Poisson Distribution Question- Viral Vector Integrations

Question

The number of viral genomes that integrate in cells follows a poisson distribution (https://www.nature.com/articles/3302270). This assumes every target cell has the same infectivity. How does one model the scenario when the cells may have a difference in infectivity?

Answer 1

If the cells have different infectivity, with a distribution $g$ say over infectivity, the resulting distribution of number of viral genomes (if I am understanding the situation correctly ...) will be a Poisson mixture (or compound) distribution, where the Poisson mean $\theta$ itself has the distribution $g$. See Wikipedia

So the pmf (probability mass function) will be given by $$ \DeclareMathOperator{\P}{\mathbb{P}} \P(Y=k) = \int_0^\infty \P_\text{Poisson}(Y=k \mid \theta) g(\theta)\; d\theta $$ If $g$ is a Gamma distribution, the result is a negative-binomial distribution, see Poisson Gamma Mixture = Negative Binomially Distributed?.