We present a new proliferation model of cells living within a colony that is a non-local equation with a discontinuous interaction kernel. We select data that are suitable to perform Bayesian inference for unknown parameters and we provide a discussion on the range of applicability of the model. We discuss proof of the well-posedness of the problem and we investigate the convergence of the EBT algorithm applied to solve the equation. The main difficulty lies in the low regularity of the kernel which is not Lipschitz continuous, thus preventing the application of standard arguments. Therefore, we use the radial symmetry of the problem instead and transform it using spherical coordinates. The resulting equation has a Lipschitz kernel with only one singularity at zero. We introduce a new weighted flat norm and prove that the particle method converges in this norm. We present numerical simulations confirming the theoretical results. Finally, we prove the stability of posterior distributions in the total variation norm which exploits the theory of spaces of measures equipped with the weighted flat norm.
References
[1] P. Gwiazda, B. Miasojedow, J. Skrzeczkowski, Z. Szymańska, Convergence of the EBT method for a non-local model of cell proliferation with discontinuous interaction kernel, IMA Journal of Numerical Analysis, 10.1093/imanum/drab102, 2022.
[2] Z. Szymańska, J. Skrzeczkowski, B. Miasojedow, P. Gwiazda, Bayesian inference of a non-local proliferation model, Royal Society Open Science 8 (11), 211279, 0.1098/rsos.211279, 2021.