  Two factors that are often ignored but could play a crucial role in the
progression of an infectious disease are the distributions of inherent
susceptibility ($\sigma_{inh}$) and external infectivity ($\iota_{ext}$), in a
given population. While the former is determined by the immunity of an
individual towards a disease, the latter depends on the duration of exposure to
the infection. We model the spatio-temporal propagation of a pandemic using a
generalized SIR (Susceptible-Infected-Removed) model by introducing the
susceptibility and infectivity distributions to understand their combined
effects, which appear to remain inadequately addressed till date. We consider
the coupling between $\sigma_{inh}$ and $\iota_{ext}$ through a new Critical
Infection Parameter (CIP) ($\gamma_c$). We find that the neglect of these
distributions, as in the naive SIR model, results in an overestimation of the
amount of infection in a population, which leads to incorrect (higher)
estimates of the infections required to achieve the herd immunity threshold.
Additionally, we include the effects of seeding of infection in a population by
long-range migration. We solve the resulting master equations by performing
Kinetic Monte Carlo Cellular Automata (KMC-CA) simulations. Importantly, our
simulations can reproduce the multiple infection peak scenario of a pandemic.
The latent interactions between disease migration and the distributions of
susceptibility and infectivity can render the progression a character vastly
different from the naive SIR model. In particular, inclusion of these
additional features renders the problem a character of a living percolating
system where the disease cluster survives by migrating from region to region.
