Global well-posedness and pattern formations of the immune system induced by chemotaxis

Infectious disease model generalization based on diffuse perturbations under conditions of body's temperature reaction

The infectious disease mathematical model is generalized based on the influence of diffuse perturbations on the development of the disease under conditions of the body's temperature reaction. The singularly perturbed model problem was reduced with delay to a sequence of problems without delay, for which the corresponding asymptotic expansions of solutions are obtained. The presented results of computer modeling in various situational states illustrate the expected decrease in the growth rate of the number of viral particles as a result of the action of the body's protective temperature reaction. The results of numerical experiments demonstrate the influence of the diffuse effect of "scattering" of forcing factors on the dynamics of a viral disease under conditions of the body's temperature reaction are presented too. It is noted that the decrease of the model amount of antigens in the epicenter of infection to a non-critical level caused by diffuse "scattering" over a relatively short time period makes them further destroyed by immune agents presented in the body, or requires the introduction of an injection solution with a smaller amount of donor antibodies.

Space-velocity thermostatted kinetic theory model of tumor growth

The competition between cancer cells and immune system cells in inhomogeneous conditions is described at cell scale within the framework of the thermostatted kinetic theory. Cell learning is reproduced by increased cell activity during favorable interactions. The cell activity fluctuations are controlled by a thermostat. The direction of cell velocity is changed according to stochastic rules mimicking a dense fluid. We develop a kinetic Monte Carlo algorithm inspired from the direct simulation Monte Carlo (DSMC) method initially used for dilute gases. The simulations generate stochastic trajectories sampling the kinetic equations for the distributions of the different cell types. The evolution of an initially localized tumor is analyzed. Qualitatively different behaviors are observed as the field regulating activity fluctuations decreases. For high field values, i.e. efficient thermalization, cancer is controlled. For small field values, cancer rapidly and monotonously escapes from immunosurveillance. For the critical field value separating these two domains, the 3E's of immunotherapy are reproduced, with an apparent initial elimination of cancer, a long quasi-equilibrium period followed by large fluctuations, and the final escape of cancer, even for a favored production of immune system cells. For field values slightly smaller than the critical value, more regular oscillations of the number of immune system cells are spontaneously observed in agreement with clinical observations. The antagonistic effects that the stimulation of the immune system may have on oncogenesis are reproduced in the model by activity-weighted rate constants for the autocatalytic productions of immune system cells and cancer cells. Local favorable conditions for the launching of the oscillations are met in the fluctuating inhomogeneous system, able to generate a small cluster of immune system cells with larger activities than those of the surrounding cancer cells.