The coronavirus, COVID-19, pandemic is the biggest global health catastrophe of our time and greatest challenge since World War II. Since its first appearance in December last year, the virus has spread to each and every continent except Antarctica. Countries are in a hurdle to slow down the spread by testing, identifying, and providing treatment in containment zones, also bringing nation-wide lock-downs.
COVID-19 is not only a health crisis but it can create devastating economic, social and political crisis that pushes the world years back in advancement. Most of the world’s greatest cities are deserted as people stay indoors, either by choice or by government order. People are losing jobs and income, with no way of knowing when normality will return.
It’s very important to model and predict when the pandemic ends and how it affects across regions for the Government. It helps to take better preventive measures before hand and to minimize the overall loss. The best and easiest way is to use compartment models. Compartment models simplify mathematical modelling of infectious diseases. SIRD is one of the Compartment models.
What is SIRD?
The SIR model is one of the simplest compartmental models, and many models are derivatives of this basic form. The SIRD model is derived from SIR model. This model consists of four compartments: S for the number of susceptible, I for the number of infectious, R for the number of recovered (or immune) individuals, and D for the number of deceased individuals.
How does SIRD model works?
In SIRD model, we assume Population to be constant during the pandemic time. Total population is constant (N), at any time instant, the sum of susceptible, infectious, recovered and deceased individuals is equal to total population(N).
(Eq.1) S + I + R + D = N
Without vital dynamics, let’s start our model with initial conditions. Initially, let’s say we have I0 infected cases. There will be no recovered or deceased cases initially. So initial susceptible cases is equal to the difference between initial infected cases and total population.
(Eq.2) D0 = 0; Re0 = 0; I0; S0 = N - I0
From the figure, we can understand that people will move from susceptible to infectious at transmission/contact rate (α). To find out how many people have moved from susceptible to infectious at any time, we find total interactions possible i.e.,
S*I and we multiply total interactions with contact/transmission rate
-α*S*I (Negative sign is because people will decrease in susceptible compartment over time). Similarly, for recovered or deceased, people come from infectious compartment at recovery rate (β) and death rate (μ) respectively. The change in recovered or deceased compartment is given by
μ*I recpectively. Finally, for infectious compartment, people come from susceptible compartment and also go to recovered/deceased compartment, so change in infectious compartment is given by
α*S*I - β*I - μ*I.
We know that
S≤S0, by substituting S with S0 ,
dI/dt turns out to be an inequality. From this inequality, we deduce
R0,which are contact ratio and basic reproduction ratio.
R0 is called basic reproduction number (also called basic reproduction ratio).
R0 is the expected number of new infections from a single infection in a population where all subjects are susceptible.
Now, we got our differential equations, initial conditions and constants. We solve this differential equations and get output by integrating the SIR equations over the time. Since COVID-19 is an ongoing pandemic, we have to guess and find out transmission rate, recovery rate and death rate, or basic reproduction number based on the effects of lock-down, social distance practicing and other preventive measures taken.
Case-Study of SIRD model
We have used SIRD model to predict COVID-19 outbreak in India. We have taken India’s COVID-19 data from https://www.covid19india.org/ . By analyzing the COVID-19 historical data of India from 30th January till 23rd April, we found out that outbreak is spreading after a community event in Delhi. We have taken reasonable assumptions on the hyper parameters to suit the COVID-19 outbreak in India.
- Initially taking total India population as susceptible is not very good. So we have taken total population as the existing infected cases multiplied by factor of 10. The total population we have taken is 275498
- Taking the effect of early lock down, social distancing into consideration and performing regression tests, we found out the transmission rate (α) to be 0.277
- Finally after analyzing historical data, we have taken mean recovery rate(β) to be 0.138 and death rate( μ) to be 0.0125
Below is the SIRD model simulation for India’s hyper parameters.
The final prediction of Infected cases in India using SIRD model in given below along with actual number of infected cases till now.
As you can see from the above graph, the predicted maximum number of infected cases are nearly 46,000 which occurs on 9th May. The estimated end date is the time for realizing 97% of the total expected epidemic cases. The estimated end for India is around 15th June.
Even though COVID-19 is an on-going pandemic, it is still possible to model SIRD hyper parameters and estimate the pandemic over time. It is seen that when hyper parameters are tuned reasonably well, the SIRD model is estimating the COVID-19 outbreak good enough to take better preventive measures. Exafluence, as a data driven company, believes in producing more realistic insights on any social issue and there by helping the community with solutions.