Bayesian adaptive model selection design for optimal biological dose finding in phase I/II clinical trials.

Identification of the optimal dose presents a major challenge in drug development with molecularly targeted agents, immunotherapy, as well as chimeric antigen receptor T-cell treatments. By casting dose finding as a Bayesian model selection problem, we propose an adaptive design by simultaneously incorporating the toxicity and efficacy outcomes to select the optimal biological dose (OBD) in phase I/II clinical trials. Without imposing any parametric assumption or shape constraint on the underlying dose-response curves, we specify curve-free models for both the toxicity and efficacy endpoints to determine the OBD. By integrating the observed data across all dose levels, the proposed design is coherent in dose assignment and thus greatly enhances efficiency and accuracy in pinning down the right dose. Not only does our design possess a completely new yet flexible dose-finding framework, but it also has satisfactory and robust performance as demonstrated by extensive simulation studies. In addition, we show that our design enjoys desirable coherence properties, while most of existing phase I/II designs do not. We further extend the design to accommodate late-onset outcomes which are common in immunotherapy. The proposed design is exemplified with a phase I/II clinical trial in chronic lymphocytic leukemia.

Clinical outcomes and predictors of delayed echocardiographic response to cardiac resynchronization therapy.

INTRODUCTION: The clinical outcomes and mechanisms of delayed responses to cardiac resynchronization therapy (CRT) remain unclear. We aimed to investigate the differences in outcomes and gain insight into the mechanisms of early and delayed responses to CRT. METHODS: This retrospective study included 110 patients who underwent CRT implantation. Positive response to CRT was defined as ≥15% reduction of left ventricular (LV) end-systolic volume on echocardiography at 1 year (early phase) and 3 years (delayed phase) after implantation. The latest mechanical activation site (LMAS) of the LV was identified using two-dimensional speckle-tracking radial strain analysis. RESULTS: Seventy-eight (71%) patients exhibited an early response 1 year after CRT implantation. Of 32 non-responders in the early phase, 12 (38%) demonstrated a delayed response, and 20 (62%) were classified as non-responders after 3 years. During the follow-up time of 10.3 ± 0.5 years, the delayed and early responders had a similar prognosis of mortality and heart failure (HF) hospitalization. In contrast, non-responders had a worse prognosis. Multivariate analysis revealed that a longer duration (months) between initial HF hospitalization and CRT (odds ratio [OR]: 1.126; 95% confidence interval [CI]: 1.036-1.222; p = .005), non-exact concordance of LV lead location with LMAS (OR: 32.744; 95% CI: 1.101-973.518; p = .044), and pre-QRS duration (OR: 0.901; 95% CI: 0.827-0.981; p = .016) were independent predictors of delayed response to CRT compared with early response. CONCLUSION: The prognoses were similar regardless of the response time after CRT. A longer history of HF, suboptimal LV lead position, and shorter pre-QRS duration were related to delayed response than early response.

A two-stage group-sequential design for delayed treatment responses with the possibility of trial restart.

Common statistical theory applicable to confirmatory phase III trial designs usually assumes that patients are enrolled simultaneously and there is no time gap between enrollment and outcome observation. However, in practice, patients are enrolled successively and there is a lag between the enrollment of a patient and the measurement of the primary outcome. For single-stage designs, the difference between theory and practice only impacts on the trial duration but not on the statistical analysis and its interpretation. For designs with interim analyses, however, the number of patients already enrolled into the trial and the number of patients with available outcome measurements differ, which can cause issues regarding the statistical analyses of the data. The main issue is that current methodologies either imply that at the time of the interim analysis there are so-called pipeline patients whose data are not used to make a statistical decision (like stopping early for efficacy) or the enrollment into the trial needs to be at least paused for interim analysis to avoid pipeline patients. There are methods for delayed responses available that introduced error-spending stopping boundaries for the enrollment of patients followed by critical values to reject the null hypothesis in case the stopping boundaries have been crossed beforehand. Here, we will discuss other solutions, considering different boundary determination algorithms using conditional power and introducing a design allowing for recruitment restart while keeping the type I error rate controlled.

Delay-related activity in marmoset prefrontal cortex.

Persistent delay-period activity in prefrontal cortex (PFC) has long been regarded as a neural signature of working memory (WM). Electrophysiological investigations in macaque PFC have provided much insight into WM mechanisms; however, a barrier to understanding is the fact that a portion of PFC lies buried within the principal sulcus in this species and is inaccessible for laminar electrophysiology or optical imaging. The relatively lissencephalic cortex of the New World common marmoset (Callithrix jacchus) circumvents such limitations. It remains unknown, however, whether marmoset PFC neurons exhibit persistent activity. Here, we addressed this gap by conducting wireless electrophysiological recordings in PFC of marmosets performing a delayed-match-to-location task on a home cage-based touchscreen system. As in macaques, marmoset PFC neurons exhibited sample-, delay-, and response-related activity that was directionally tuned and linked to correct task performance. Models constructed from population activity consistently and accurately predicted stimulus location throughout the delay period, supporting a framework of delay activity in which mnemonic representations are relatively stable in time. Taken together, our findings support the existence of common neural mechanisms underlying WM performance in PFC of macaques and marmosets and thus validate the marmoset as a suitable model animal for investigating the microcircuitry underlying WM.

Delayed complete remission of hemifacial spasms following microvascular decompression and the implications for optimal time of revision surgery.

BACKGROUND: Microvascular decompression (MVD) is an effective method for directly treating hemifacial spasms (HFS). The timing for the consideration of failed MVD and reoperation has been paradoxical. OBJECTIVE: This study aimed to investigate the delayed complete remission of HFS in terms of prevalence rate, duration between surgery and delayed complete remission, and predictive factors. METHODS: A hundred patients with HFS who underwent MVD from 2012-2021 were enrolled in the study. All HFS occurred as a result of compression of the facial nerve by adjacent blood vessels. Clinical information, intraoperative findings, and surgical outcomes were incorporated for data analysis. RESULTS: In the first week after MVD, 67 of 100 patients achieved complete remission of HFS, while the remaining 33 had incomplete remission. In long-term follow-up, 26 individuals gradually developed delayed complete remission with a median duration of 9.1 months. Finally, 86 of 100 patients achieved complete long-term remission. Recurrent HFS and incomplete remission were found in 7 and 7 patients, respectively. Factors associated with postoperative complete remission in the first week were a severe degree of facial nerve compression (p = 0.047, OR 2.75, 95% CI 1.01-7.40), with long-term complete remission was left-sided HFS (p = 0.012, OR 5.73, 95% CI 1.47-22.36), and with the appearance of delayed complete remission was the prolonged duration of HFS at least 3 years before MVD (p = 0.046, OR 3.75, 95% CI 1.03-13.76). Transient facial paresis was found in 11% of the patients. Of them, facial nerve function recovered completely in all cases. CONCLUSIONS: A delayed complete remission of HFS could be expected in long-term follow-up after MVD and is probably related to a longer duration of HFS before surgery. Unnecessary reoperation should be avoided in the early years following the first surgery.

Identification of a Brain Network Underlying the Execution of Freely Chosen Movements.

Action selection refers to the decision regarding which action to perform in order to reach a desired goal, that is, the "what" component of intention. Whether the action is freely chosen or externally instructed involves different brain networks during the selection phase, but it is assumed that the way an action is selected should not influence the subsequent execution phase of the same movement. Here, we aim to test this hypothesis by investigating whether the modality of movement selection influences the brain networks involved during the execution phase of the movement. Twenty healthy volunteers performed a delayed response task in an event-related functional magnetic resonance imaging design to compare freely chosen and instructed unimanual or bimanual movements during the execution phase. Using activation analyses, we found that the pre-supplementary motor area (preSMA) and the parietal and cerebellar areas were more activated during the execution phase of freely chosen as compared to instructed movements. Connectivity analysis showed an increase of information flow between the right posterior parietal cortex and the cerebellum for freely chosen compared to instructed movements. We suggest that the parieto-cerebellar network is particularly engaged during freely chosen movement to monitor the congruence between the intentional content of our actions and their outcome.

Extending the two-stage single arm phase II clinical trial design to the delayed response scenario.

Two-stage single arm designs are widely used in phase II clinical trials with binary endpoints. The trial may be stopped early due to insufficient positive responses in the first stage. There may be some enrolled subjects who have yet to respond by the end of the first stage, and their data are ignored if the first stage results in rejection of the trial. It is possible that the result after the first stage is rejection by a slim margin, while the results of pipeline subjects are quite positive. In this case, combining the data from the two sources may provide a valuable opportunity to rescue a promising treatment that was mistakenly rejected. We propose a novel double-check design to take advantage of the pipeline subjects' data to establish a rescue criterion based on two-stage design. When the rescue criterion is met, the decision to reject the trial at the end of the first stage can be reversed, allowing the trial to continue. A derivation based on a binomial distribution shows that the double-check strategy can strictly preserve the type I error rate. Further examination shows that the strategy can provide a slight increase in overall power and a substantial increase in conditional power when the proportion of positive response at the end of the first stage is at the margin. The extra rescue opportunity's cost is pretty low, only a slight increasing in the expected sample size.

Robust group sequential designs for trials with survival endpoints and delayed response.

Randomized clinical trials in oncology typically utilize time-to-event endpoints such as progression-free survival or overall survival as their primary efficacy endpoints, and the most commonly used statistical test to analyze these endpoints is the log-rank test. The power of the log-rank test depends on the behavior of the hazard ratio of the treatment arm to the control arm. Under the assumption of proportional hazards, the log-rank test is asymptotically fully efficient. However, this proportionality assumption does not hold true if there is a delayed treatment effect. Cancer immunology has evolved over time and several cancer vaccines are available in the market for treating existing cancers. This includes sipuleucel-T for metastatic hormone-refractory prostate cancer, nivolumab for metastatic melanoma, and pembrolizumab for advanced nonsmall-cell lung cancer. As cancer vaccines require some time to elicit an immune response, a delayed treatment effect is observed, resulting in a violation of the proportional hazards assumption. Thus, the traditional log-rank test may not be optimal for testing immuno-oncology drugs in randomized clinical trials. Moreover, the new immuno-oncology compounds have been shown to be very effective in prolonging overall survival. Therefore, it is desirable to implement a group sequential design with the possibility of early stopping for overwhelming efficacy. In this paper, we investigate the max-combo test, which utilizes the maximum of two weighted log-rank statistics, as a robust alternative to the log-rank test. The new test is implemented for two-stage designs with possible early stopping at the interim analysis time point. Two classes of weights are investigated for the max-combo test: the Fleming and Harrington (1981) Gρ,γ$G^{\rho , \gamma }$ weights and the Magirr and Burman (2019) modest (τ∗)$ (\tau ^{*})$  weights.

Case report: delayed response after electroconvulsive therapy in a patient with major depressive disorder.

BACKGROUND: Major depressive disorder and associated mood syndromes are amongst the most common psychiatric disorders. To date, electroconvulsive therapy (ECT) is considered the most effective short-term treatment for patients with severe or treatment-resistant depression. In clinical practice, there is considerable variation in the ECT dosing schedule, with the number of sessions typically ranging from 6 to 12, with early antidepressant effects being predictive of increased positive outcomes. We describe here an unusual case of a female patient with severe depression who did not respond to ECT until the 11th session, after which she had shown a drastic improvement in her mental state. CASE PRESENTATION: A 75-year-old female presented to the old age psychiatry inpatient unit with new onset dysphoric mood, anhedonia, and severe negativity. She scored 23 on the 17-item Hamilton Rating Scale for Depression (HAM-D), and was rated 6 on Clinical Global Impression severity (CGIS) by the responsible clinician. She suffered from post-natal depression fifty years ago and was successfully treated with ECT. She was therefore initiated on a course of ECT treatment. Her condition initially deteriorated, displaying features of catatonia and psychosis, unresponsive to ECT treatment or concurrent psychotropic medications. After 11th ECT session, she started to show signs of clinical improvement and returned close to her baseline mental state after a total of 17 ECT sessions. She remained well 3 months post-treatment, scoring 4 on HAM-D, Clinical Global Improvement or change (CGI-C) rated as 1 (very much improved). The diagnosis was ICD-10 F32.3 severe depressive episode with psychotic symptoms. CONCLUSIONS: we describe here an unusual case of delayed response to electroconvulsive therapy in the treatment of severe depressive disorder. Studies have shown the number of acute ECT treatments to be highly variable, affected by a number of factors including treatment frequency, condition treated and its severity, the ECT technical parameters, as well as concurrent use of pharmacological treatment. This may call for re-consideration of the current ECT treatment guidelines, requiring more research to help stratify and standardize the treatment regime.

Response and sample bridging in a primate short-term memory task.

Freely-moving rodents can solve short-term memory (STM) tasks using "response bridging" strategies, relying on motor patterns instead of mnemonic functions. This limits the interpretational power of results yielded by some STM tasks in rodents. To determine whether head-fixed monkeys can employ parallel non-mnemonic strategies, we measured eye position and velocity of two head-fixed monkeys performing a delayed response reaching and grasping task. We found that eye position during the delay period was correlated with reach direction. Moreover, reach direction as well as grasp object could be predicted from eye kinematics during the delay. Both eye velocity and eye position contributed to the prediction of reach direction. These results show that motor signals carry sufficient information to allow monkeys to solve STM tasks without using any mnemonic functions. Thus, the potential of animals to solve STM tasks using motor patterns is more diverse than previously recognized.

The incidence and outcomes of delayed response to cardiac resynchronization therapy.

BACKGROUND: The incidence and clinical outcomes of delayed response to cardiac resynchronization therapy (CRT) have not been well clarified. We aimed to observe the incidence and prognosis of delayed response and to identify its possible mechanisms. METHODS: A total of 115 CRT patients were retrospectively analyzed in our study. Patients who met the enrollment criteria were divided into two groups: group A, conventional responders who showed response at 1-year follow-up, and group B, delayed responders who showed response after 1-year follow-up. CRT response was defined as an absolute increase of ≥10% in left ventricular ejection fraction. RESULTS: Fifty-two patients (61 ± 12 years, 37 male) experienced conventional response to CRT and 17 patients (63 ± 11 years, 10 male) experienced delayed response. The mean follow-up time was 5.2 ± 2.4 years. The incidence of delayed response was 14.8% (17/115). All-cause mortality and hospitalization rates for heart failure were similar for delayed and conventional responders. Multivariate logistic regression analysis revealed that scar burden > 35% was an independent predictor of CRT delayed response (odds ratio 8.794, P  =  0.038). CONCLUSIONS: A significant proportion of patients demonstrated delayed response to CRT. The delayed responders had a good prognosis that was similar to that of conventional responders. More scar burden might be related to the incidence of delayed response.

Diverse Synaptic Distributions of G Protein-coupled Estrogen Receptor 1 in Monkey Prefrontal Cortex with Aging and Menopause.

Age- and menopause-related impairment in working memory mediated by the dorsolateral prefrontal cortex (dlPFC) occurs in humans and nonhuman primates. Long-term cyclic 17ß-estradiol treatment rescues cognitive deficits in aged ovariectomized rhesus monkeys while restoring highly plastic synapses. Here we tested whether distributions of G protein-coupled estrogen receptor 1 (GPER1) within monkey layer III dlPFC synapses are sensitive to age and estradiol, and coupled to cognitive function. Ovariectomized young and aged monkeys administered vehicle or estradiol were first tested on a delayed response test of working memory. Then, quantitative serial section immunoelectron microscopy was used to determine the distributions of synaptic GPER1. GPER1-containing nonperforated axospinous synapse density was reduced with age, and partially restored with estrogen treatment. The majority of synapses expressed GPER1, which was predominately localized to presynaptic cytoplasm and mitochondria. GPER1 was also abundant at plasmalemmas, and within cytoplasmic and postsynaptic density (PSD) domains of dendritic spines. GPER1 levels did not differ with age or treatment, and none of the variables examined were tightly associated with cognitive function. However, greater representation of GPER1 subjacent to the PSD accompanied higher synapse density. These data suggest that GPER1 is positioned to support diverse functions key to synaptic plasticity in monkey dlPFC.

Estrogen Restores Multisynaptic Boutons in the Dorsolateral Prefrontal Cortex while Promoting Working Memory in Aged Rhesus Monkeys.

Humans and nonhuman primates are vulnerable to age- and menopause- related decline in working memory, a cognitive function reliant on area 46 of the dorsolateral prefrontal cortex (dlPFC). We showed previously that presynaptic mitochondrial number and morphology in monkey dlPFC neurons correlate with working memory performance. The current study tested the hypothesis that the types of synaptic connections these boutons form are altered with aging and menopause in rhesus monkeys and that these metrics may be coupled with mitochondrial measures and working memory. Using serial section electron microscopy, we examined the frequencies and characteristics of nonsynaptic, single-synaptic, and multisynaptic boutons (MSBs) in the dlPFC. In contrast to our previous observations in the monkey hippocampal dentate gyrus, where MSBs comprised ∼40% of boutons, the vast majority of dlPFC boutons were single-synaptic, whereas MSBs constituted a mere 10%. The frequency of MSBs was not altered by normal aging, but decreased by over 50% with surgical menopause induced by ovariectomy in aged monkeys. Cyclic estradiol treatment in aged ovariectomized animals restored MSB frequencies to levels comparable to young and aged premenopausal monkeys. Notably, the frequency of MSBs positively correlated with working memory scores, as measured by the average accuracy on the delayed response (DR) test. Furthermore, MSB incidence positively correlated with the number of healthy straight mitochondria in dlPFC boutons and inversely correlated with the number of pathological donut-shaped mitochondria. Together, our data suggest that MSBs are coupled to cognitive function and mitochondrial health and are sensitive to estrogen. Significance statement: Many aged menopausal individuals experience deficits in working memory, an executive function reliant on recurrent firing of prefrontal cortex (PFC) neurons. However, little is known about the organization of presynaptic inputs to these neurons and how they may be altered with aging and menopause. Multisynaptic boutons (MSBs) were of particular interest, because they form multiple synapses and can enhance coupling between presynaptic and postsynaptic neurons. We found that higher MSB frequency correlated with better working memory performance in rhesus monkeys. Additionally, aged surgically menopausal monkeys experienced a 50% loss of MSBs that was restored with cyclic estradiol treatment. Together, our findings suggest that hormone replacement therapy benefits cognitive aging, in part by retaining complex synaptic organizations in the PFC.

Investigation of timing preparation during response initiation and execution using a startling acoustic stimulus.

The purpose of the current study was to examine the processes involved in the preparation of timing during response initiation and execution through the use of a startling acoustic stimulus (SAS). In Experiment 1, participants performed a delayed response task in which a two key-press movement was to be initiated 200 ms after an imperative signal (IS) with either a short (200 ms) or long (500 ms) interval between key-presses. On selected trials, a SAS was presented to probe the preparation processes associated with the initiation delay and execution of the inter-key interval. The SAS resulted in a significant decrease in the initiation time, which was attributed to a speeding of pacemaker pulses used to time the delay interval, caused by an increased activation due to the SAS. Conversely, the SAS delayed the short inter-key interval, which was attributed to temporary interference with cortical processing. In Experiment 2, participants performed a 500-ms delayed response task involving two key-presses 200 ms apart. In this condition, the SAS resulted in significantly decreased initiation time and a delayed inter-key interval (p = .053). Collectively, these results support a different timeline for the preparation of the delay interval, which is thought to be prepared in advance of the IS, and the inter-key interval, which is thought to be prepared following the IS. This conclusion provides novel information with regard to timing preparation that is consistent with models in which response preparation, initiation, and execution are considered separate and dissociable processes.

Stigmergy, collective actions, and animal social spacing.

Collective animal behavior studies have led the way in developing models that account for a large number of individuals, but mostly have considered situations in which alignment and attraction play a key role, such as in schooling and flocking. By quantifying how animals react to one another's presence, when interaction is via conspecific avoidance rather than alignment or attraction, we present a mechanistic insight that enables us to link individual behavior and space use patterns. As animals respond to both current and past positions of their neighbors, the assumption that the relative location of individuals is statistically and history independent is not tenable, underscoring the limitations of traditional space use studies. We move beyond that assumption by constructing a framework to analyze spatial segregation of mobile animals when neighbor proximity may elicit a retreat, and by linking conspecific encounter rate to history-dependent avoidance behavior. Our approach rests on the knowledge that animals communicate by modifying the environment in which they live, providing a method to analyze social cohesion as stigmergy, a form of mediated animal-animal interaction. By considering a population of animals that mark the terrain as they move, we predict how the spatiotemporal patterns that emerge depend on the degree of stigmergy of the interaction processes. We find in particular that nonlocal decision rules may generate a nonmonotonic dependence of the animal encounter rate as a function of the tendency to retreat from locations recently visited by other conspecifics, which has fundamental implications for epidemic disease spread and animal sociality.

A spiking neural integrator model of the adaptive control of action by the medial prefrontal cortex.

Subjects performing simple reaction-time tasks can improve reaction times by learning the expected timing of action-imperative stimuli and preparing movements in advance. Success or failure on the previous trial is often an important factor for determining whether a subject will attempt to time the stimulus or wait for it to occur before initiating action. The medial prefrontal cortex (mPFC) has been implicated in enabling the top-down control of action depending on the outcome of the previous trial. Analysis of spike activity from the rat mPFC suggests that neural integration is a key mechanism for adaptive control in precisely timed tasks. We show through simulation that a spiking neural network consisting of coupled neural integrators captures the neural dynamics of the experimentally recorded mPFC. Errors lead to deviations in the normal dynamics of the system, a process that could enable learning from past mistakes. We expand on this coupled integrator network to construct a spiking neural network that performs a reaction-time task by following either a cue-response or timing strategy, and show that it performs the task with similar reaction times as experimental subjects while maintaining the same spiking dynamics as the experimentally recorded mPFC.

Time of onset and factors associated with delayed response post intradetrusor injection of onabotulinumtoxin a in patients with neurogenic and idiopathic overactive bladder syndrome.

Objective: The objective of this study was to determine risk factors for delayed response in patients with neurogenic and idiopathic overactive bladder (OAB) after intradetrusor onabotulinumtoxin A injection. Subjects and Methods: This is a retrospective study that included 87 patients who underwent onabotulinumtoxin A intradetrusor injection from October 2011 to November 2019. Patients were followed up at 2, 4, and 12 weeks post intervention in the outpatient clinic and over the phone. The data of patients with early response were compared with those with late response using univariate and multivariate analyses. Results: The study included 87 patients. The mean age was 41 ± 15.3 standard deviation, and 69% of the participants were female. Fifty-one percent were diagnosed with neurogenic OAB. A median response time to onabotulinumtoxin A injection of 7 days was demonstrated, and patients who responded during the first 7 days post procedure were considered early responders. Independent predictors for late response include diabetes (Relative risk: 3.89, P = 0.018, and 95% confidence interval [CI]: 1.26-11.98), >1 BTX-A session (Relative risk: 4, P = 0.011, and 95% CI: 1.38-11.6), and wet OAB (RR: 9.94, P = 0.002, and 95% CI: 2.31-42.17). Conclusions: The median time of onset post intradetrusor injection of onabotulinumtoxin A was found to be 7 days. Diabetes mellitus, wet OAB, and <1 Botox sessions were independent risk factors for late onset of response.

Separating Inhibitory and Excitatory Responses of Epileptic Brain to Single-Pulse Electrical Stimulation.

To enable an accurate recognition of neuronal excitability in an epileptic brain for modeling or localization of epileptic zone, here the brain response to single-pulse electrical stimulation (SPES) has been decomposed into its constituent components using adaptive singular spectrum analysis (SSA). Given the response at neuronal level, these components are expected to be the inhibitory and excitatory components. The prime objective is to thoroughly investigate the nature of delayed responses (elicited between 100[Formula: see text]ms-1 s after SPES) for localization of the epileptic zone. SSA is a powerful subspace signal analysis method for separation of single channel signals into their constituent uncorrelated components. The consistency in the results for both early and delayed brain responses verifies the usability of the approach.

[Numerical Response Analysis of PM2.5-O3 Compound Pollution in Beijing].

The multi-scale variation trend of PM2.5-O3 compound pollution events was analyzed based on air quality data, meteorological data, and COVID-19 data in Beijing from 2015 to 2020. For the threshold of compound pollution, a compound pollution index was proposed, and the numerical response trend was evaluated based on the generalized additive model. A distributed lag nonlinear model was introduced to analyze the risk response relationship between compound pollution and influencing factors. The results showed that the events of PM2.5-O3 compound pollution in Beijing decreased annually. At the same time, due to the influence of pollutant emissions and meteorological conditions, there were obvious seasonal effects, week effects, holiday effects, and epidemic effects. The composite pollution index had no correlation with rainfall but had a linear positive correlation with O3 and air temperature and a nonlinear correlation with other explanatory variables. Air pollutants and meteorological conditions had obvious lag effects on the composite pollution index, and the lag effects were mainly concentrated in 1-3 d. PM2.5, PM10, O3, SO2, and air temperature in high-value areas significantly increased the risk of compound pollution. The CO (1-6 mg·m-3), NO2 (38-118 µg·m-3), and relative humidity (54%-87%) in the median section would also increase the risk of compound pollution, as would low wind speed. The compound pollution events showed a trend of multi-day continuous pollution in the numerical response. Compared with PM2.5 and PM10, compound pollution events were more dependent on O3, and the compound pollution rate in high-value areas was 30.7%-47.5%. CO and relative humidity had little effect on compound pollution events. The air temperature had the greatest impact, and 84.7% of the composite pollution incidents occurred at 20-30℃.

A Model for Reinfections and the Transition of Epidemics.

Reinfections of infected individuals during a viral epidemic contribute to the continuation of the infection for longer periods of time. In an epidemic, contagion starts with an infection wave, initially growing exponentially fast until it reaches a maximum number of infections, following which it wanes towards an equilibrium state of zero infections, assuming that no new variants have emerged. If reinfections are allowed, multiple such infection waves might occur, and the asymptotic equilibrium state is one in which infection rates are not negligible. This paper analyzes such situations by expanding the traditional SIR model to include two new dimensionless parameters, ε and θ, characterizing, respectively, the kinetics of reinfection and a delay time, after which reinfection commences. We find that depending on these parameter values, three different asymptotic regimes develop. For relatively small θ, two of the regimes are asymptotically stable steady states, approached either monotonically, at larger ε (corresponding to a stable node), or as waves of exponentially decaying amplitude and constant frequency, at smaller ε (corresponding to a spiral). For θ values larger than a critical, the asymptotic state is a periodic pattern of constant frequency. However, when ε is sufficiently small, the asymptotic state is a wave. We delineate these regimes and analyze the dependence of the corresponding population fractions (susceptible, infected and recovered) on the two parameters ε and θ and on the reproduction number R0. The results provide insights into the evolution of contagion when reinfection and the waning of immunity are taken into consideration. A related byproduct is the finding that the conventional SIR model is singular at large times, hence the specific quantitative estimate for herd immunity it predicts will likely not materialize.