Dynamics of Corona virus
At what speed it has spread. In Exponential or Reproduction Number R0.
How much time it takes to get normal.
What percentage of infected people get vaccinated.
Hard Immunity will reduce the spread of virus.
When is next epidemic? (Prediction of novel virus).
This are no specific type of problem statements, the model only cares of spread of the disease. Using mathematical analysis to understanding the behavior of novel corona virus. Mathematics and Statistics are cutting edge of how Hospital and Government deals with such pandemic.
An epidemic happens when any two or more larger groups has been infected or spread with in certain time(here time varies for urban between rural population) frame and when the disease is spread global is called pandemic.
Note : Time taken to spread and infection rate are proportional.
Using Epidemiological Model.
- The Basic Exponential Model
In such scenario the number of cases multiplied by the infection cases this is called exponential growth. Using this simplistic model it can be traced down how the infection rate can be controlled and how infection is potentially spread through the population.
Problem Statement 1: The City of Mumbai which is densely populated on Indian west cost. which as has population about 18.41 million and areas of 603.4 Km Square. How many people will get infected at the end of first week( 7 days) since the initial out brake.
From above, in a span of week 64 people will be infected.
Problem statement 2: If the total susceptible people were 18410000, how long it would take before entire population become infected.
y = 18410000 and 2 power (t-1). Applying logarithms on both sides.
log(18410000) = (t-1)log(2)
It takes 25 days to spread the infection to entire population.
2. The SIR Model
SIR model is more accurately represented it show exactly how an infection would spread with in population because it taken into account some people will recover from the infection and no longer be susceptible. In this model assumes people who recover from infection become immune and cannot spread it second time.
This model is divided into three parts
- Susceptible (s) (not infected population)
- Infectious (I)
- Recovered(R) (this are vaccinated, recovered with immunity or including death).
The number of infected people will increase at the rate proportional to both the number of infected and number of susceptible people and susceptible people decreases same rate. This ratio represented as transmission rate β ( beta), because a susceptible person who catches infection will be infectious immediately.
Also the number of recover ratio will increases at a rate proportional to number of infected people. This ratio is called recovery rate γ (gamma).
In model S,I,R are relevant to function variable t time, which are measure in days.
S = S(t) which is number of susceptible individuals at time t.
I = I(t) which is number of infected individuals at time t.
R=R(t) which is number of recovered individuals at time t.
All the above three function are consider to proportions each of individuals and if the total population has size N.
s(t) = S(t)/N , proportion of susceptible individuals at time t.
i(t) = I(t)/N, proportion of infected individuals at time t.
r(t) = R(t)/N, proportion of recovered individuals at time t.
Rate of infection = β x ( proportion of susceptible) x (proportion of infected)
ROF = βs(t)i(t), β is nothing but transmission rate and γ recovery rate.
Beta =Contacts per infects per day, Gama = recoveries per person per day
Problem Statement : From the population of Mumbai city where we had outbreak of Corona virus within population of 18.41 million, when infection was first recognized and analysis there are already 4000 people infected and 350 recovered.
Consider other variables β = 0.00005 and γ = 0.08.
Initializing values of S,I,R from the SI model s,i,r. that is at time t =0 at initial stage of identification.
And this model is continued until the spread of this particular infection over a period of time. Performing Herd Immunity might over load health case system.
About Author : Raghu Bayya, Data Scientist ML/Deep Learning.
Expert in Big Data
