Mathematical model study of a pandemic: Graded lockdown approach

A kinetic approach is developed, in a “tutorial style” to describe the evolution of an epidemic with spread taking place through contact. The “infection - rate” is calculated from the rate at which an infected person approaches an uninfected susceptible individual, i.e. a potential recipient of the disease, up to a distance p , where the value of p may lie between p _min ≤ p ≤ p _max . We consider a situation with a total population of N individuals, living in an area A, x(t) amongst them being infected while x _d (t) = β′x(t) is the number that have died in the course of transmission and evolution of the epidemic. The evolution is developed under the conditions (1) a faction α (t) of the [N-x(t) – x _d (t)] uninfected individuals and (2) a β(t) fraction of the x(t) infected population are quarantined, while the “source events” that spread the infection are considered to occur with frequency υ ₀ . The processes of contact and transmission are considered to be Markovian. Transmission is assumed to be inhibited by several processes like the use of “masks”, “hand washing or use of sanitizers” while “physical distancing” is described by p . The evolution equation for x ( t ) is a Riccati - type differential equation whose coefficients are time-dependent quantities, being determined by an interplay between the above parameters. A formal solution for x(t) is presented, for a “graded lockdown” with the parameters, 0≤ α(t), β(t)≤1 reaching their respective saturation values in time scales, τ ₁ , τ ₂ respectively, from their initial values α(0)=β(0)=0 . The growth is predicted for several BBMP wards in Bengaluru and in urban centers in Chikkaballapur district, as an illustrative case. Above selections serve as model cases for high, moderate and thin population densities. It is seen that the evolution of [x(t)/N] with time depends upon (a) the initial time scale of evolution, (b) the time scale of cure and (c) on the time dependence of the Lockdown function Q(t) = {[1-α(t)]·[1-β(t)]} . The formulae are amenable to simple computations and show that in order to curb the spread one must ensure that Q(∞) must be below a critical value and the vigilance has to be continued for a long time (at least 100 to 150 days) after the decay starts, to avoid all chances of the infection reappearing.