Mathematical modeling of spatio-temporal population dynamics and application to epidemic spreading

The paper’s Theorem 3 gives exactly the projected rates for second-order adoptions under the Doi-type kernel and defines b_kl, f̂, and ε; its Appendix 5 proof computes Ĝ by projecting with Q onto indicator ansatz functions, decomposing G into outflow/inflow, and using uniform averaging over A_k×A_l. The candidate solution reproduces the same projector, indicator basis, structural sparsity, conditional averaging with b_kl, the f̂+ε off-diagonals, and the conservative diagonal. Minor wording differences aside, the logical steps and result coincide.