Predictive Model on the Spreading of COVID-19: SEIR

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COVID-19 has affected everyone in the world, and unfortunately it is unlikely to end soon. Many research groups worldwide are making efforts to control the trend of COVID-19 and provide forecast data to facilitate policy making. The Centers for Disease Control and Prevention in the United States (CDC) lists 24 groups who have been actively contributing to predict the spreading of COVID-19. These models provide the forecast on the number of total infected, total death, the duration of COVID-19, and the effect of policy intervention. Based on the description of predictive models from the CDC website, these predictive models can be categorised into: epidemic model, agent-based model, statistical model and deep learning. SEIR is the most popular one in the model selection. In the following, we are going to focus on SEIR model:

  • Basic concepts of SEIR model
  • Different SEIR models listed by CDC
  • SEIR model developed by MIT (DELPHI)

Basic Concept of SEIR Model

SEIR is short for Susceptible-Exposed-Infection-Recover. It is a compartmental model in epidemiology, which assigns people into different compartments and people could progress between compartments. I find it helpful to learn compartmental model in epidemiology via Wikipedia.

The classical SEIR model separates people into 4 compartments:

  • Susceptible (S): the individuals who are not infected but susceptible to the disease
  • Exposed (E): the individuals who are infected but not yet contagious
  • Infectious (I): the individuals who are infected and contagious
  • Recovered (R): the individuals who recovered and are immune.

In the classical SEIR model, it is assumed that the number of population is constant where the death rate equals to the birth rate, and all the newborn are the susceptible by default. As shown in Figure 1, S may be infected and become E first. The conversion rate of S → E depends on the contact rate (i.e. how many persons one may contact in a period) and the fraction of I in the population. The government impacts on the contact rate and the hospital impacts on the recovery rate, in order to control the spread of the virus.

Figure 1. Classical SEIR Model

The whole process of virus spreading above can be described by a number of differential equations. Given the initial condition and relevant parameters, we will know the values of S, E, I and R in any time period. Also, given the history of S, E, I and R, we will know the optimal values of relevant parameters to describe virus spreading.

SEIR is the most appropriate model to describe the spreading of COVID-19 currently, even though some evidence shows that an asymptomatic patient (E) may be as contagious as the infectious (I), and a recovered patient (R) may not be immune to COVID-19. Of course, with more robust evidence to reveal the traits of COVID-19, we need to modify the model to make it fit the actual spreading. If it is more like measles which becomes infectious almost immediately, then SIR model may be more appropriate. If it is more like flu which a recovered patient may still get infected later, then SEIS model may be more appropriate.

Summary on Different SEIR Models

Among the 24 modelling groups listed by CDC, there are at least 8 groups using SEIR. The table below summaries different SEIR models used/modified by these groups. (Some groups are not included because their websites show no evidence or details on their SEIR models, although CDC listed them as SEIR groups.)

Table 1. Summary on SEIR Models Developed by Different Groups

SEIR Model Developed by MIT (DELPHI)

Based on the availability of clear description and source code on model, we pick and focus on the SEIR model developed MIT this section. Their SEIR model is called DELPHI (Differential Equations Leads to Predictions of Hospitalisations and Infections).

The structure of DELPHI is as below. It is a complicated version of SEIR.

Figure 2. Structure of DELPHI

Mathematically, DELPHI is composed by the following differential equations.

where:

Each equation tells a different part of the whole story. Let’s have a closer look at them.

  • Equation (1): the susceptible does not increase and is only decreased by the number of infection.
  • Equation (2): every patient starts from the incubation period which is not contagious (i.e. E). The actual infection rate is influenced by how quickly and effectively a government responses to control contact between the infected and uninfected. Besides, the virus cannot be detected during the exposed stage (E).
  • Equation (3): the change in the number of infectious who have not been tested yet. For those who are tested (i.e. d I(t)), they will progress into 6 subgroups: AR, DHR, DQR, AD, DHD, DQD. This number can be influenced by government intervention such as the coverage of swab tests.
  • Equation (4): the number of the infectious who have been tested and are still alive (i.e. d (1-m)), not detected (i.e. 1-p_d), and not recovered yet (i.e. -r_q).
  • Equation (5): the change in the number of the infectious who have been tested and still alive (i.e. d (1-m)), detected (i.e. p_d), hospitalised (i.e. p_h), and not recovered yet (i.e. -r_h).
  • Equation (6): the change in the number of the infectious who have been tested and still alive, detected, not hospitalised, and not recovered yet.
  • Equation (7): the change in the number of the infectious who have been tested but not detected (i.e. d (1-p_d)), dead (i.e. m) but not due to COVID-19 (i.e. -l).
  • Equation (8): the change in the number of the infectious who have been tested and detected, hospitalised, dead but not due to COVID-19.
  • Equation (9): the change in the number of the infectious who have been tested and detected, not hospitalised, dead but not due to COVID-19.
  • Equation (10): the change in the number of total hospitalised.
  • Equation (11): the change in the number of detected death due to COVID-19.
  • Equation (12): the change in the number of total detected.
  • Equation (13): the change in the number of total recovered.
  • Equation (14): the change in the number of total death due to COVID-19, regardless whether they are detected or not.