Inflammation is a physiological process aimed to protect the organism in various diseases and injuries. This work presents a generic inflammation model based on the reaction-diffusion equations for the concentrations of uninflamed cells, inflamed cells, immune cells and the inflammatory cytokines. The analysis of the model shows the existence of three different regimes of inflammation progression depending on the value of a parameter R called the inflammation number. If R>1, then inflammation propagates in cell culture or tissue as a reaction-diffusion wave due to diffusion of inflammatory cytokines produced by inflamed cells. If 0<R<1, then inflammation vanishes and the system converges to the stable inflammation-free equilibrium. Finally if R<0, inflammation also propagates as a reaction-diffusion wave, but the mechanism of propagation is different, it is determined by positive feedback between inflammation and immune response. From the biological point of view, these three regimes correspond to acute inflammation resolved due to the immune response, to the disappearance of inflammatory reaction, and to an auto-immune inflammatory reaction or cytokine storm. We focus on finding the wave speed and other characteristics of inflammation progression by analytical and numerical methods, in order to deduce a qualitative understanding of various inflammatory reactions.

We develop and analyze a mathematical model of oncolytic virotherapy in the treatment of melanoma. We begin with a special, local case of the model, in which we consider the dynamics of the tumour cells in the presence of an oncolytic virus at the primary tumour site. We then consider the more general regional model, in which we incorporate a linear network of lymph nodes through which the tumour cells and the oncolytic virus may spread. The modelling also considers the impact of hypoxia on the disease dynamics. The modelling takes into account both the effects of hypoxia on tumour growth and spreading, as well as the impact of hypoxia on oncolytic virotherapy as a treatment modality. We find that oxygen-rich environments are favourable for the use of adenoviruses as oncolytic agents, potentially suggesting the use of complementary external oxygenation as a key aspect of treatment. Furthermore, the delicate balance between a virus' infection capabilities and its oncolytic capabilities should be considered when engineering an oncolytic virus. If the virus is too potent at killing tumour cells while not being sufficiently effective at infecting them, the infected tumour cells are destroyed faster than they are able to infect additional tumour cells, leading less favourable clinical results. Numerical simulations are performed in order to support the analytic results and to further investigate the impact of various parameters on the outcomes of treatment. Our modelling provides further evidence indicating the importance of three key factors in treatment outcomes: tumour microenvironment oxygen concentration, viral infection rates, and viral oncolysis rates. The numerical results also provide some estimates on these key model parameters which may be useful in the engineering of oncolytic adenoviruses.