Role of healthcare workers in the SIR epidemiological model

Common agents considered in such epidemiological models include the susceptible populations, infected cases and recovered patients. Of course, these are the main ingredients of an epidemic but there are other key agents that can regulate the dynamics of the epidemic. In this research-oriented article, we’ll look at the impact of healthcare system in an epidemic outbreak.

In this first of the two posts on the topic, We’ll be discussing an approach to mathematically model the impact of healthcare system in a pandemic. We’ll start with an overview of the basic Susceptible Infected Recovered (SIR) epidemiological model to set the stage. We shall then discuss the role of healthcare professionals (H) in a disease outbreak. Next, we’ll try to introduce ‘H’ into an SIR model and create slightly more advanced models of the current coronavirus pandemic.

This post is a summary of the more detailed analysis using Python (in Jupyter notebook) that can be found here. Interested readers are encouraged to delve into the details. For others, you can rest assured that you won’t be seeing any equations here, as I shall refrain from any mathematical details.

SIR model

The very popular SIR model gives key information on the number of patients at any point in time, maximum number of patients and peak time of infections. The model treats the three (‘S’, ‘I’ and ‘R’) classes in a population as separate entities that are interacting between themselves. The rate of interaction between ‘S’ and ‘I’ is decided by constraints like physical distancing measures and the rate of transition from ‘I’ to ‘R’ depends on disease treatment efficacy and/or natural recovery. When we plot these different classes, we observe, as expected, that ‘S’ decreases over time while ‘R’ increases with time. ‘I’ has a bell-shaped curve reaching a peak value and then dropping. There are great blogs and videos that explain these concepts in detail and will not be covered here.

Role of healthcare professionals

For our modelling purposes, let’s define a healthcare professional (H) as anyone who is in contact or works in close range to patients to support their recovery. This includes nurses, doctors, cleaning staffs, assistant nurses and other hospital staffs. The role of ‘H’ in disease control is twofold.

1. They treat patients and support their recovery. The efficiency of this process depends on the availability of medical equipment (ventilators) and an effective cure for the disease.

2. They need to safeguard themselves from patients to avoid becoming one themselves and this depends on the availability of personal protective equipment (PPE).

To understand how much PPE is required, I referred to the guidelines from the European Centre for Disease Prevention and Control (ECDPC). According to their document (available here), we require 3–6 PPE for a suspected case to test for the virus, 14–15 PPE/day for cases with mild symptoms and 15–24 PPE/day for cases with severe symptoms. The number of ventilators needed would depend on patients with severe symptoms.

Model integration

To include the dual effect of ‘H’ in a disease outbreak, let’s use the following approach.

1. Let’s assume two kinds of patient recovery process. A natural recovery process that does not require any hospitalization/treatment and second recovery process that demands medical support. The rate of the latter recovery will be proportional to the available medical equipment (ventilators) in the hospital.

2. Similarly, let’s assume two routes to get infected — ‘S’ that come in contact with ‘I’ and ‘H’ that have poor PPE and come in contact with ‘I’. The rate of the latter infection will be proportional to the lack of PPE.

SIRH model

Let’s include this twofold role of ‘H’ in the SIR model and call the resulting new model — ‘SIRH’. To simulate this model, let’s assumed arbitrary parameter values and a one-year time period. To replicate unfortunate real-world scenarios, let’s assume a lack of PPE and ventilators. Model simulations show that in addition to the typical trends for an SIR model we observe a decreasing ‘H’ curve, which indicates that with time ‘H’ are infected due to the lack of PPE.

COVID19 version

The SIRH model is quite generic in nature and would need to be altered to account for the current pandemic. There are two main events of COVID19 pandemic that are missing in the SIRH model — death of infected patients and chances re-infection of recovered patients. To include these factors, let’s create a new version with a new class for dead patients (D) and assume that ‘R’ can go back to being ‘S’. Let’s call this model SIRDH. Simulating the model with arbitrary parameter values for a longer period (3 years) suggests the possibility of a second weaker wave of infections. The second wave arises due to the reinfections of recovered patients.

Elaborate COVID19 model

The SIRDH model can further be made closer to reality by including the different categories of infected cases. There are individuals who are exposed to the infection (E) but are not yet infectious. Then there are individuals with mild symptoms of the disease (I1) and ones with more severe symptoms (I2). Infections can be spread by ‘I1’ and ‘I2’ and the progression of the disease goes from exposed (E) to mild symptoms (I1) to severe symptoms (I2) to dead (D). Let’s factor these in and create yet another new model. Long-term simulations show a complex dynamics of ‘E’, ‘I1’ and ‘I2’ and suggest a second infection wave.

Summary

In this article, we discussed concepts of including elements of healthcare system in an epidemiological model. We integrated the twofold role of healthcare professionals as recovery agents as well as highly susceptible individuals into the basic SIR model. Next, we created slightly more advanced COVID19-specific models that capture different aspects of the pandemic.

What’s next?

There might be an interesting underlying behaviour of these models because of the following unfortunate irony — Healthcare workers who are struggling to help patients recover are sometimes infected and become patients themselves in the process. When PPE and medical equipment are insufficient, can there be a situation where having more healthcare workers will result in more patients than having fewer healthcare workers?

In other words, can it be counter-productive to have more healthcare workers under some circumstances?

In the sequel article we will explore an interesting implicit behaviour of the SIRH model :)