Infectious Disease Modelling最新文献

{"title":"Understanding human mobility patterns under a public health emergency","authors":"Cheng Peng ,&nbsp;Nana Chen ,&nbsp;Bo-Wen Ming ,&nbsp;Anqi Zhang ,&nbsp;Yao Zuo ,&nbsp;Paulo C. Ventura ,&nbsp;Hongjie Yu ,&nbsp;Marco Ajelli ,&nbsp;Juanjuan Zhang","doi":"10.1016/j.idm.2025.10.009","DOIUrl":"10.1016/j.idm.2025.10.009","url":null,"abstract":"<div><h3>Background</h3><div>Understanding human mobility changes during epidemics is critical for predicting disease spread and planning interventions. However, capturing fine-scale dynamics is challenging.</div></div><div><h3>Methods</h3><div>This study analyzed high-resolution human mobility patterns in Shanghai, China, during the 2022 SARS-CoV-2 Omicron BA.2 outbreak using large-scale anonymized cellular signaling data. We investigated mobility shifts across five distinct epidemic phases (pre-outbreak, targeted interventions, citywide lockdown, targeted lifting, and reopening) stratified by age, sex, and travel purpose. A comprehensive evaluation of four gravity and four radiation spatial interaction models was conducted to assess their ability to explain the observed mobility patterns under varying demographic and behavioral conditions.</div></div><div><h3>Results</h3><div>Population size and distance were found to be primary drivers of mobility, with notable variations across demographic groups and travel purposes. During the lockdown, mobility significantly decreased, particularly for social-related trips and the working-age population, while the effect of distance was substantially higher. Although mobility volumes recovered post-lockdown, a larger effect of distance persisted, implying long-lasting behavioral changes. Our comparative analysis showed that while several variants of gravity and radiation models captured overall patterns effectively, their performance was context-dependent, varying significantly across epidemic phases, population subgroups, and travel purposes.</div></div><div><h3>Conclusion</h3><div>These findings highlight the importance of integrating different mobility models to capture the complex human mobility picture by different population groups during an epidemic outbreak. Overall, this study advances our understanding of behavioral adaptations during crises, enhancing preparedness and response planning.</div></div>","PeriodicalId":36831,"journal":{"name":"Infectious Disease Modelling","volume":"11 1","pages":"Pages 241-255"},"PeriodicalIF":2.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dynamics and optimal control for tuberculosis transmission via a data-validated periodic model","authors":"Chenkai Guo,&nbsp;Peng Wu","doi":"10.1016/j.idm.2025.09.002","DOIUrl":"10.1016/j.idm.2025.09.002","url":null,"abstract":"<div><div>China is the third-largest contributor to the global incidence of tuberculosis (TB), and there are significant differences in the prevalence of TB among different age groups. Therefore, it is necessary to study the contribution of adolescents to the transmission of tuberculosis. Given that tuberculosis in mainland China exhibits periodic transmission characteristics, a non-autonomous differential equation model that considers age stage and periodic transmission has been proposed. We derived the basic reproduction number <em>R</em>₀ of this model and proved the global asymptotic stability of the disease-free equilibrium when <em>R</em>₀ &lt; 1, as well as the persistence of the disease when <em>R</em>₀ &gt; 1. We estimated the basic reproduction number <em>R</em>₀ = 1.18, which indicates that tuberculosis in mainland China is of low endemicity. Sensitivity analysis tells us that the adolescent group has a significant impact on the transmission of tuberculosis and is an indispensable force. Furthermore, we constructed a tuberculosis transmission control model and proposed four optimal control strategies, calculated the strategy-related benefits (ACER) and the incremental benefits between strategies (ICER), and further provided targeted recommendations for controlling tuberculosis transmission among different groups.</div></div>","PeriodicalId":36831,"journal":{"name":"Infectious Disease Modelling","volume":"11 1","pages":"Pages 121-142"},"PeriodicalIF":2.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145097456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal prevention and control strategy of infectious disease: Cost-effectiveness analysis based on a modified dynamic model with economic loss","authors":"Wenjun Liu ,&nbsp;Guohua Zou ,&nbsp;Qin Bao ,&nbsp;Shouyang Wang","doi":"10.1016/j.idm.2025.11.001","DOIUrl":"10.1016/j.idm.2025.11.001","url":null,"abstract":"<div><div>The large-scale outbreaks of novel infectious diseases threaten public health, while strict intervention measures might slow down the economic activity. The effective prevention and control measures should balance cost and benefit. This study aims to explore the optimal intervention strategy for the infectious diseases by proposing a dynamic model with economic cost based on the modified SEIR model. Seven compartments were expanded as QSEAIRD model according to China's real practice in COVID-19. The parameters were estimated by minimizing the prediction error, and the GDP loss coefficients were introduced to quantify the economic costs of different measures. Thereafter, we formulated a corresponding algorithm to solve for the optimal prevention policies, which could control the epidemic within a specified time with minimized economic loss. Using Shanghai as a case study, we simulated the epidemic trends from March 2022 under different policy scenarios. We found that the government interventions effectively shortened the peak time by 60 % and significantly reduced its magnitude by 90 %. Without these measures, we predicted that Shanghai would reach the peaks of the first and second waves of infections at the end of 2022 and in June 2023, respectively, with the number of infections during the second peak being about 1/7 of that during the first. These results demonstrate that the government's prevention and control measures were effective in containing the epidemic. If relatively loose measures were adopted, the epidemic would not be controlled within one month, which would prolong the implementation of the prevention measures and increase economic loss. By conducting a cost-effectiveness analysis, the proposed model and algorithm can be flexibly applied to optimize the design of infectious disease prevention and control schemes under different scenarios, systematically enhancing the capacity to respond to the novel infectious diseases.</div></div>","PeriodicalId":36831,"journal":{"name":"Infectious Disease Modelling","volume":"11 1","pages":"Pages 377-388"},"PeriodicalIF":2.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145568709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dengue forecasting and outbreak detection in Brazil using LSTM: integrating human mobility and climate factors","authors":"Xiang Chen,&nbsp;Paula Moraga","doi":"10.1016/j.idm.2025.11.002","DOIUrl":"10.1016/j.idm.2025.11.002","url":null,"abstract":"<div><h3>Background</h3><div>Dengue fever is a major global health concern, with Brazil experiencing recurrent and severe outbreaks due to its favorable climate factors, socio-environmental conditions, and increasing human mobility. Accurate forecasting of dengue cases and outbreak risk is essential for early warning systems and effective public health interventions. Traditional forecasting models primarily rely on historical case data and climate variables, often neglecting the role of human movement in virus transmission. This study addresses this gap by incorporating human mobility data into a deep learning-based dengue forecasting framework.</div></div><div><h3>Method</h3><div>An LSTM-based model was developed to forecast weekly dengue cases and detect outbreaks across selected Brazilian cities. The model integrates historical dengue cases, lagged climate variables (temperature and humidity), and human mobility-adjusted imported cases to capture both temporal trends and spatial transmission dynamics. Its performance was evaluated against three alternative models: (1) an LSTM using only dengue case data, (2) an LSTM incorporating climate variables, and (3) an LSTM integrating climate and geographic neighborhood effects. Forecasting accuracy was assessed using Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Continuous Ranked Probability Score (CRPS), while outbreak classification was evaluated using accuracy, sensitivity, specificity, and the F1 score.</div></div><div><h3>Results</h3><div>The proposed mobility-enhanced LSTM model consistently outperformed all baselines in both dengue case forecasting and outbreak detection. Across all cities, it achieved lower MAE and MAPE values, indicating improved accuracy, while also demonstrating superior CRPS performance, reflecting well-calibrated uncertainty estimates. In outbreak classification, the model achieved the highest sensitivity and F1 scores, highlighting its effectiveness in detecting outbreak periods compared to models that relied solely on case trends, climate variables, or geographic proximity. The results underscore the importance of integrating mobility data in dengue forecasting, particularly in urban centers with high population movement.</div></div><div><h3>Conclusion</h3><div>By incorporating human mobility dynamics into deep learning-based forecasting, this study presents a scalable and adaptable framework for enhancing dengue early warning systems. The proposed model provides more accurate case predictions and outbreak classifications, offering actionable insights for public health planning and resource allocation. Beyond dengue, this approach can be extended to other vector-borne diseases influenced by mobility and climate factors, supporting more effective epidemic preparedness strategies worldwide.</div></div>","PeriodicalId":36831,"journal":{"name":"Infectious Disease Modelling","volume":"11 1","pages":"Pages 338-354"},"PeriodicalIF":2.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145519782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Predicting the burden of co-infections in seasonally driven dynamics of pediatric rotavirus and norovirus","authors":"Mohammadi Qurratul Ain,&nbsp;Angela Peace","doi":"10.1016/j.idm.2025.10.005","DOIUrl":"10.1016/j.idm.2025.10.005","url":null,"abstract":"<div><div>Rotavirus and norovirus are principal viral agents of acute gastroenteritis, primarily transmitted through close contact. Although each virus has its own capability to spread the epidemic, rotavirus and norovirus infection simultaneously is known to have more serious repercussions for children. To examine this coalition, we construct a robust co-infection mathematical model to analyze the prevalence of these diseases and the factors influenced by varying seasonal transmission rates. We conduct numerical simulations and equilibria analyses to capture the dynamics of the individual diseases and their co-infections. We investigate basic and seasonal reproductive numbers, perform sensitivity analysis on key parameters, and numerically explore shifts in the timing and strength of seasonal transmission rates. The research demonstrates how seasonal dynamics significantly impact reproductive numbers, as well as drive the potential burden of co-infection.</div></div>","PeriodicalId":36831,"journal":{"name":"Infectious Disease Modelling","volume":"11 1","pages":"Pages 278-302"},"PeriodicalIF":2.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quantifying mpox transmission and control: A regional analysis of vaccination strategies in East Africa","authors":"Musa Rabiu ,&nbsp;Bosede Fagbemigun ,&nbsp;Sunday Fadugba ,&nbsp;Michael Shatalov ,&nbsp;Kekana Malesela ,&nbsp;Adejimi Adeniji","doi":"10.1016/j.idm.2025.09.001","DOIUrl":"10.1016/j.idm.2025.09.001","url":null,"abstract":"<div><div>Africa is home to the endemic mpox disease, especially in the tropical rain-forest regions of Central and West Africa. Although it is mostly found in the Democratic Republic of the Congo, reports of it have also come from other neighboring African nations. To understand the dynamics of mpox, we studied its spread in Burundi, Uganda, Rwanda, Congo, and Kenya before and after the implementation of interventions. Using a Bayesian framework, a simple mathematical model of Susceptible-Infected-Recovered type was calibrated and fitted to the 2022 mpox data covering the period before the introduction of intervention strategies. The model was then re-stratified to incorporate key epidemiological features, including vaccination with imperfect efficacy, partial immunity, exposure, and demographics. The transmission of mpox varied throughout East Africa, with Uganda exhibiting the highest basic reproduction number <span><math><msub><mrow><mi>R</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> = 2.51, suggesting the possibility of a rapid spread. Despite having the highest initial infection count and the lowest <span><math><msub><mrow><mi>R</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> (1.23), Congo may have had delayed detection. The moderate <span><math><msub><mrow><mi>R</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> values (1.35 and 1.88) in Rwanda and Burundi have implications for prompt intervention to control epidemics. Transmission and vaccination rates have a non-linear relationship with the thresholds required to contain mpox outbreaks. Our model shows that in high-transmission settings, substantially higher vaccination coverage (exceeding 80 % at an effectiveness of 70 %) is required to reduce the control reproduction number below unity, whereas in moderate-transmission contexts, coverage above 40 % may suffice. These quantitative thresholds provide actionable guidance for tailoring vaccination strategies to different epidemiological conditions. In particular, sustained vaccination strategies that achieve coverage above the threshold predicted by our model (approximately 80 %) can guarantee mpox eradication, even in situations with strong transmission rates. While real-world complexities such as heterogeneous risk groups and behavioral factors may affect outcomes, these findings shed light on potential quantitative thresholds and provide a foundation for more detailed, population-specific modeling of mpox interventions.</div></div>","PeriodicalId":36831,"journal":{"name":"Infectious Disease Modelling","volume":"11 1","pages":"Pages 29-46"},"PeriodicalIF":2.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145019350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A multi-method study evaluating the inference of compartmental model parameters from a generative agent-based model","authors":"Elizabeth Hunter ,&nbsp;Jim Duggan","doi":"10.1016/j.idm.2025.10.002","DOIUrl":"10.1016/j.idm.2025.10.002","url":null,"abstract":"<div><div>Calibrating process models such as compartmental SIR Models to real data can be performed using either optimization or Bayesian techniques. To accurately assess the performance of these methods, synthetic outbreak data can be used. All information about the data generative process is known for synthetic data, while when using real data there are many unknowns such as under-reporting of cases or real parameter values. We propose using an agent-based model to generate synthetic data. Calibrating to synthetic datasets created using different agent contact structures can provide us with information on how changes in contact structures impact SIR model parameters. We compare results for two calibration methods: Nelder-Mead an optimization technique and HMC, a Bayesian technique. The analysis finds that the two calibration methods perform similar in terms of accuracy when looking at the Mean Absolute Error, Mean Absolute Scaled Error, and Relative Root Mean Squared Error. Looking at the model parameters, HMC is better able to capture the ground truth parameters then Nelder-Mead. The results of the calibration additionally show that the effective infectious period is sensitive to the changes in contact patterns and the proportion of susceptible individuals in the population. For choosing a calibration method, if overall accuracy is the desired outcome, either method should perform equally well, however, if the aim is to understand and analyse the model parameters HMC is a better choice. Understanding how the effective parameters such as the infectious period changes as contact patterns and vaccination rates change can provide valuable information in understanding how to interpret parameters calibrated from real world data that captures both isolation and vaccination.</div></div>","PeriodicalId":36831,"journal":{"name":"Infectious Disease Modelling","volume":"11 1","pages":"Pages 218-240"},"PeriodicalIF":2.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Vaccination games of boundedly rational parents toward new childhood immunization","authors":"Wei Yin ,&nbsp;Martial L. Ndeffo-Mbah ,&nbsp;Tamer Oraby","doi":"10.1016/j.idm.2025.09.004","DOIUrl":"10.1016/j.idm.2025.09.004","url":null,"abstract":"<div><div>Infectious diseases harm societies through disease-induced morbidity, mortality, loss of productivity, and inequality. Thus, controlling and preventing them is critical for public health and societal well-being. However, societies can hinder efforts to control the spread of diseases by failing to adhere to public health recommendations, such as through vaccine hesitancy. Various disease-transmission models have been utilized to help policymakers respond to (re)emerging outbreaks. The usefulness of such models in assessing the effectiveness of public health policies is significantly dependent on human behavior. This paper introduces a new model of parental behavior toward a new childhood immunization. The model incorporates societal features, social norms, and bounded rationality. We integrate this model with the dynamics of childhood disease, as depicted by a standard susceptible-infected-recovered model, to offer a detailed perspective on vaccine acceptance dynamics. We found that the behavioral model provides a new population game theory's replicator dynamical equation with an entropy-like term. Interestingly, societal norms and bounded rationality play a crucial role in shaping vaccine uptake through a novel function, which we term the critical societal vaccine cost. The results suggest that reduced vaccine costs below the critical societal vaccine cost and higher initial acceptance rates increase the probability of disease elimination. A gradual increase in vaccination costs, as an adaptive dynamic policy for disease eradication, is also possible. In particular, strong social norms and low levels of bounded rationality positively contribute to disease eradication even when the basic reproduction number of the disease in that society is large.</div></div>","PeriodicalId":36831,"journal":{"name":"Infectious Disease Modelling","volume":"11 1","pages":"Pages 150-164"},"PeriodicalIF":2.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145158855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Modeling the transmission dynamics and control strategies during the 2017 diphtheria outbreak in Jakarta, Indonesia","authors":"Bimandra A. Djaafara ,&nbsp;Verry Adrian ,&nbsp;Etrina Eriawati ,&nbsp;Iqbal R.F. Elyazar ,&nbsp;Raph L. Hamers ,&nbsp;J. Kevin Baird ,&nbsp;Guy E. Thwaites ,&nbsp;Hannah E. Clapham","doi":"10.1016/j.idm.2025.08.004","DOIUrl":"10.1016/j.idm.2025.08.004","url":null,"abstract":"<div><div>Diphtheria has resurged globally, including in Indonesia, despite widespread vaccination since the 1970s. Knowledge gaps persist in understanding contemporary transmission drivers and effective outbreak control, especially in densely populated areas like Jakarta. We analyzed the 2017 Jakarta outbreak data and developed a compartmental model incorporating estimates of population susceptibility and asymptomatic carriers. Key epidemiological parameters were estimated, and various control measures were simulated. Our study found overall diphtheria susceptibility at 12.9 % (95 % CrI: 8.6 %–19.0 %) and 28.0 % (95 % CrI: 20.5 %–36.0 %) in children under 5 under different modeling scenarios, which were below the 'herd immunity threshold'. We estimated asymptomatic carriers to be highly prevalent, substantially contributing to the reproduction number. The model indicated that contact tracing and treating suspected cases and their contacts were more effective in preventing new cases than catch-up vaccination alone. These findings provide valuable insights for future outbreak management strategies in similar settings.</div></div>","PeriodicalId":36831,"journal":{"name":"Infectious Disease Modelling","volume":"11 1","pages":"Pages 1-15"},"PeriodicalIF":2.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145019348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Impact of high-order time-delayed information on epidemic propagation in multiplex networks","authors":"Zehui Zhang ,&nbsp;Fang Wang ,&nbsp;Lilin Liu ,&nbsp;Lin Wang","doi":"10.1016/j.idm.2025.08.007","DOIUrl":"10.1016/j.idm.2025.08.007","url":null,"abstract":"<div><div>Traditional epidemic models often overlook disease incubation periods and high-order social interactions, limiting their ability to capture real-world transmission dynamics. To address these gaps, we develop a stochastic model that integrates both factors, investigating their combined effects on information diffusion and disease spread. Our framework consists of a two-layer network: an awareness layer, where disease-related information propagates through high-order delayed interactions, and an epidemic layer, where disease transmission follows an SIS model with incubation delays. Using a Markov chain approach, we derive outbreak thresholds and perform numerical simulations to assess the impact of delayed awareness adoption on epidemic outcomes. High-order delayed interactions accelerate information spread compared to traditional pairwise models. Interestingly, while incubation periods increase the risk of hidden transmission, they also provide a crucial window for awareness diffusion, potentially mitigating outbreaks. This dual role of incubation prolonging undetected transmission while enabling proactive awareness dissemination underscores the importance of synchronizing public health interventions with disease incubation phases.</div></div>","PeriodicalId":36831,"journal":{"name":"Infectious Disease Modelling","volume":"11 1","pages":"Pages 72-86"},"PeriodicalIF":2.5,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145047953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}