Mathematical Modeling And Dynamical Analysis Of Innate Immune Responses

Innate immune system is a complex network of passages which include not only feed-back or feedforward circuits, but also cross-talks or transcriptional and post-translational modifications. On the other hand, the interactions of host and various pathogens, in-volved the induction and identification of innate immune receptor signaling, cell antiviral responses and viral escape mechanism, also increase the complexity of the system. The traditional method, analyzing the innate immune responses of viral infection from a candidate gene, is feasible, but no doubt restricts understanding of host-pathogen com-plex relationship from system level and has no way to integrate the known information. Thus, the simple method can not elucidate the integration effect and system charac-teristics about the molecular system, and we can only get a few isolated conclusions on the innate immune responses. To the contrast, systems biology is more concerned about the complex interactions between biological system components, as well as the system behaviors and biological functions arising from the interactions. Therefore, to understand the innate immune responses comprehensively, through the analysis of the host-pathogen interactions is helpful from the perspective of system theory.By systems biology methods, the dissertation establishes the mathematical model of the innate immune responses based interferon as core component, describing the innate immune related pathways and carrying out dynamic behavior analysis. This will undoubtedly enhance understanding the interactions between main components in the innate immune responses, so that it is possible to reveal the innate immune system antiviral mechanism from the system level. Therefore, the main works of the dissertation are focused on the following aspects:First, in order to understand the whole process of the innate immune responses, we propose a model with three order delays differential equations about virus, interferon and antiviral protein based the mass action law, considering the generation phase and the effect phase of interferon together. In accordance with the Hill coefficient of n2=1and n2>1, we analyze the stability of the model, discuss the parameter space when the innate immune system clears all virus, plays part role or fails antiviral ability and investigate the behaviors under different time delays. We found that the innate immune system can guarantee to remove virus gradually if the relative strength of interferon exceeds a certain threshold. The synergistic effect of interferon self-feedback can induce bistability and increasing the viral fatality rate can cause oscillation by a Hopf bifurcation. Some delays can not only induce the oscillation of the system but also calm instable system within a certain range, switching the system from an unstable or oscillatory state to a stable state. These results show that some delays of innate immune responses are beneficial to reduce the pathological injury caused by viral infection and this interesting phenomenon has not been previously described in the literatures. This helps us to understand the antiviral mechanism of innate immune system. In order to illustrate the correctness of the theoretical analysis, we carry out numerical simulation for all results and validate the model combined with the biological experiments.Second, the model can be viewed as a regulatory system with a negative feedback coupled with two positive auto-feedback loops if ignoring the specific biological back-ground, which can switch in a single stable, bistable or oscillation state. In order to better illustrate the positive and negative feedback how to induce complex dynamics, we carry out two-parameter bifurcation analysis and numerical simulation and we find that the system exhibits rich bifurcation phenomena (for example, saddle node bifur-cation, transcritical bifurcation, and supercritical or subcritical Hopf bifurcation). And the auto-positive feedback and negative feedback strength (σ1and σ2) plays an impor-tant role in the induction of complicated dynamic behaviors. However, other system parameters, such as relative reactive coefficient K and the relative degradation rates of α2and α4, have no significant effects on the system behaviors. At the same time, we again confirm that the positive feedback is just the necessary condition for bistability and synergistic effect (the Hill coefficient n2≥2) is helpful to induce bistability by strengthening the nonlinear of system. Negative feedback can cause oscillation under appropriate positive feedback strength and we can adjust the amplitude and period of oscillation by regulating strength of positive or negative feedback. The model with the complex dynamic behaviors can be used to design a network with specific biological function.Finally, establishing the mathematical model not only can be used to predict the behaviors of the system, but also can help us to find a proper method to intervene and control the behaviors of the system. Therefore, in the last part of the dissertation, we discuss how to adopt optimal control strategy to obtain a good therapeutic effect when innate immune system failure in protection by using the optimal control theory. We find that in the basic case, the three control strategies can effectively kill all viruses, but Strategy1will be the best treatment option, which not only lead to the smallest cost but also make the antiviral protein and interferon quickly returning to normal levels. When the weight changes, Strategy1or Strategy2will be the best optional control. When the control law weights are small or virus-weight becomes large, Strategy1will be the best treatment option. On the contrast, when the weight of the control law weights increase or virus-weight decreases, the optimal control will be Strategy2. When the treatment efficiency factor decreases, the control u2by enhancing interferon activity in Strategy2, becoming the best treatment options, not only minimizes the cost, but also reduces control volatility throughout the treatment period. When treatment efficiency factor increases, Strategy1again becomes the best choice. In all our discussions cases, Strategy3is unable to become the best treatment, indicating that a separate treatment sometimes has more advantages, including the smallest cost and operability, in the treatment of the disease. And improving efficiency factor for treatment will not only reduce the cost, but also provide more the treatment choices in the actual treatment of diseases.