Global Existence, Uniqueness, and Non-negativity of the Solutions to Stochastic Partial Differential Equations of Infectious Diseases

We consider stochastic partial differential equations of infectious disease with linear multiplicative random noise. We transform these systems in a very simple way so that the resulting equations can be considered path-wise, and we can apply techniques from a deterministic approach to prove the global existence, uniqueness, and non-negativity of the solutions to the transformed systems and consequently the global existence, uniqueness, and non-negativity of the solutions to the original systems. Applications of our approach seem universal so that if any deterministic systems of partial differential equations of infectious disease admit the existence and uniqueness of their non-negative solutions, then so do their stochastic counterparts under the forcing by linear multiplicative random noise or their slight variants. We apply this approach to the models of Ebola virus disease and cholera as examples. Finally, we make comments on potential studies toward stability issues.