, early vs late treatment), and many other sanitary decisions and policies.

What were the measures that could have possibly been taken to minimize the spread of the corona virus and limit its impact?

Government Responses to the Corona Virus

Countries in South East Asia had some experience with SARS and MERS to draw on. The rest of the world had very little. Different ideas were fiercely debated. Some countries like China imposed draconian measures in areas of outbreaks like Wuhan. The objective being the total eradication the virus.

Sweden (and the UK at the very beginning) went for something akin to herd immunity, believing that lockdowns are unsustainable and will only serve to delay what is going to happen anyway.

France, Italy, Germany, and Spain, to name a few, followed a suppression/containment strategy trying to protect the vulnerable groups and keeping the National Healthcare System (NHS) from being overwhelmed by imposing lockdowns and strict social distancing measures.

Marseilles in France, lead by IHU – Méditerranée Infection, adopted a “Test, Diagnose, Treat” methodology, using the drug combination Azythromycin – Hydroxychloroquine as a treatment in the early phases of the disease.

Which policy was the best? Which saved more lives while preserving the economy? Was there any scientific evidence at all behind any of the policies implemented?

The fact is: nobody knows. Answers will only be available after the pandemic is over, when its too late.

Understanding the dynamics of the corona virus and its spread is vital to formulating appropriate responses.

What we needed was a way to quickly and unequivocally determine which policy would deliver an optimal solution (i.e. minimal loss of lives, better sustainability, and minimal damage to the economy). What is it that prevents us from running these different scenarios with computer simulations, measure the outcome, perform logical comparisons, and finally derive scientific conclusions? Is that not a sensible approach?

Like most of the people I know, I struggled to obtain clarity around the topics being debated. I was more attracted to two subjects in particular: one was treatment, two was modeling. The latter will be the main topic of discussion in this article but before we dive in, lets try to articulate the problem a bit more.

Problem Complexity

At the early stages of the pandemic, there was a widespread belief that locking down the entire world for 14 days would eradicate the virus. This was and still is, in most cases, not a viable solution for the following reasons:

  • Individual countries (as in Europe) or states (as in the US) could not simply seal off their borders for economical and socio-political reasons
  • Critical services provided by financial institutions, hospitals, food chains, and utilities would absolutely need to remain available
  • Allowing front line workers to continue doing their jobs required essential services such as schools and transport systems to remain open
  • There was no scientific evidence that the virus would be eradicated in 14 days
  • In today’s highly connected and interdependent global economy, the complete lockdown of an entire country means cutting vital exports and imports without which the economy might collapse.

On the other end of the spectrum, countries like Great Britain (initially) and Sweden – although in a lighter version – adopted a “herd immunity” approach. That also presented issues of its own:

  • The anticipated cost in lives, as put forward by modeling exercises (such as the one completed by the Imperial College in London), was enormous and would not be acceptable by the public
  • These decisions are unpopular as they gave the impression that not enough effort was being made to fight the virus and that the easy way out was adopted
  • These decisions could not be ethically defended; Italy’s medical staff was having to choose between patients on who would have priority over ventilation machines and ICU beds.

In fact, both these solutions are theoretical at best as they are either A) not viable, sustainable, and/or practical or B) come at a tremendous price in human lives.

The optimal sanitary policy to fight COVID-19 would have to be specifically tailored for each country based on its medical and economical infrastructure as well as its socio-political system.

Spectrum of possible responses to COVID-19

The human-virus ecosystem can be safely considered a highly complex physical system. The precise solution to an optimization problem on this system is, very likely, difficult to obtain. What this means is, no matter how hard we try, we are quite unable to come up with a mathematical formula having one or more exact solutions which would then be used to predict fatality rates, total duration, peak infections, and other outputs, to an arbitrary level of precision. Any model would certainly be a simplified version of the reality.

The notion of a complex physical system is quite essential for this discussion and deserves some more attention before we proceed any further.

Complex, Non-Linear, Multi-Dimensional Physical Systems

Lets start with some definitions borrowed from Wikipedia.

In physics, a physical system is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the system.

In our case, the physical system can be defined as a village, state, country, island, or the planet.

complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth’s global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication systems, social and economic organizations (like cities), an ecosystem, a living cell, and ultimately the entire universe. Complex systems are systems whose behavior is intrinsically difficult to model due to the dependencies, competitions, relationships, or other types of interactions between their parts or between a given system and its environment.

Using “country” as our system of interest, we can break it down into smaller components or “clusters”. These can be regions, services (hospitals, transport systems, supermarkets), or congregations of people (workplace, household). The highly dynamic and strongly interconnected clusters will produce a system whose properties cannot be directly inferred by observing its individual components.

Systems that are “complex” have distinct properties that arise from these relationships, such as nonlinearityemergencespontaneous orderadaptation, and feedback loops, among others… In many cases, it is useful to represent such a system as a network where the nodes represent the components and links to their interactions.

One example that is especially important and immediately applicable for our exercise here is non-linearity. Its meaning is quite simple: a small change in a stimulus on the system can produce a disproportionate outcome. For example, the dynamics of the virus spread will change significantly if the percentage of asymptomatic cases were null rather than any other small number as the presence of asymptomatic cases increases community spread and goes undetected and unmitigated.

A good model should be able to display the same properties observed in the real world.

Modeling the Spread of Infectious Diseases

Lets review some useful terms and concepts present today in the literature of infectious diseases.

Basic Reproduction Number or R0

The Basic Reproduction Number or R0 can be defined as the average number of new infections generated by one case in a population that is wholly susceptible to the virus (i.e. not immune or vaccinated). A good intro on the definition and calculation of this mathematical number can be found here and in this article for the more tech savvies.

In essence, R0 can be written as:

R_0 = \beta \times \tau

Where \beta  is the average number of infections produced per unit time and \tau  is the average infection period. This sounds deceptively easy but in fact social distancing, lockdowns, good hygiene, isolation, or any other measures put in place would produce a different (generally smaller) value that could be a bit harder to calculate.

Impact of the Basic Reproduction Number
Evolution of the Basic Reproduction Number

The inherent difficulty in estimating R0 under different circumstances makes any model that’s highly sensitive to it extremely fragile. We will break it down into more manageable pieces which will allow us to manipulate the model in a better way.

Herd Immunity

As the virus spreads through a community, more people will get sick and develop antibodies for that virus and eventually become immune. If the number of new cases generated from a single case is less than or equal to one, the disease will eventually die out.

Assuming there are enough immune cases (and consequently less and less susceptible cases) in a population to considerably slow down the virus transmission, the virus will run its course and disappear. This state of the population is referred to as Herd Immunity.

Herd immunity does not necessarily require that all the population become immune.

Achieving Herd Immunity is a bit tricky as we shall see in the following articles in this series.

Stages of the Epidemic

The following diagram depicts the four different stages of the epidemic.

Different Stages of an Epidemic

Stage 1: Susceptibility

During this stage:

  • This is where the majority of the population is susceptible to the virus
  • The ecosystem is prone to outbreaks of major proportions.
  • The population has not been vaccinated against the new virus or rendered immune through prior contacts with viruses of the same family.
  • At this stage, the population does not have any cases of infection.
  • This is the typical starting point in a model of a population hit by the novel corona virus.

Stage 2: Outbreaks

During this stage:

  • The population one or more outbreaks
  • Immunity is is negligible and R0 is greater than 1
  • The number of infections doubles at a constant rate implying exponential growth (Dr. Micheal Levitt argues against exponential growth in this interesting video)

From this stage, two possible paths can be taken depending on the policies enforced by sanitary authorities. These policies are either aimed towards herd immunity or containment, suppression, and eradication.

Stage 3: Fragile Equilibrium

During this stage:

  • Infections rates are constantly decreasing due to imposed measures
  • Immunity levels remain typically low
  • Potential outbreaks cannot be ruled out once the restrictions have been lifted.
  • The system is in a fragile or unstable state of equilibrium

The path leading to this stage from the previous is one aimed at eradication, suppression, or containment. Sanitary measures such as quarantine, isolation, social distancing, hygiene practices, and lockdowns will reduce the value of R0 which in turn will reduce the number of infected cases.

Forcibly suppressing the disease necessarily means a smaller immunity ratio; the overall population remains susceptible to potential outbreaks.

Stage 3 is more of a fragile or unstable state of equilibrium rather than an endemic steady state. If the sanitary measures are relaxed or an additional influx of cases are introduced into the population, the situation may very well return to Stage 2.

Stage 4: Steady State

At this stage:

  • The virus has run its course, infecting a maximum number of people
  • The population is very much immune
  • Sporadic new cases do not lead to outbreaks
  • The system is in a state of stable equilibrium.
  • The following equation holds (S being the ratio of the population still susceptible to the virus, see SIR model).

R_0 \times S = 1

A population of individuals could also change over time as infants are born, children become adults, and old people die. This will shift the balance back and forth in favor of new infections. The system, though, will remain in the steady state if the number of new infections and the number of susceptible individuals have a constant ratio.

In an idealized model, the system would stay in Stage 4 indefinitely.

Whats Next?

In Part II of this series, we will look at how to create a software model of the epidemic using computer simulations.


Leave a Reply