The advances in genetics and biochemistry that have taken place over the last 10 years led to significant advances in experimental and clinical immunology. In turn, this has led to the development of new mathematical models to investigate qualitatively and quantitatively various open questions in immunology. In this study we present a review of some research areas in mathematical immunology that evolved over the last 10 years. To this end, we take a step-by-step approach in discussing a range of models derived to study the dynamics of both the innate and immune responses at the molecular, cellular and tissue scales. To emphasise the use of mathematics in modelling in this area, we also review some of the mathematical tools used to investigate these models. Finally, we discuss some future trends in both experimental immunology and mathematical immunology for the upcoming years.

We present previously undescribed spatial group patterns that emerge in a one-dimensional hyperbolic model for animal group formation and movement. The patterns result from the assumption that the interactions governing movement depend not only on distance between conspecifics, but also on how individuals receive information about their neighbors and the amount of information received. Some of these patterns are classical, such as stationary pulses, traveling waves, ripples, or traveling trains. However, most of the patterns have not been reported previously. We call these patterns zigzag pulses, semizigzag pulses, breathers, traveling breathers, and feathers.

The anti-tumour and pro-tumour roles of Th1/Th2 immune cells and M1/M2 macrophages have been documented by numerous experimental studies. However, it is still unknown how these immune cells interact with each other to control tumour dynamics. Here, we use a mathematical model for the interactions between mouse melanoma cells, Th2/Th1 cells and M2/M1 macrophages, to investigate the unknown role of the re-polarisation between M1 and M2 macrophages on tumour growth. The results show that tumour growth is associated with a type-II immune response described by large numbers of Th2 and M2 cells. Moreover, we show that (i) the ratio k of the transition rates k12 (for the re-polarisation M1→M2) and k21 (for the re-polarisation M2→M1) is important in reducing tumour population, and (ii) the particular values of these transition rates control the delay in tumour growth and the final tumour size. We also perform a sensitivity analysis to investigate the effect of various model parameters on changes in the tumour cell population, and confirm that the ratio k alone and the ratio of M2 and M1 macrophage populations at earlier times (e.g., day 7) cannot always predict the final tumour size.