Understanding spread of Covid-19 using Math models

On March 11, 2020 Coronavirus or Covid-19 has been declared a pandemic by WHO. As of today, the pandemic/virus has spread to 145 countries all across the globe (in every continent except Antarctica), with 156,738 reported cases of which ~50% have already recovered, ~3.7% dead and another ~3.6% in critical state. Every government is taking action to suppress transmission, improve containment efforts and increase preparedness for the possible eventualities.

It is important to understand how the infectious disease such as Covid-19 spreads in the community, especially about:

  • What can happen if we did not take any action?
  • Will social distancing be effective? if so, how much?
  • Will early detection be effective? if so, how much?
  • How timely we need to decide to take action?
  • How effective should the isolation/ quarantining of the (infected) patients need to be?
  • … and others

Now, let’s begin by listing out things we know about the spread of Covid-19.

About Covid-19

Covid-19 is a new disease, although 2003/04 SARS was caused by a different type of coronavirus. Covid-19 spreads person-to-person, either (i) through respiratory droplets when an infected person coughs/ sneezes and it gets inhaled by persons nearby, or (ii) by touching a surface which has the virus and touching ones own nose, mouth or eyes. [Source]

Once a person is exposed to the virus, it can take 2-14 days for the symptoms to appear. The common symptoms include fever, dry cough, and shortness of breath/fatigue, which may lead to severe illness and death. It is thought that people are most contagious when they show the symptoms. This is the primary form of transmission. It is also suspected that it may be possible to catch the infection from persons not yet exhibiting the symptoms (we are still awaiting conclusive evidence on this).[Source]

As of today, there is no specific anti-viral treatment for Covoid-19. Patients are typically treated in isolation (either at home or hospitals) or in extreme case in quarantine. Most patients do not need any hospitalisation. Isolation is being recommended for ~14 days for patients, to recover completely (virus fully shedded). The patient is also said to be fully recovered if successive tests in a 24 hour period are negative.

Model

A common approach to build a community level model of the spread of infectious diseases is the compartment models. In this, the population is divided into compartments, with the assumption that each person in a compartment have the same characteristics. The most common compartment model is what is known as S-I-R model, where S is the number of susceptible population, I is the number of infectious population, and R is the number of recovered population. Since there are plenty of sources out there to teach you the basic SIR model, we will delve right into our model of the spread of coronavirus.

Considering the nature of the coronavirus, we can divide the population into Susceptible population (S), Exposed/ Asymptomatic infected population (E), Symptomatic infected population (I), under Treatment population (T), and Recovered population (R) (see Figure 1). Initially, the healthy individual is said to be part of the Susceptible population. If the individual gets exposed to virus, he/she ‘moves’ to Asymptomatic infected population. After an average Incubation period, the individual will show symptoms and hence ‘moved’ to Symptomatic infected population. Given the nature of the symptoms (fatigue, dry cough, fever), it may take some time (indicated as Delay in discovery) before the individual seeks medical help/ begin treatment. Once treatment starts, the patient ‘moves’ to Under Treatment population; and after the illness duration, the patient recovers (or dies), thus ‘moving’ to Recovered population. The following assumptions are also made in the model:

  • Only symptomatic infected population transmits the virus. That is, the susceptible population can get the virus only if they come in contact (directly or indirectly) with symptomatic individuals, and NOT from asymptomatic individuals.
  • Once treatment begins, the patient is in isolation/ quarantine and stops contributing to the spread of the virus.
  • Once the patient recovers, they become immune to the virus, and also do not spread the virus.
  • Population is well mixed
Figure 1: Stock-flow model of spread of Covid-19 (SEITR model)

Model Parameters

Let’s consider a community of 10,000 individuals. Initially (at time = 0), let’s assume that 10 infected but asymptomatic individuals are there in the community, and there are no symptomatic infected, or under treatment or recovered population.

Since it said that it takes 2-14 days for symptoms to appear in an infected individual, lets assume the average Incubation Period to be 7 days. After onset of symptoms let’s assume it takes a total of 14 days to recover. Suppose it takes 2 days for the patient to begin treatment, we can set Delay in discovery = 2 days, and Duration of illness= 12 days.

The basic reproduction number (R0) for coronavirus is expected to be between 2 and 3. This represents the number of newly infected individual from one case. Let’s take R0=2.5. In our model, since the transmission is by the symptomatic infected population only, we can define transmission rate = R0/Delay in discovery = 2.5/2 = 1.25 (new) persons/ (infected) persons/ day.

The model has been built and simulated in Vensim software.

Do nothing scenario

Figure 2a-c shows what will happen if we did not take any measure to mitigate the spread. The virus will wreck havoc on the population, with ~90% of the population getting infected within 120 days (4 months!), and only 10% of the population do not get the virus (Fig. 2a). Figure 2b shows that the infection rate increases exponentially peaking at 300 persons/day in day 46. However, due to incubation delay and the delay in seeking treatment, the peak of discovery rate of new cases happens around day 55, with 260 (new) patients/ day. The under treatment population reaches a peak of ~2500 patients, which is 25% of the community population! This will put enormous pressure on our healthcare system, on doctors, nurses, diagnostics, etc (more on this later).

Social distancing and Shorter discovery period

Social distancing helps to reduce the transmission rate and infection rate, and thus can prevent the spread of the virus. Early discovery of the symptoms can help in getting quicker treatment and isolation of the patients from the general population.

We model social distancing as follows: Reduce the transmission rate by 20% (from 1.25 to 1), indicating that the population in general reduce their contacts/ interactions by 20%.

We model shorter discovery period as follows: Reduce the discovery delay by 20% (from 2 to 1.6 days).

The scenarios are assumed applicable from day 0. A total three scenarios are simulated (OnlySocialDistancing, OnlyEarlyDetection, and BothSocialDistancingAndEarlyDetection) and compared with the DoNothing scenario. The resulting behaviours are discussed in the following figures.

Figure 3: Dynamics of Total active infective over time. Total Active Infectives are the sum of Symptomatic Infected Population and UnderTreatment Population. Compared to the DoNothing (grey line) scenario, OnlySocialDistancing (green line) and OnlyEarlyDetection (red line) shows a 28.5% reduction, while BothSDandED (blue line) shows a 59% reduction in the peak number of infected persons. Also, observe the delay in the occurrence of the peaks
Figure 4: Dynamics of Recovered population over time. Recovered population also represents the cumulative number of persons who actually get the virus/ disease. At the end of 120 days, compared to the DoNothing (grey line) scenario, OnlySocialDistancing (green line) shows a 16.67% reduction, OnlyEarlyDetection (red line) shows a 14.7% reduction, and BothSDandED (blue line) shows a 51.4% reduction in the cumulative recovered patients.

Social distancing and/or early detection can significantly reduce the peak infection rate, help ‘flatten the curve’, and reduce the eventual size of the cumulative number of people who get the virus. This notion of ‘flattening the curve’ is quite important as it help reduce the peak burden on our healthcare system, allowing for better care for existing patients.

How early to take decisions?

Typically, efforts to reduce social distancing and/ or discovery time are taken after a few cases of the disease is reported. The community (government) becomes worried mainly when there is an exponential increase in the daily reported cases over time. Now, what is exponential increase? Well, it means that the number of new cases doubles after each interval of time. The doubling time of Covid-19 is ~7 days. This means that, if on week 1 we had 10 cases, it will become 20 in week 2, 40 in week 3, 80 in week 4, …, and eventually 327,680 in week 16 (4 months time)!

Let’s consider the following decisions:

  • Beginning on Day 0, for a duration of 30 days, intense social distancing and early detection is emphasised in the community.
  • Beginning on Day 10, for a duration of 30 days, intense social distancing and early detection is emphasised in the community.
  • Beginning on Day 20, for a duration of 30 days, intense social distancing and early detection is emphasised in the community.

Let’s suppose social distancing and discovery time are reduced by 50% to indicate the intensity of the intervention during the above periods. We model this by setting, from the said start day (for the duration of 30 days) the transmission rate from 1.25/day to 0.625/day; and the Discovery delay from 2 days to 1 day. Note that beyond the 30 day duration, the transmission rate and Discovery delay comes back to its original values of 1.25/day and 2 days, respectively.

Interestingly, as seen from Figures 5 and 6, the infection did not seem have a disappeared after the period of 30 days of intense intervention The intervention only seems to have offset the peak infections by 40-50 days. This is apparent in Figure 6 where the total recovered (cumulative people who get the disease coverage to the same value).

Figure 5: Dynamics of Total Active Infective Population. The grey curve shows the DoNothing scenario. The blue, red and green curves show the dynamics when the intervention starts at day 0, 10 and 20 respectively. Observe that the peak number of infectives are quite comparable with the DoNothing scenario in all cases, albeit with a 40 day shift
Figure 6: Dynamics of Recovered Population. The grey curve shows the DoNothing scenario. The blue, red and green curves show the dynamics when the intervention starts at day 0, 10 and 20 respectively. Observe that the all scenarios show that ~90% of the population will eventually get infected.

This seemingly non-impact of the intense intervention (social distancing and early detection) on epidemics spread seems counter-intuitive. This is because, once the intervention was over, i.e, the restrictions on social distancing and eagerness for early detection was removed, the transmission rate was back to the typical level of the disease. And subsequently the disease again starts to spread vigorously.

Having said this, it is still useful to carry out a 30 days (or suitable block period) intervention. This will delay the growth of the disease, during which time the community can build required healthcare capacity and other resources to help better handle the epidemic when it hits.

What do we need to do to completely eliminate the epidemic?

The transmission rate has to come down and stay down. Figure 7 shows the dynamics of total infected population when the transmission rate stays down, from day 10 onwards. You can observe that almost immediately the epidemic spread is arrested. This will require not only to reduced people-to-people contacts, but also more steps to reduce the infectivity of the diseases as well.

Figure 7: Dynamics of total active infectives population in DoNothing (red line) scenario and scenario when transmission rate stays down (blue line).

So, we need to do the following to mitigate spread of Covid-19:

  • Awareness campaign to encourage people with symptoms to report to health clinics (reduce discovery delay).
  • Isolate and treat patients appropriately.
  • Reduce/ minimise internal interactions of the community. Conduct a 15 to 30 day ‘lock-in’ period when public gatherings/ events/ schools/ etc are closed. Consider work-from-home for offices (reduce social distance).
  • Reduce/ minimise external interactions of the community. All such interactions must be ‘sanitised’ to avoid creation of new cases from external influences.
  • Quarantine/ isolate people who are visiting the community from another location where cases of coronavirus has been reported.
  • Sanitise (clean with disinfectant), every single public place regularly (hourly if needed). Public places includes malls, parks, bus, bus stations, sidewalks, railway stations, trains, airports, plans, public universities, public library, classrooms, restaurants, supermarkets, playgrounds, gymnasiums, elevators, parking lots, etc – basically any place where large number of people frequently visit.
  • Use history of movement of current patients to identify potential cases.
  • Build capacity of your healthcare facilities immediately.
  • And do all the above regularly and continuously until Covid-19 disappears from all communities, cities, regions and states.

I feel all or most of the measures will need to continue for next 3-4 months, at least until July 2020.

4 comments on “Understanding spread of Covid-19 using Math models

  1. Manjesh Hanawal says:

    Nice article. What is the difference between community-based model and agent-based model? Which are the important feature each capture or miss?

    Like

    • Jayendran V says:

      Compartment-based model considers the population at an aggregate level, and models the behaviour as set of differential equations. In agent-based modeling (in this case), we model each individual person as an agent, with each having their own movements, immunity, interactions etc. As these agents interact, the disease spreads from one person (agent) to another. For generic models with homogenous well-mixed population, we can show that the the resulting aggregate behaviour of an agent based approach is equivalent to the compartment based models. Perhaps I can discuss this is another post. Compartment-based models gives sufficient insight to take key decisions in many scenarios.
      In case we need a higher fidelity model (and we have sufficient data to build the same), we can use agent-based approach. And of course, the animation of agent-based models is better!

      Liked by 1 person

  2. Nandyala Hemachandra says:

    Sounds good. Eradication seems possible with lockdown, even without medication, as no medicine or vaccination is available now.

    Like

  3. Aba says:

    Nice article. It is easy to understand the basic knowledge of COVID-19 & also many points mentioned to get control on this.

    Like

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s