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Compartmental models are popular in the mathematics of epidemiology for their simplicity and wide range of applications. Although they are typically solved as initial value problems for a system of ordinary differential equations, the observed data is typically akin of a boundary value type problem: we observe some of the dependent variables at given times, but we do not know the initial conditions. In this paper, we reformulate the classical Susceptible-Infectious-Recovered system in terms of the number of detected positive infected cases at different times, we then prove the existence and uniqueness of a solution to the derived boundary value problem and then present a numerical algorithm to approximate the solution.

The Covid-19 pandemic has taken millions of lives, demonstrating the tragedy and disruption of respiratory diseases, and how difficult they can be to manage. However, there is still significant debate in the scientific community as to which transmission pathways are most significant and how settings and behaviour affect risk of infection, which all have implications for which mitigation strategies are most effective. This study presents a general model to estimate the rate of viral transfer between individuals, objects, and the air. The risk of infection to individuals in a setting is then computed considering the behaviour and interactions of individuals between themselves and the environment in the setting, survival times of the virus on different surface types and in the air, and mitigating interventions (ventilation, hand disinfection, surface cleaning, etc.). The model includes discrete events such as touch events, individuals entering/leaving the setting, and cleaning events. We demonstrate the model capabilities on three case studies to quantify and understand the relative risk associated with the different transmission pathways and the effectiveness of mitigation strategies in different settings. The results show the importance of considering all transmission pathways and their interactions, with each scenario displaying different dominant pathways depending on the setting and behaviours of individuals therein. The flexible model, which is freely available, can be used to quickly simulate the spread of any respiratory virus via the modelled transmission pathways and the efficacy of potential mitigation strategies in any enclosed setting by making reasonable assumptions regarding the behaviour of its occupants. It is hoped that the model can be used to inform sensible decision-making regarding viral infection mitigations that are targeted to specific settings and pathogens., Comment: 49 pages, 7 figures, 3 tables