Mathematical modeling of immune responses to Viral pathogens

Abstract

The adaptive immune system, made up of a network of cells, tissues and organs, protect the body against infections and maintain overall health.Immunological research has identified two types of cell interactions: thymus(T) and bone marrow(B) derived cell interaction to explain immune responsiveness. Adaptive immune responses are highly specific to pathogens that induce them. Cell Mediated Immunity defend against intracellular pathogens such as viruses, intracellular bacteria and protozoa.The T cell function lies in the heart of an efficient cytotoxic response.The cells activation is highly regulated and is important to ensure that activation occurs in the right context to prevent development of harmful conditions. With some key processes of the immune system still poorly understood, construction of mathematical models of the immune responses provide the researchers and clinicians powerful tool for the simulation of immune system in order to increase its efficiency in the struggle against pathogens.The purpose of this study was to establish the interplay between the immune responses and viral pathogens.The study offers an innovative, analytical and methodological approach in elucidating key processes of the immune responses.The primary objectives ware: to develop a mathematical model that simulates immune system responses to viral pathogens, analyze the stability of the model in order to get some important ideas about the proliferation of the pathogen and to estimate the range of parameter combinations required to mount an immune response. To achieve this a mathematical model containing five variables: purely susceptible host cells, virus infected cells, free virions, antibody responses and cytotoxic T lymphocyte (CTL) response was formulated using differential equations. The simulations and analysis was done using Matlab & Mathematica softwares, available experimental data on Hepatitis C Virus is used to validate the analytical results. Stability analysis was carried out using the Routh -Hurwitz method and the theory of next generation matrix was used to determine the basic reproductive ratio as R0 = blk adw . Threshold values of parameters that influence immune responsiveness were also determined using the same method. Three possible outcomes in the activation of immune responses were considered: CTL response is established & antibody response fail, antibody response is established & CTL response fail and both CTL & antibody responses become established. Critical bounds are established to determine the threshold requirement for establishment or failure of either CTL or antibody. Analytical results have shown that when antibody response is established and CTL response fail this will represent stable equilibrium while when both CTL and antibody responses are established it will represent an unstable equilibrium. The numerical simulations results showed that dominant CTL establishment is likely to clear a viral pathogen while dominant antibody response alone may not clear the pathogen. In the case that the virus is not cleared, viral evolution was considered, to examine how virus variation affects viral and host survival and to understand viral disease. It was found that that CTL-induced pathology is observed if the rate of viral replication is fast relative to the CTL responsiveness of the host and CTL activation at this stage is not beneficial to the host but can actually be harmful. In conclusion it is important for CTL and antibody responses mount at the right time and strength to reduce the chances of antigenic escape. It is recommended that the model be adopted as a tool to simulate different treatment protocols before administering them patients.