Adaptive therapy improves cancer treatment by controlling the competition between sensitive and resistant cells through treatment holidays. This study highlights the role of treatment-holidays and the treatment-restarting thresholds in adaptive therapy for tumours composed of drug-sensitive and resistant cells. Using a Lotka-Volterra model, adaptive therapy outcomes are compared with maximum tolerated dose therapy and intermittent therapy outcomes, showing that adaptive therapy success depends critically on the thresholds for pausing and resuming treatment and on competitive interactions between cell populations. Three comparison scenarios between adaptive therapy and other therapies emerge, including uniform-decline where adaptive therapy underperforms regardless of threshold, conditional-improve where efficacy requires threshold optimisation, and uniform-improve where adaptive therapy consistently outperforms alternatives. Tumour initial conditions such as initial burden and initial resistant cell proportion influence outcomes. Threshold adjustments enable adaptive therapy to suppress resistant subclones while preserving sensitive cells, extending progression-free survival. Crucially, this work establishes an optimal control problem for time-to-progression and mathematically proves that under biological constraints like neutral competition or low initial burden, the theoretically optimal strategy is unrealisable as it requires infinitely many treatment holidays, rendering it clinically impractical. These findings emphasize personalised treatment strategies for enhancing long-term therapeutic outcomes.

Viral infections trigger complex immune responses with heterogeneous outcomes shaped by nonlinear feedback. A system of ordinary differential equations is developed to investigate immune response dynamics during viral infection, incorporating six modules, including viral load, innate immunity, cellular immunity, humoral immunity, immune suppression, and IL-6. Bifurcation analysis reveals that under continuous viral exposure, when viral clearance rate and intrinsic viral death rate satisfy specific conditions, the system exhibits up to five stable equilibria. This indicates that different health and disease states may coexist depending on initial conditions, while severe inflammation mainly arises from strong activation of cellular immunity, highlighting the complexity of immune responses. Simulations of finite-time viral exposure demonstrate multi-timescale recovery characteristics, with viral load and IL-6 levels declining rapidly, while humoral immune activation and immunosuppression show delayed and sustained patterns. Furthermore, analysis of infectious period and disease duration also indicates that during transition from early acute response to chronic disease, viral replication rate plays a critical role, while immune response intensity is sensitive to both viral clearance and immune self-activation. Subsystem analysis identifies the three-component subsystem of viral load, innate immunity, and cellular immunity as core drivers of bistability and oscillations, while humoral immunity, immune suppression, and IL-6 primarily modulate response amplitude and timing. This work establishes a theoretical framework for analyzing immune response and chronic risks through feedback dynamical modelling, providing insights for intervention strategies.

The interactions between tumor cells and the immune system play a crucial role in cancer evolution. In this study, we explore how these interactions influence cancer progression by modeling the relationships among naive T cells, effector T cells, and chronic myeloid leukemia cells. We examine the existence of equilibria, the asymptotic stability of the positive steady state, and the global stability of the tumor-free equilibrium. Additionally, we develop a partial differential equation to describe the conditions under which the concentration of cancer cells reaches a level that allows for effective control of cancer evolution. Finally, we apply the proposed model to investigate optimal treatment strategies that aim to minimize both the concentration of cancer cells at the end of treatment and the accumulation of tumor burden, as well as the cost associated with treatment during the intervention period. Our study reveals an optimal therapeutic protocol using optimal control theory. We perform numerical simulations to illustrate our theoretical results and to explore the dynamic behavior of the system and optimal therapeutic protocols. The simulations indicate that the optimal treatment strategy can be more effective than a constant treatment approach, even when applying the same treatment interval and total drug input.

Background: Sepsis is a life-threatening organ dysfunction caused by a dysregulated host response to infection. There is currently no simple immune-imbalance-driven indicator for patients with sepsis. Methods: This study was conducted in Peking Union Medical College Hospital. Patients with sepsis were identified according to Sepsis 3.0 after reviewing patient data from May 2018 through October 2022. Least absolute shrinkage and selection operator logistic regression was used for features selection. Receiver operating characteristic curves for 28-day mortality were used to compare the predictive performance of level of interleukin 6 (IL-6) and lymphocyte count (LY#) with that of the combined ratio, namely, the IL-6/LY# ratio. A Cox hazard model was also used to test the predictive performance of IL-6/LY# versus several other measurements. The dynamic trend of IL-6/LY# based on day 1 IL-6/LY# level was analyzed. Results: The mortality rate was 24.5% (220/898) in the study cohort. The LY#, IL-6 level, blood platelet count, Sequential Organ Failure Assessment score, Acute Physiology and Chronic Health Evaluation II score, heart rate, age and Fio(2) level were identified as key factors for predicting 28-day mortality. IL-6/LY# was identified as a core indicator according to Least absolute shrinkage and selection operator logistic regression analysis. IL-6/LY# was significantly higher in nonsurvivors than in survivors (348 [154.6-1371.7] vs. 42.3 [15.4-117.1]). IL-6/LY# yielded a higher area under the curve (0.852 [95% CI = 0.820-0.879]) than the level of IL-6 (0.776 [95% CI = 0.738-0.809]) and LY# (0.719 [95% CI = 0.677-0.755]) separately. Survival analysis of mortality risk versus the IL-6/LY# ratio suggested that IL-6/LY# was significantly more predictive of patient risk than the Sequential Organ Failure Assessment score or the other factors (P = 1.5 x 10(-33)). In trend analysis, as the trend of D1-D3-D7 IL-6/LY# decreases, the morality rate is lower than increase or fluctuate group (42.1% vs. 58.3%, 37.9% vs. 43.8%, 37.5% vs. 38.5% in high, moderate, and low D1 IL-6/LY# group separately). Conclusion: IL-6/LY# examined on first day in intensive care unit can be used as an immune-imbalance alert to identify sepsis patients with higher risk of 28-day mortality. Decreasing trend of IL-6/LY# suggests a lower 28-day mortality rate of sepsis patients.

By collecting various control policies taken by 127 countries/territories during the first wave of COVID-19 pandemic until July 2nd, 2020, we evaluate their impacts on the epidemic dynamics quantitatively through a combination of the multiple linear regression, neural-network-based nonlinear regression and sensitivity analysis. Remarkable differences in the public health policies are observed across these countries, which affect the spreading rate and infected population size to a great extent. Several key dynamical features, like the normalized cumulative numbers of confirmed/cured/death cases on the 100th day and the half time, show statistically significant linear correlations with the control measures, which thereby confirms their dramatic impacts. Most importantly, we perform the mediation analysis on the SEIR-QD model, a representative of general compartment models, by using the structure equation modeling for multiple mediators operating in parallel. This, to the best of our knowledge, is the first of its kind in the field of epidemiology. The infection rate and the protection rate of the SEIR-QD model are confirmed to exhibit a statistically significant mediation effect between the control measures and dynamical features of epidemics. The mediation effect along the pathway from control measures in Category 2 to four dynamical features through the infection rate, highlights the crucial role of nucleic acid testing and suspected cases tracing in containing the spread of the epidemic. Our data-driven analysis offers a deeper insight into the inherent correlations between the effectiveness of public health policies and the dynamic features of COVID-19 epidemics.

IntroductionThe immune responses play important roles in the course of disease initiation and progression upon virus infection such as SARS-CoV-2. As the tissues consist of spatial structures, the spatial dynamics of immune responses upon viral infection are essential to the outcome of infection.MethodsA hybrid computational model based on cellular automata coupled with partial differential equations is developed to simulate the spatial patterns and dynamics of the immune responses of tissue upon virus infection with several different immune movement modes.ResultsVarious patterns of the distribution of virus particles under different immune strengths and movement modes of immune cells are obtained through the computational models. The results also reveal that the directed immune cell wandering model has a better immunization effect. Several other characteristics, such as the peak level of virus density and onset time and the onset of the diseases, are also checked with different immune and physiological conditions, for example, different immune clearance strengths, and different cell-to-cell transmission rates. Furthermore, by the Lasso analysis, it is identified that the three main parameters had the most impact on the rate of onset time of disease. It is also shown that the cell-to-cell transmission rate has a significant effect and is more important for controlling the diseases than those for the cell-free virus given that the faster cell-to-cell transmission than cell-free transmission the rate of virus release is low.DiscussionOur model simulates the process of viral and immune response interactions in the alveola repithelial tissues of infected individuals, providing insights into the viral propagation of viruses in two dimensions as well as the influence of immune response patterns and key factors on the course of infection.

Background:The relationship between perfusion index (PI) and organ dysfunction in patients in the intensive care unit (ICU) is not clear. This study aimed to explore the relationship between PI and renal function in the perioperative critical care setting and evaluate the predictive efficiency of PI on patients with acute kidney injury (AKI) in the ICU. Methods:This retrospective analysis involved 12,979 patients who had undergone an operation and were admitted to the ICU in Peking Union Medical College Hospital from January 2014 to December 2019. The distribution of average PI in the first 24 h after ICU admission and its correlation with AKI was calculated by Cox regression. Receiver operating characteristic (ROC) curves were generated to compare the ability of PI, mean arterial pressure (MAP), creatinine, blood urea nitrogen (BUN), and central venous pressure (CVP) to discriminate AKI in the first 48 h in all perioperative critically ill patients. Results:Average PI in the first 24 h served as an independent protective factor of AKI (Odds ratio [OR]=0.786, 95% confidence interval [CI]: 0.704-0.873, P <0.0001). With a decrease in PI by one unit, the incidence of AKI increased 1.74 times. Among the variables explored for the prediction of AKI (PI, MAP, creatine, BUN, and CVP), PI yielded the highest area under the ROC curve, with a sensitivity of 64.34% and specificity of 70.14%. A cut-off value of PI ≤2.12 could be used to predict AKI according to the Youden index. Moreover, patients in the low PI group (PI ≤2.12) exhibited a marked creatine elevation at 24-48 h with a slower decrease compared with those in the high PI group (PI >2.12). Conclusions:As a local blood flow indicator, the initial 24-h average PI for perioperative critically ill patients can predict AKI during their first 120 h in the ICU.

In this paper, based on the Akaike information criterion, root mean square error and robustness coefficient, a rational evaluation of various epidemic models/methods, including seven empirical functions, four statistical inference methods and five dynamical models, on their forecasting abilities is carried out. With respect to the outbreak data of COVID-19 epidemics in China, we find that before the inflection point, all models fail to make a reliable prediction. The Logistic function consistently underestimates the final epidemic size, while the Gompertz's function makes an overestimation in all cases. Towards statistical inference methods, the methods of sequential Bayesian and time-dependent reproduction number are more accurate at the late stage of an epidemic. And the transition-like behavior of exponential growth method from underestimation to overestimation with respect to the inflection point might be useful for constructing a more reliable forecast. Compared to ODE-based SIR, SEIR and SEIR-AHQ models, the SEIR-QD and SEIR-PO models generally show a better performance on studying the COVID-19 epidemics, whose success we believe could be attributed to a proper trade-off between model complexity and fitting accuracy. Our findings not only are crucial for the forecast of COVID-19 epidemics, but also may apply to other infectious diseases.

During the study of epidemics, one of the most significant and also challenging problems is to forecast the future trends, on which all follow-up actions of individuals and governments heavily rely. However, to pick out a reliable predictable model/method is far from simple, a rational evaluation of various possible choices is eagerly needed, especially under the severe threat of COVID-19 epidemics which is spreading worldwide right now. In this paper, based on the public COVID-19 data of seven provinces/cities in China reported during the spring of 2020, we make a systematical investigation on the forecast ability of eight widely used empirical functions, four statistical inference methods and five dynamical models widely used in the literature. We highlight the significance of a well balance between model complexity and accuracy, over-fitting and under-fitting, as well as model robustness and sensitivity. We further introduce the Akaike information criterion, root mean square errors and robustness index to quantify these three golden means and to evaluate various epidemic models/methods. Through extensive simulations, we find that the inflection point plays a crucial role in the choice of the size of dataset in forecasting. Before the inflection point, no model considered here could make a reliable prediction. We further notice the Logistic function steadily underestimate the final epidemic size, while the Gomertz's function makes an overestimation in all cases. Since the methods of sequential Bayesian and time-dependent reproduction number take the non-constant nature of the effective reproduction number with the progression of epidemics into consideration, we suggest to employ them especially in the late stage of an epidemic. The transition-like behavior of exponential growth method from underestimation to overestimation with respect to the inflection point might be useful for constructing a more reliable forecast. Towards the dynamical models based on ODEs, it is observed that the SEIR-QD and SEIR-PO models generally show a better performance than SIR, SEIR and SEIR-AHQ models on the COVID-19 epidemics, whose success could be attributed to the inclusion of self-protection and quarantine, and a proper trade-off between model complexity and fitting accuracy.

基于经典动力学模型和模型参数自动优化算法, 本文分析了全国当前累计确诊感染病例数大于100（截止到2020年2月16日）的24个省市自治区, 以及湖北省除神农架以外的16个地市从2020年1月20日至2月16日的累计确诊病例数, 并对相应省市地区疫情可能的结束时间和总感染人数的进行了长期预测. 我们的研究表明, 在目前严防严控措施下, 全国大部分省市的疫情将于2月底前基本结束, 而湖北省内疫情也有望于3月中旬结束, 但是武汉市的疫情可能要持续到4月初. 通过对比公开数据与预测值, 我们建议加强对黑龙江、河北、江西、安徽、贵州和四川六省, 以及湖北省内武汉、荆州、鄂州、随州、天门和恩施六个地市的的监控, 以防疫情死灰复燃. 此外, 分析结果提示, 在疫情发展前期, 天津、河北、重庆、四川、海南和广西等省, 以及湖北省下辖多个地市可能存在着聚集性感染, 这有待于疫后进一步的流行病学调查确认.

The outbreak of novel coronavirus-caused pneumonia (COVID-19) in Wuhan has attracted worldwide attention. Here, we propose a generalized SEIR model to analyze this epidemic. Based on the public data of National Health Commission of China from Jan. 20th to Feb. 9th, 2020, we reliably estimate key epidemic parameters and make predictions on the inflection point and possible ending time for 5 different regions. According to optimistic estimation, the epidemics in Beijing and Shanghai will end soon within two weeks, while for most part of China, including the majority of cities in Hubei province, the success of anti-epidemic will be no later than the middle of March. The situation in Wuhan is still very severe, at least based on public data until Feb. 15th. We expect it will end up at the beginning of April. Moreover, by inverse inference, we find the outbreak of COVID-19 in Mainland, Hubei province and Wuhan all can be dated back to the end of December 2019, and the doubling time is around two days at the early stage.

Cyclical thrombocytopenia (CT) is a rare hematological disease characterized by periodic oscillations in circulating platelet counts. In almost all CT patients, other cell lines show no sign of oscillation, but recently a CT patient was reported with significant oscillations in circulating neutrophils (in the same period as the platelets). In this paper, we attempt to understand this phenomenon through a previously published model of human hematopoiesis. We have investigated a variety of possible oscillation patterns that may appear when alterations occur in the control parameters in the platelet regulatory dynamics. Our results indicate that the platelet maturation time and the differentiation rate from hematopoietic stem cells (HSCs) into the platelet cell line play important roles in the emergence of various types of CT like oscillations. Moreover, we find different oscillation patterns, including CT and cyclical neutropenia like oscillations, with certain parameter values in the platelet compartment. A bifurcation analysis revealed the different origins of these oscillation patterns. We also identified bistable dynamics which indicate the potential importance of system history in the treatment of these diseases. Together, these results demonstrate the possible origins for various oscillation patterns dependent on alterations in the platelet cell line control mechanisms. One of the important origins may be related to the regulation of apoptosis in platelet precursors.

The tumor suppressor p53 plays a central role in cell fate decisions after DNA damage. Programmed Cell Death 5 (PDCD5) interacts with the p53 pathway to promote cell apoptosis. Recombinant human PDCD5 can significantly sensitize different cancers to chemotherapies. In the present paper, we construct a computational model that includes PDCD5 interactions in the p53 signaling network and study the effects of PDCD5 on p53-mediated cell fate decisions during the DNA damage response. Our results revealed that PDCD5 functions as a co-activator of p53 and regulates p53-dependent cell fate decisions via the mediation of p53 dynamics. The effects of PDCD5 are dose-dependent, such that p53 activity exhibits sustained low level, pulsed oscillations, or sustained high level dynamics depending on the PDCD5 level following DNA damage. Moreover, PDCD5 regulates caspase-3 activation via two mechanisms during the two phases of sustained and pulsed p53 dynamics. This study provides insights regarding how PDCD5 functions as a regulator of the p53 pathway and might be helpful for increasing our understanding of the molecular mechanisms by which PDCD5 can be used to treat cancers.

The tumor suppressor p53 plays a central role in cell fate decisions after DNA damage. Programmed Cell Death 5 (PDCD5) is known to interact with the p53 pathway to promote cell apoptosis. Recombinant human PDCD5 can significantly sensitize different cancers to chemotherapies. In the present paper, we construct a computational model that includes PDCD5 interactions in the p53 signaling network and study the effects of PDCD5 on p53-mediated cell fate decisions during the DNA damage response. Our results revealed that PDCD5 functions as a co-activator of p53 that regulates p53-dependent cell fate decisions via the mediation of p53 dynamics. The effects of PDCD5 are dose-dependent such that p53 can display either sustained or pulsed dynamics at different PDCD5 levels. Moreover, PDCD5 regulates caspase-3 activation via two mechanisms during the two phases of sustained and pulsed p53 dynamics. This study provides insights regarding how PDCD5 functions as a regulator of the p53 pathway and might be helpful for increasing our understanding of the molecular mechanisms by which PDCD5 can be used to treat cancers.

The dynamics of p53 play important roles in the regulation of cell fate decisions in response to various stresses, and programmed cell death 5 (PDCD5) functions as a co-activator of p53 that modulates p53 dynamics. In the present paper, we investigated how p53 dynamics are modulated by PDCD5 during the deoxyribose nucleic acid damage response using methods of bifurcation analysis and potential landscape. Our results revealed that p53 activities display rich dynamics under different PDCD5 levels, including monostability, bistability with two stable steady states, oscillations, and the coexistence of a stable steady state (or two states) and an oscillatory state. The physical properties of the p53 oscillations were further demonstrated by the potential landscape in which the potential force attracts the system state to the limit cycle attractor, and the curl flux force drives coherent oscillation along the cyclic trajectory. We also investigated the efficiency with which PDCD5 induced p53 oscillations. We show that Hopf bifurcation can be induced by increasing the PDCD5 efficiency and that the system dynamics exhibited clear transition features in both barrier height and energy dissipation when the efficiency was close to the bifurcation point.

This paper considers adiabatic reduction in a model of stochastic gene expression with bursting transcription considered as a jump Markov process. In this model, the process of gene expression with auto-regulation is described by fast/slow dynamics. The production of mRNA is assumed to follow a compound Poisson process occurring at a rate depending on protein levels (the phenomena called bursting in molecular biology) and the production of protein is a linear function of mRNA numbers. When the dynamics of mRNA is assumed to be a fast process (due to faster mRNA degradation than that of protein) we prove that, with appropriate scalings in the burst rate, jump size or translational rate, the bursting phenomena can be transmitted to the slow variable. We show that, depending on the scaling, the reduced equation is either a stochastic differential equation with a jump Poisson process or a deterministic ordinary differential equation. These results are significant because adiabatic reduction techniques seem to have not been rigorously justified for a stochastic differential system containing a jump Markov process. We expect that the results can be generalized to adiabatic methods in more general stochastic hybrid systems.

This paper studies positive solutions of a class of delay differential equations of two delays that are originated from a mathematical model of hematopoietic dynamics. We give an optimal condition on initial conditions for t <= 0 such that the solutions are positive for t > 0. Long time behaviors of these positive solutions are also discussed through a dynamical system defined at a space of continuous functions. Characteristic description of the omega limit set of this dynamical system is obtained. This omega limit set provides informations for the long time behaviors of positive solutions of the delay differential equation.