Infectious Disease Modelling

Globally rising cases of malaria have prompted concentrated efforts to control malaria transmission, utilising various mathematical models to support the Roll Back Malaria agenda. Many existing models with their specific modifications exhibit rigidity, limiting their application to inform malaria control interventions. This study addresses this limitation by employing a reduction technique on a comprehensive malaria control model to derive a simplified system that preserves the essential dynamics of the original system. We validate the accuracy of the reduced model by comparing the two models via Bayesian MCMC. Based on a simulation study, parameter identifiability analysis and sensitivity analysis, we compare the two models and show that the reduced system exhibits similar transmission characteristics as the full model. Our results demonstrate that the reduced model effectively captures the essential behaviour of the comprehensive model, while providing flexibility and computational efficiency, making it a valuable tool for evaluating and implementing malaria control strategies.

The effective reproduction number, Reff, is an important measure of transmission potential in the modelling of epidemics. It measures the average number of people that will be infected by a single contagious individual. A value of Reff > 1 suggests that an outbreak will occur, while Reff < 1 suggests the virus will die out. In response to the COVID-19 pandemic, countries worldwide are implementing a range of intervention measures, such as population-wide social distancing and case isolation, with the goal of reducing Reff to values below one, to slow or eliminate transmission. We analyse case data from 25 international locations to estimate their Reff values over time and to assess the effectiveness of interventions, equivalent to New Zealands Alert Levels 1-4, for reducing transmission. Our results show that strong interventions, equivalent to NZs Alert Level 3 or 4, have been successful at reducing Reff below the threshold for outbreak. In general, countries that implemented strong interventions earlier in their outbreak have managed to maintain case numbers at lower levels. These estimates provide indicative ranges of Reff for each Alert Level, to inform parameters in models of COVID-19 spread under different intervention scenarios in New Zealand and worldwide. Predictions from such models are important for informing policy and decisions on intervention timing and stringency during the pandemic. Executive SummaryO_LIIn response to the COVID-19 pandemic, countries around the world are implementing a range of intervention measures, such as population-wide social distancing and case isolation, with the goal of reducing the spread of the virus. C_LIO_LIReff, the effective reproduction number, measures the average number of people that will be infected by a single contagious individual. A value of Reff > 1 suggests that an outbreak will occur, while Reff < 1 suggests the virus will die out. C_LIO_LIComparing Reff in an early outbreak phase (no or low-level interventions implemented) with a later phase (moderate to high interventions) indicates how effective these measures are for reducing Reff. C_LIO_LIWe estimate early-phase and late-phase Reff values for COVID-19 outbreaks in 25 countries (or provinces/states). Results suggest interventions equivalent to NZs Alert Level 3-4 can successfully reduce Reff below the threshold for outbreak. C_LI

BackgroundThe limitations of forecasting (real-time statistical) and predictive (dynamic epidemiological) models have become apparent as COVID-19 has progressed from a rapid exponential ascent to a slower decent, which is dependent on unknowable parameters such as extent of social distancing and easing. We present a means to optimize a forecasting model by functionalizing our previously reported Asymmetric Gaussian model with SEIR-like parameters. Conversely, SEIR models can be adapted to better incorporate real-time data. MethodsOur previously reported asymmetric Gaussian model was shown to greatly improve on forecasting accuracy relative to use of symmetric functions, such as Gaussian and error functions for death rates and cumulative deaths, respectively. However, the reported asymmetric Gaussian implementation, which fitted well to the ascent and much of the recovery side of the real death rate data, was not agile enough to respond to changing social behavior that is resulting in persistence of infections and deaths in the later stage of recovery. We have introduced a time-dependent {sigma}(t) parameter to account for transmission rate variability due to the effects of behavioral changes such as social distancing and subsequent social easing. The {sigma}(t) parameter is analogous to the basic reproduction number R0 (infection factor) that is evidently not a constant during the progression of COVID-19 for a particular population. The popularly used SEIR model and its many variants are also incorporating a time dependent R0(t) to better describe the effects of social distancing and social easing to improve predictive capability when extrapolating from real-time data. ResultsComparisons are given for the previously reported Asymmetric Gaussian model and to the revised, what we call, SEIR Gaussian model. We also have developed an analogous model based on R0(t) that we call SEIR Statistical model to show the correspondence that can be attained. It is shown that these two models can replicate each other and therefore provide similar forecasts based on fitting to the same real-time data. We show the results for reported U.S. death rates up to June 12, 2020 at which time the cumulative death count was 113,820. The forecasted cumulative deaths for these two models and compared to the University of Washington (UW) IHME model are 140,440, 139,272, and 149,690 (for 8/4/20) and 147,819, 148, 912, and 201,129 (for 10/1/20), respectively. We also show how the SEIR asymmetric Gaussian model can also account for various scenarios of social distancing, social easing, and even re-bound outbreaks where the death and case rates begin climbing again. ConclusionsForecasting models, based on real-time data, are essential for guiding policy and human behavior to minimize the deadly impact of COVID-19 while balancing the need to socialize and energize the economy. It is becoming clear that changing social behavior from isolation to easing requires models that can adapt to the changing transmission rate in order to more accurately forecast death and case rates. We believe our asymmetric Gaussian approach has advantages over modified SEIR models in offering simpler governing equations that are dependent on fewer variables.

This study is based on the design and analysis of a novel age- and dose-structured model for assessing the population-level impact of the recently-approved R21/Matrix-M malaria vaccine (which is administered in three doses followed by a booster dose) on controlling the spread of malaria in children under five in Burkina Faso. While the current malaria vaccination program in Burkina Faso prioritizes children 0-3 years of age (Group 1 in our model), we also assessed a hypothetical scenario where children 3-5 years of age (Group 2 in our model) are also vaccinated (since children under five years of age suffer the brunt of malaria morbidity and mortality). The vaccination-free version of the model was calibrated using yearly cumulative malaria mortality data for children in Burkina Faso. In addition to establishing well-posedness, we showed that the disease-free equilibrium of the model is locally-asymptotically stable whenever the control reproduction number ([R]v) is below one. Conditions for achieving vaccine-induced herd immunity (needed for disease elimination) under varying age-group structures and dosage schedules were derived, and a global sensitivity analysis was conducted to identify the parameters of the model that most strongly influence [R]v. Simulations of a homogeneous model including only Group 1 indicate that administering only the first dose of the vaccine with baseline bednet usage requires an impractically high herd immunity threshold of 97%. However, with all four doses, herd immunity is achievable without bednet when the required coverage ratios receiving doses 2, 3, and the booster dose are 73% to 90%. With baseline bednets, these ratios drop to just 10%-30%, dramatically improving elimination prospects. In a heterogeneous setting incorporating both Groups 1 and 2, herd immunity can be achieved (with bednet at baseline) by vaccinating either 46% of the total population of Groups 1 and 2 or 75% of individuals in Group 1 alone. Simulations of the full two-group model (with bednet at baseline) show that vaccinating only children in Group 1 with the first dose reduces the cumulative number of new malaria cases and malaria-induced deaths in Group 1 by about 19%-20%, and produces spillover reductions of about 11%-12% in the unvaccinated Group 2, indicating a moderate indirect benefit across groups. If children in Group 1 receive all four doses, the reductions in Group 1 increase to about 36%-38%, with larger spillover reductions of about 25%-26% in Group 2. When both groups receive only the first dose, the malaria burden decreases by about 24%-26% in each group. The greatest reductions occur when both groups receive all four doses, yielding decreases of about 43%-46%. These results show that extending Burkina Fasos current vaccination program to include children in the 3-5-year age group can substantially improve malaria elimination prospects, particularly when combined with bednet usage at baseline levels or higher.

SUMMARY Background and Objectives The island of Mauritius experienced a COVID-19 outbreak from mid-March to end April 2020. The first three cases were reported on March 18 (Day 1) and the last locally transmitted case occurred on April 26 (Day 40). An island confinement was imposed on March 20 followed by a sanitary curfew on March 23. Supermarkets were closed as from March 25 (Day 8). There were a total of 332 cases including 10 deaths from Day 1 to Day 41. Control of the outbreak depended heavily on contact tracing, testing, quarantine measures and the adoption of personal protective measures (PPMs) such as social distancing, the wearing of face masks and personal hygiene by Mauritius inhabitants. Our objectives were to model and understand the evolution of the Mauritius outbreak using mathematical analysis, a logistic growth model and an SEIR compartmental model with quarantine and a reverse sigmoid effective reproduction number and to relate the results to the public health control measures in Mauritius. Methods The daily reported cumulative number of cases in Mauritius were retrieved from the Worldometer website at https://www.worldometers.info/coronavirus/country/mauritius/. A susceptible-exposed-infectious-quarantined-removed (SEIQR) model was introduced and analytically integrated under the assumption that the daily incidence of infectious cases evolved as the derivative of the logistic growth function. The cumulative incidence data was fitted using a logistic growth model. The SEIQR model was integrated computationally with an effective reproduction number (R_e) varying in time according to a three-parameter reverse sigmoid model. Results were compared with the retrieved data and the parameters were optimised using the normalised root mean square error (NRMSE) as a comparative statistic. Findings A closed-form analytical solution was obtained for the time-dependence of the cumulative number of cases. For a small final outbreak size, the solution tends to a logistic growth. The cumulative number of cases was well described by the logistic growth model (NRMSE = 0.0276, R^2=0.9945) and by the SEIQR model (NRMSE = 0.0270, R^2=0.9952) with the optimal parameter values. The value of R_e on the day of the reopening of supermarkets (Day 16) was found to be approximately 1.6. Interpretation A mathematical basis has been obtained for using the logistic growth model to describe the time evolution of outbreaks with a small final outbreak size. The evolution of the outbreak in Mauritius was consistent with one modulated by a time-varying effective reproduction number resulting from the epidemic control measures implemented by Mauritius authorities and the PPMs adopted by Mauritius inhabitants. The value of R_e{approx}1.6 on the reopening of supermarkets on Day 16 was sufficient for the outbreak to grow to large-scale proportions in case the Mauritius population did not comply with the PPMs. However, the number of cases remained contained to a small number which is indicative of a significant contribution of the PPMs in the public health response to the COVID-19 outbreak in Mauritius.

A simple and well known epidemiological deterministic model was selected to estimate the main results for the basic dynamics of the Covid-19 epidemic breakout in the city of Sao Paulo - Brazil. The methodology employed the SEIR Model to characterize the epidemics outbreak and future outcomes. A time-dependent incidence weight on the SEIR reproductive basic number accounts for local Mitigation Policies (MP). The insights gained from analysis of these successful interventions were used to quantify shifts and reductions on active cases, casualties, and estimatives on required medical facilities (ITU). This knowledge can be applied to other Brazilian areas. The analysis was applied to forecast the consequences of releasing the MP over specific periods of time. Herd Immunity (HI) analysis allowed estimating how far we are from reaching the HI threshold value, and the price to be paid.

The transmission of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) becomes pandemic but presents different incidences in the world. Mathematical models were formulated to describe the coronavirus disease 2019 (CoViD-19) epidemic in each country or region. At the beginning of the pandemic, many authors used the SIR (susceptible, infectious, and recovered compartments) and SEIR (including exposed compartment) models to estimate the basic reproduction number R0 for the CoViD-19 epidemic. These simple deterministic models assumed that the only available collection of the severe CoViD-19 cases transmitted the SARS-CoV-2 and estimated lower values for R0, ranging from 1.5 to 3.0. However, the major flaw in the estimation of R0 provided by the SIR and SEIR models was that the severe CoViD-19 patients were hospitalized, and, consequently, not transmitting. Hence, we proposed a more elaborate model considering the natural history of CoViD-19: the inclusion of asymptomatic, pre-symptomatic, mild and severe CoViD-19 compartments. The model also encompassed the fatality rate depending on age. This SEAPMDR model estimated R0 using the severe CoViD-19 data from Sao Paulo State (Brazil) and Spain, yielding higher values for R0, that is, 6.54 and 5.88, respectively. It is worth stressing that this model assumed that severe CoViD-19 cases were not participating in the SARS-CoV-2 transmission chain. Therefore, the SIR and SEIR models are not suitable to estimate R0 at the beginning of the epidemic by considering the isolated severe CoViD-19 data as transmitters.

Projections of the near future of daily case incidence of COVID-19 are valuable for informing public policy. Near-future estimates are also useful for outbreaks of other diseases. Short-term predictions are unlikely to be affected by changes in herd immunity. In the absence of major net changes in factors that affect reproduction number (R), the two-parameter exponential model should be a standard model - indeed, it has been standard for epidemiological analysis of pandemics for a century but in recent decades has lost popularity to more complex compartmental models. Exponential models should be routinely included in reports describing epidemiological models as a reference, or null hypothesis. Exponential models should be fitted separately for each epidemiologically distinct jurisdiction. They should also be fitted separately to time intervals that differ by any major changes in factors that affect R. Using an exponential model, incidence-count half-life (t1/2) is a better statistic than R. Here an example of the exponential model is applied to King County, Washington during Spring 2020. During the pandemic, the parameters and predictions of this model have remained stable for intervals of one to four months, and the accuracy of model predictions has outperformed models with more parameters. The COVID pandemic can be modeled as a series of exponential curves, each spanning an interval ranging from one to four months. The length of these intervals is hard to predict, other than to extrapolate that future intervals will last about as long as past intervals.

AO_SCPLOWBSTRACTC_SCPLOWThe determination of the infection fatality rate (IFR) for the novel SARS-CoV-2 coronavirus is a key aim for many of the field studies that are currently being undertaken in response to the pandemic. The IFR together with the basic reproduction number R0, are the main epidemic parameters describing severity and transmissibility of the virus, respectively. The IFR can be also used as a basis for estimating and monitoring the number of infected individuals in a population, which may be subsequently used to inform policy decisions relating to public health interventions and lockdown strategies. The interpretation of IFR measurements requires the calculation of confidence intervals. We present a number of statistical methods that are relevant in this context and develop an inverse problem formulation to determine correction factors to mitigate time-dependent effects that can lead to biased IFR estimates. We also review a number of methods to combine IFR estimates from multiple independent studies, provide example calculations throughout this note and conclude with a summary and "best practice" recommendations. The developed code is available online.

COVID-19 global dynamics is modeled by an adaptation of the deterministic SEIR Model, which takes into account two dominant lineages of the SARS-CoV-2, and a time-varying reproduction number to estimate the disease transmission behavior. Such a methodology can be applied worldwide to predict forecasts of the outbreak in any infected country. The pandemic in Brazil was selected as a first study case. Brazilian official published data from February 25th to August 30th, 2020 was used to adjust a few epidemiologic parameters. The estimated time-dependence mean value to the infected individuals (confirmed cases) presents - in logarithmic scale - standard deviation SD = 0.08 for over six orders of magnitude. Data points for additional three weeks were added after the model was complete, granting confidence on the outcomes. By the end of 2020, the predicted numbers of confirmed cases in Brazil, within 95% credible intervals, may reach 6 Million (5 -7), and fatalities would accounts for 180 (130 - 220) thousands. The total number of infected individuals is estimated to reach 13 {+/-} 1 Million, 6.2% of the Brazilian population. Regarding the original SARS-CoV-2 form and its variant, the only model assumption is their distinct incubation rates. The variant form reaches a maximum of 96% of exposed individuals as previously reported for South America.

Nipah virus (NiV) is a sporadic yet extremely deadly zoonotic pathogen, with reported case fatality rates of 40%-75% in impacted areas. Prolonged incubation, documented relapse, and delayed-onset encephalitis following apparent recovery indicate that NiV dynamics are influenced by intricate temporal processes. However, mechanistic contributions of these processes to epidemic persistence remain poorly understood. In this study, we develop and analyze a delay differential equation model for NiV transmission that explicitly incorporates incubation delay, relapse, and post-recovery delay effects. We compute a primary-transmission reproduction threshold (R0), characterize the disease-free and endemic equilibria, and analyze their stability, including delay-induced Hopf bifurcations. We show that relapse modifies the endemic-equilibrium existence condition, so an endemic equilibrium is not determined solely by the classical threshold criterion R0 = 1. We calibrate the model to NiV incidence data from Bangladesh (2001-2024) and perform simulations and sensitivity analyses to evaluate the effects of relapse and delays across epidemiological scenarios. Results indicate that sustained oscillations occur only under hypothetical parameter regimes, suggesting that delay-induced periodic outbreaks are unlikely under empirically informed conditions. Scenario analyses demonstrate that relapse and encephalitis-related delays predominantly influence post-peak dynamics, while incubation delay alters the time and intensity of the epidemic peak. We also introduce a relapse-driven replenishment fraction to quantify contribution of relapse to continued transmission, demonstrating its growing significance following the first outbreak peak. Overall, our results identify relapse as a key mechanism for epidemic persistence and underscore the importance of incorporating relapse and biological time delays into epidemiological modeling and public health strategies.

Real-time assessments of the reproduction rates are crucial for timely responses to changes in epidemic dynamics. Known effective reproduction numbers Rt are based on registered (visible) cases despite that asymptomatic and unregistered patients occur in all epidemics and need to be corrected to take into account the number of hidden cases. Since newborns and re-infections significantly affect the dynamics of epidemics, they should also be taken into account in calculations of Rt and recently proposed reproduction rates R{tau} - the ratios of the real numbers of infectious persons at different moments of time. The numbers of cases generated by symptomatic and asymptomatic patients were introduced, estimated using a novel mathematical model and compared with the results of classical SIR (Susceptible-Infectious-Removed) model for the COVID-19 pandemic dynamics in Austria. Reproduction rates R{tau} were estimated with the use of visible accumulated numbers of COVID-19 cases in Austria and Tanzania (including real-time approach). The proposed methods for calculating reproduction numbers may better reflect the COVID-19 pandemic dynamics than the results listed by John Hopkins University.

The first case of COVID-19 was reported in Kenya in March 2020 and soon after non-pharmaceutical interventions (NPIs) were established to control the spread of the disease. The NPIs consisted, and continue to consist, of mitigation measures followed by a period of relaxation of some of the measures. In this paper, we use a deterministic mathematical model to analyze the dynamics of the disease, during the first wave, and relate it to the intervention measures. In the process, we develop a new method for estimating the disease parameters. Our solutions yield a basic reproduction number, R0 = 2.76, which is consistent with other solutions. The results further show that the initial mitigation reduced disease transmission by 40% while the subsequent relaxation increased transmission by 25%. We also propose a mathematical model on how interventions of known magnitudes collectively affect disease transmission rates. The modelled positivity rate curve compares well with observations. If interventions of unknown magnitudes have occurred, and data is available on the positivity rate, we use the method of planar envelopes around a curve to deduce the modelled positivity rate and the magnitudes of the interventions. Our solutions deduce mitigation and relaxation effects of 42.5% and 26%, respectively; these percentages are close to values obtained by the solution of the SIRD system. Our methods so far apply to a single wave; there is a need to investigate the possibility of extending them to handle multiple waves.

Modeling is an important tool to utilize at the beginning of an infectious disease outbreak, as it allows estimation of parameters--such as the basic reproduction number, R0--that can be used to postulate how the outbreak may continue to spread. However, there exist many challenges that need to be accounted for, such as an unknown first case date, retrospective reporting of probable cases, changing dynamics between case count and death count trends, and the implementation of multiple control efforts and their delayed or diminished effects. Using the near-daily data provided from the recent outbreak of Sudan ebolavirus in Uganda as a case study, we create a model and present a framework aimed at overcoming these aforementioned challenges. The impact of each challenge is examined by comparing model estimates and fits throughout our framework. Indeed, we found that allowing for multiple fatality rates over the course of an outbreak generally resulted in better fitting models. On the other hand, not knowing the start date of an outbreak appeared to have large and non-uniform effects on parameter estimates, particularly at the beginning stages of an outbreak. While models that did not account for the decaying effect of interventions on transmission underestimated R0, all decay models run on the full dataset yielded precise R0 estimates, demonstrating the robustness of R0 as a measure of disease spread when examining data from the entire outbreak.

The recent outbreak of the novel coronavirus (COVID-19) pandemic which originated from the Wuhan City of China has devastated many parts of the globe. At present, non-pharmaceutical interventions are the widely available measures being used in combating and controlling this disease. There is great concern over the rampant unaccounted cases of individuals skipping the border during this critical period in time. We develop a deterministic compartmental model to investigate the impact of escapees on the transmission dynamics of COVID-19 in Zimbabwe. A suitable Lyapunov function has been used to show that the disease-free equilibrium is globally asymptotically stable provided [R]0 < 1. We performed global sensitivity analysis using the Latin-hyper cube sampling method and partial rank correlation coefficients to determine the most influential model parameters on the short and long term dynamics of the pandemic, so as to minimize uncertainties associated with our variables and parameters. Results confirm that there is a positive correlation between the number of escapees and the reported number of COVID-19 cases. It is shown that escapees are largely responsible for the rapid increase in local transmissions. Also, the results from sensitivity analysis show that an increase in the governmental role actions and a reduction in immigration rate will help to control and contain the disease spread.

This study aims to understand the features of the COVID-19 spread in the South Region of Brazil by estimating the Effective Reproduction Number (ERN)[R] e for the states of Parana (PR), Rio Grande do Sul (RS), and Santa Catarina (SC). We used the SIRD (Susceptibles-Infectious-Recovered-Dead) model to describe the past data and to simulate strategies for the gradual mitigation of the epidemic curve by applying non-pharmacological measures. Besides the SIRD model does not include some aspects of COVID-19, as the symptomatic and asymptomatic subgroups of individuals and the incubation period, for example, in this work we intend to use a classical and easy to handle model to introduce a thorough method of adjustment that allows us to achieve reliable fitting for the real data and to obtain insights about the current trends for the pandemic in each locality. Our results demonstrate that for localities for which the ERN is about 2, only rigid measures are efficient to avoid overwhelming the health care system. These findings corroborate the relevance of keeping the value of[R] e below 1 and applying containment measures early.

The agent-based model CovidSIMVL (github.com/ecsendmail/MultiverseContagion) is employed in this paper to delineate different network structures of transmission chains in simulated COVID-19 epidemics, where initial parameters are set to approximate spread from a single transmission source, and R0ranges between 1.5 and 2.5. The resulting Transmission Trees are characterized by breadth, depth and generations needed to reach a target of 50% infected from a starting population of 100, or self-extinction prior to reaching that target. Metrics reflecting efficiency of an epidemic relate closely to topology of the trees. It can be shown that the notion of superspreading individuals may be a statistical artefact of Transmission Tree growth, while superspreader events can be readily simulated with appropriate parameter settings. The potential use of contact tracing data to identify chain length and shared paths is explored as a measure of epidemic progression. This characterization of epidemics in terms of topological characteristics of Transmission Trees may complement equation-based models that work from rates of infection. By constructing measures of efficiency of spread based on Transmission Tree topology and distribution, rather than rates of infection over time, the agent-based approach may provide a method to characterize and project risks associated with collections of transmission events, most notably at relatively early epidemic stages, when rates are low and equation-based approaches are challenged in their capacity to describe or predict. MOTIVATION - MODELS KEYED TO CONTEMPLATED DECISIONSOutcomes are altered by changing the processes that determine them. If we wish to alter contagion-based spread of infection as reflected in curves that characterize changes in transmission rates over time, we must intervene at the level of the processes that are directly involved in preventing viral spread. If we are going to employ models to evaluate different candidate arrays of localized preventive policies, those models must be posed at the same level of granularity as the entities (people enacting processes) to which preventive measures will be applied. As well, the models must be able to represent the transmission-relevant dynamics of the systems to which policies could be applied. Further, the parameters that govern dynamics within the models must embody the actions that are prescribed/proscribed by the preventive measures that are contemplated. If all of those conditions are met, then at a formal or structural level, the models are conformant with the provisions of the Law of Requisite Variety1 or the restated version of that law - the good regulator theorem.2 On a more logistical or practical level, the models must yield summary measures that are responsive to changes in key parameters, highlight the dynamics, quantify outcomes associated with the dynamics, and communicate that information in a form that can be understood correctly by parties who are adjudicating on policy options. If the models meet formal/structural requirements regarding requisite variety, and the parameters have a plausible interpretation in relationship to real-world situations, and the metrics do not overly-distort the data contents that they summarize, then the models provide information that is directly relevant to decision-making processes. Models that meet these requirements will minimize the gap that separates models from decisions, a gap that will otherwise be filled by considerations other than the data used to create the models (for equation-based models) or the data generated by the simulations. In this work, we present an agent-based model that targets information requirements of decision-makers who are setting policy at a local level, or translate population level directives to local entities and operations. We employ an agent-based modeling approach, which enables us to generate simulations that respond directly to the requirements of the good regulator theorem. Transmission events take place within a spatio-temporal frame of reference in this model, and rates are not conditioned by a reproduction rate (R0) that is specified a priori. Events are a function of movement and proximity. To summarize dynamics and associated outcomes of simulated epidemics, we employ metrics reflecting topological structure of transmission chains, and distributions of those structures. These measures point directly to dynamic features of simulated outbreaks, they operationalize the "efficiency" construct, and they are responsive to changes in parameters that govern dynamics of the simulations.

In early 2020, the government of Rio Grande do Sul established a public-health assessment-response framework to halt the spread of SARS-CoV-2, called controlled-distancing model (CDM). This framework subdivided the state in 21 regions where it evaluated a composite index of disease transmission and health-service capacity. Updated on a weekly basis, the index placed regions on a color-coded scale of flags, which guided adoption of non-pharmaceutical interventions. We aim to evaluate the extent to which the CDM accurately assessed transmission and the effectiveness of its responses throughout 2020. We estimated the weekly effective reproduction number (Rt) of SARS-CoV-2, for each region, using a renewal-equation-based statistical model of notified COVID-19 deaths. Using Rt estimates, we explored whether flag colors assigned by the CDM either reflected or affected SARS-CoV-2 dissemination. Flag assignments did reflect variations in Rt, to a limited extent, but we found no evidence that they affected Rt in the short term. Medium-term effects were apparent in only four regions after eight or more weeks of red-flag assignment. Analysis of Google movement metrics showed no evidence that people moved differently under different flags. The dissociation between flag colors and the propagation of SARS-CoV-2 does not support the claim that non-pharmaceutical interventions are ineffective. Our results show, however, that decisions made under the CDM framework were ineffective both for influencing the movement of people and for halting the spread of the virus.

Brazils continental dimension poses a challenge to the control of the spread of COVID-19. Due to the country specific scenario of high social and demographic heterogeneity, combined with limited testing capacity, lack of reliable data, under-reporting of cases, and restricted testing policy, the focus of this study is twofold: (i) to develop a generalized SEIRD model that implicitly takes into account the quarantine measures, and (ii) to estimate the response of the COVID-19 spread dynamics to perturbations/uncertainties. By investigating the projections of cumulative numbers of confirmed and death cases, as well as the effective reproduction number, we show that the model parameter related to social distancing measures is one of the most influential along all stages of the disease spread and the most influential after the infection peak. Due to such importance in the outcomes, different relaxation strategies of social distancing measures are investigated in order to determine which strategies are viable and less hazardous to the population. The results highlight the need of keeping social distancing policies to control the disease spread. Specifically, the considered scenario of abrupt social distancing relaxation implemented after the occurrence of the peak of positively diagnosed cases can prolong the epidemic, with a significant increase of the projected numbers of confirmed and death cases. An even worse scenario could occur if the quarantine relaxation policy is implemented before evidence of the epidemiological control, indicating the importance of the proper choice of when to start relaxing social distancing measures.

With significant population fractions in many societies who refuse vaccines, it is important to reconsider how vaccination is incorporated into compartmental epidemiology models. It is still most common to apply the vaccination rate to the entire class of susceptibles, rather than to use the more realistic assumption that the vaccination rate function should depend only on the population of susceptibles who are willing and able to receive a vaccination. This study uses a simple generic disease model to address two questions: (1) How much error is introduced in key model outcomes by neglecting vaccine unwillingness?, and (2) Can the error be reduced by incorporating vaccine unwillingness into the vaccination rate constant rather than the rate diagram? The answers depend greatly on the time scale of interest. For the endemic time scale, where longterm behavior is studied with equilibrium point analysis, the error in neglecting unwillingess is large and cannot be improved upon by decreasing the vaccination rate constant. For the epidemic time scale, where the first big epidemic wave is studied with numerical simulations, the error can still be significant, particularly for diseases that are relatively less infectious and vaccination programs that are relatively slow.