The Quantitative And Statistical Analysis For Critical Transitions Of The Complex Diseases

Complex diseases are rarely caused by the mutation or abnormality of a single gene,but by the complicated interactions among many genes or even between genetics and the environment,the sophisticated mechanisms bring certain difficulty to prevent,diagnose and treat complex diseases effectively.Specifically,the occurrence and development of complex diseases,often abrupt and unexpected.This phenomenon,can be viewed as a bifurcation or a critical transition of a complex system from one stable equilibrium state to another stable equilibrium state at a tipping point.Therefore,analyzing the critical transition of such nonlinear systems is critical to reveal the multiple genetic or environmental factors that affect normal biological functions and to identify the key drivers.Using the qualitative theory of ordinary differential equations(ODEs)and statis-tical analysis of stochastic differential equations(SDEs),this dissertation proposes the new indexes for predicting the critical transitions and identifying dynamical network biomarkers(DNBs)and investigate the change tendency of these indexes.The research work in this dissertation will provide new approaches and new ideas for detecting early-warning signals for predicting the critical transition of complex diseases and identifying DNBs.The main-innovative works for this dissertation are presented as follows:?.Three new quantitative indicators are derived based on ODE model.1.We theoretically prove that the slope of the tangent at the point of the curves belonging to the bifurcation diagrams,that is,the sensitivity of the equilibrium with respect to the parameters,can be viewed as the indicator for detecting critical transition and identifying DNBs.Using the transformation of matrices,the nonlinear system is mapped onto the center manifold,and the change tendency of the sensitivity for different variables on the center manifold is rigorously derived.Then,the sensitivity for the variables on the center manifold is converted into the original variables,and further the change tendency of the biomarkers and the non-biomarkers are obtained.The numerical simulations for three biological models verify this results.2.The"recovery time" of the system is theoretically to be a qualitative indicator for DNBs.First,we take the Taylor expansion of the nonlinear ODEs at the equilibriums.Then,with the small perturbations,the linearly approximate system is taken for our investigation.The solution of linearly approximate system on the center manifold is calculated.The approximate solutions of the original system is obtained by using the transformation of matrices.Finally,the recovery time for the DNBs and the non-biomarkers are compared and analyzed.This theoretical result is validated by using the numerical simulations for three models.3.We propose the index"potential energy" as the qualitative indicator for identi-fying the DNBs.Specifically,the "potential energy" is defined as the infinite integral for the square of the righthand-side with the time.First,we take the Taylor expansion of the one-dimensional nonlinear ODEs at the equilibrium,and investigate the change tendency of the corresponding potential energy for the variables as the critical transi-tion occurs.Furthermore,based on this result and the result of the "recovery time"' we speculate that,the change tendency of the corresponding potential energy for the DNBs and the non-biomarkers for high dimensional nonlinear system when the critical tran-sition is occurred.We use the numerical simulations for three models to demonstrate our conclusion.?.The coefficient of variation(CV),tra,nsformed probability distribution(TPD)and transformed Pearson correlation coefficient(TPC)are theoretically derived to be three new indexes to predict the critical transition of complex system based on the stochastic differential equation model.In order to consider the noises of the real data,we add the Gaussian white noises to the ODEs.The approximate solution for the obtained nonlinear SDE model is calculated by mapping to the central manifold through matrix transformation.Then,the three statistical indexes are calculated and used to analyze the different trends in the critical transition between the DNBs and non-markers.Specifically,1.We theoretically deduce that as the system parameter P approaches the bifurca-tion value P*,i.e.,the critical transition is occurred,the CV for biomarkers drastically increases and is much larger than the CV for non-biomarkers.In addition,there are no drastic changes for the CV of non-biomarkers.2.We prove that when the system approach the critical transition,the TPD for biomarkers drastically increases and there are no drastic changes for the TPD of non-biomarkers.Moreover,the TPD for biomarkers is much larger than the TPD for non-biomarkers.3.We prove that when the system approach the critical transition,the indicator TPC between biomarkers drastically increases,the indicator TPC between biomarkers is much larger than TPC between non-biomarkers and the indicator TPC between a biomarker and a non-biomarker is much smaller than TPC between non-biomarkers.Finally,to verify the effectiveness of these new indexes,we use high-throughput data for three complex diseases,including influenza caused by either H3N2 or H1N1 and acute lung injury,to extract the dynamical network biomarkers(DNBs)responsible for catastrophic transition into the disease state from pre-disease state.The numerical results indicate that the derived indicators are correct.The research work in this dissertation provide a data-based quantitative analy-sis for early-warning signs for critical transitions in complex biological systems.The approaches of early warning and dynamical network biomarkers proposed in this dis-sertation can also be applied to predict the critical transition of other complex systems.