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During the SARS-CoV-2 pandemic, the effective reproduction number (R-eff) has frequently been used to describe the course of the pandemic. Analytical properties of R-eff are rarely studied. We analytically examine how and under which conditions the conventional susceptible–infected–removed (SIR) model (without infection age) serves as an approximation to the infection-age-structured SIR model. Special emphasis is given to the role of R-eff, which is an implicit parameter in the infection-age-structured SIR model and an explicit parameter in the approximation. The analytical findings are illustrated by a simulation study about an hypothetical intervention during a SARS-CoV-2 outbreak and by historical data from an influenza outbreak in Prussian army camps in the region of Arnsberg (Germany), 1918–1919. [ABSTRACT FROM AUTHOR]

A novel time-dependent deterministic SEIRS model, extended with vaccination, hospitalization, and vital dynamics, is introduced. Time-varying basic and effective reproduction numbers associated with this model are defined, which are crucial metrics in understanding epidemic dynamics. Furthermore, a parameter identification approach has been used to develop a numerical method to compute these numbers for long-term epidemics. We analyze the actual COVID-19 data from the USA, Italy, and Bulgaria to solve appropriate inverse problems and gain an understanding of the time evolution behavior of the basic and effective reproduction numbers. Moreover, an insightful comparison of key coronavirus data and epidemiological parameters across these countries has been conducted. For this purpose, while the basic and effective reproduction numbers provide insights into the virus transmission potential, we propose data-driven criteria for assessing the actual realization of the transmission potential of the SARS-CoV-2 virus and the effectiveness of the applied restrictive measures. To obtain these results, we conduct a mathematical analysis to demonstrate various biological properties of the new differential model, including non-negativity, boundedness, existence, and uniqueness of the solution. The new model and the associated numerical simulation tools proposed herein could be applied to COVID-19 data in any country worldwide and hold a promising potential for the transmission capacity and impact of the virus. [ABSTRACT FROM AUTHOR]

This paper studies a nonlinear epidemiological model called susceptible, exposed, asymptomatic infected, infected, and recovered (SEAIR) with a control input (vaccination), which helps to design a sliding mode vaccination strategy by tracking the prevalence of infectious diseases. First, the effective reproduction number is adjusted to the expected value to control infectious disease outbreaks, even with model parameter changes and systematic perturbations. Meanwhile, based on the sliding mode control theory, two control strategies, namely, first‐order sliding mode control and super‐twisting control, are proposed to control the spread of COVID‐19. Besides, an integral sliding surface associated with the effective reproduction number R0(t)$$ {R}_0(t) $$ is constructed to eliminate the chattering problem of the system. Moreover, the sliding mode controller is robust to uncertainties and external perturbations. The stability and finite‐time convergence of the closed‐loop control system are verified by Lyapunov's second method. Also, the effect of the two control strategies on the system performance is evaluated by numerical simulations. Furthermore, this paper presents a practical case simulation based on data from Wuhan. The results indicate that universal vaccination is crucial in the early stage of the COVID‐19 pandemic. As the epidemic progresses, vaccination is still an essential approach for epidemic prevention and control. [ABSTRACT FROM AUTHOR]

Monkey pox causes a rash which can be uncomfortable, itchy, and painful and its early detection is vital to every control mechanisms. Hence, the objective of this paper was the development and simulation of a mathematical model for monkey-pox transmission disease in Nigeria using Ordinary Differential Equations. The feasible region of the model was verified and solutions positivity was shown. We achieved the disease free equilibrium and computed effective reproduction number, Re of the model system. We show the global stability of disease free equilibrium and we found that the disease free equilibrium of the model system is globally asymptotically stable if Re < 1 and G(X1, X2) = 0. The model system is considered mathematically and epidemiologically well posed. Furthermore, the simulations of the model shows that the average secondary cases of disease increases as exposed individual increases and rate of infection increases. Again, the effective reproduction number reduces as vaccination increases and it is observed that as exposed nonhuman transmits at low rate than symptomatic reduced, it reduces the secondary cases of the disease. [ABSTRACT FROM AUTHOR]

Background: Tuberculosis (TB) remains a major public health issue in Iran, especially smear-positive pulmonary tuberculosis (SPPTB), due to its high transmission rate. Examining the effective reproduction number(Rt) of SPPTB and patient characteristics is crucial for crafting targeted TB control measures. This study aimed to assess the Rt of SPPTB in Iran from 2011 to 2021 and profile SPPTB patient demographics, initial smear bacilli density, diagnosis delays, and spatial distribution. Study Design: This is a historical cohort study. Methods: A time-dependent method was used to estimate Rt, and monthly data from the national TB registry were scrutinized from 2011 to 2021. Results: A decline was observed in SPPTB incidence rates of 50 909 SPPTB cases in Iran from 2011 to 2021. Approximately 29.1% of the cases were diagnosed within a month, while 44.5% experienced a one to three-month delay in diagnosis. The analysis revealed substantial heterogeneity in TB transmission dynamics across various provinces of Iran. Provinces such as Sistan and Baluchestan, Golestan, Guilan, Khuzestan, Tehran, and Khorasan Razavi exhibited the highest effective reproduction numbers. Additionally, there was a decreasing trend in the effective reproduction numbers across all provinces from 2011 to 2020. Conclusion: Effective reproduction numbers declined in most provinces from 2011 to 2020 but increased moderately after the COVID-19 pandemic, highlighting the need for targeted public health interventions. Although SPPTB incidence rates are declining nationally, elevated incidence rates and effective reproduction numbers in regions such as Sistan and Baluchestan, Golestan, Guilan, Khuzestan, Tehran, and Khorasan Razavi signify the need for persistent TB management efforts in Iran. [ABSTRACT FROM AUTHOR]

Objective: To investigate the effect of Enterovirus A71 (EV71) vaccination on the transmissibility of different enterovirus serotypes of hand, foot, and mouth disease (HFMD) in Zhejiang, China. Methods: Daily surveillance data of HFMD and EV71 vaccination from August 2016 to December 2019 were collected. Epidemic periods for each HFMD type were defined, and the time-varying effective reproduction number (Rt) was estimated, which could provide more direct evidence of disease epidemics than case number. General additive models (GAMs) were employed to analyze associations between EV71 vaccination quantity and rate and HFMD transmissibility. The epidemic prevention threshold, represented by required vaccination numbers and rates, was also estimated. Results: Vaccinating every 100,000 children ≤ 5 years could lead to a decrease in the Rt of EV71-associated HFMD by 14.44% (95%CI: 6.76%, 21.42%). Additionally, a positive correlation was observed between vaccinations among children ≤ 5 years old (per 100,000) and the increased transmissibility of other HFMD types (caused by enteroviruses other than EV71 and CA16) at 1.82% (95%CI: 0.80%, 2.84%). It was estimated that an additional 362,381 vaccinations, corresponding to increased vaccine coverage to 54.51% among children ≤ 5 years could effectively prevent EV71 epidemics in Zhejiang. Conclusions: Our findings highlight the importance of enhancing EV71 vaccine coverage for controlling the epidemic of EV71-HFMD and assisting government officials in developing strategies to prevent HFMD. [ABSTRACT FROM AUTHOR]

The Ebola virus disease (EVD) has been endemic since 1976, and the case fatality rate is extremely high. EVD is spread by infected animals, symptomatic individuals, dead bodies, and contaminated environment. In this paper, we formulate an EVD model with four transmission modes and a time delay describing the incubation period. Through dynamical analysis, we verify the importance of blocking the infection source of infected animals. We get the basic reproduction number without considering the infection source of infected animals. And, it is proven that the model has a globally attractive disease-free equilibrium when the basic reproduction number is less than unity; the disease eventually becomes endemic when the basic reproduction number is greater than unity. Taking the EVD epidemic in Sierra Leone in 2014–2016 as an example, we complete the data fitting by combining the effect of the media to obtain the unknown parameters, the basic reproduction number and its time-varying reproduction number. It is shown by parameter sensitivity analysis that the contact rate and the removal rate of infected group have the greatest influence on the prevalence of the disease. And, the disease-controlling thresholds of these two parameters are obtained. In addition, according to the existing vaccination strategy, only the inoculation ratio in high-risk areas is greater than 0.4, the effective reproduction number can be less than unity. And, the earlier the vaccination time, the greater the inoculation ratio, and the faster the disease can be controlled. [ABSTRACT FROM AUTHOR]

This article presents and analyzes a deterministic model for malaria transmission that incorporates infected individuals who are unaware of their infectious status (ignorant infected humans) and accounts for relapse dynamics. We explore the invariant region and positivity of the model and calculate the effective reproduction number using the next-generation matrix method. We demonstrate the local and global stability of disease-free equilibrium points using the Routh-Hurwitz criterion and Lyapunov function, respectively. The proposed model shows that a disease-free equilibrium point is globally asymptotically stable when the basic reproduction number Re < 1. We conducted a sensitivity analysis on the effective reproduction number to identify which basic parameter most significantly influences the increase or decrease of malaria cases. This study focuses on individuals who have been treated and cured but continue to carry dormant Plasmodium parasites in their blood, which can potentially cause relapse or reinfection. Additionally, we introduce a protected compartment to carefully evaluate how preventive measures influence the spread and persistence of malaria within the population.

Estimating effective reproduction number (Re) and predicting disease incidences are essential to formulate effective strategies for disease control. Although recent studies developed models for inferring Re from wastewater-based data, they require information on shedding dynamics. Here, we proposed a framework of Re estimation and prediction without shedding information. The framework consists of a space-state model for smoothing wastewater-based data and a renewal equation modified for wastewater-based data. The applicability of the framework was tested with simulated data and real-world data on Influenza A virus (IAV) and SARS-CoV-2 concentration in wastewater in 2022/2023 season in the USA. We confirmed the state-space model effectively fits various simulated epidemic curves and real-world data. In simulations, we found wastewater-based Re (Reww) closely aligns with instantaneous clinical Re when shedding dynamics are rapid. For more prolonged shedding, Reww approximates a smoothed Re over time. We also observed the necessary sampling frequency to trace dynamics of wastewater concentration and Reww accurately in the framework varies depending on the precision of detection methods, the epidemic status, the transmissibility of infectious diseases, and shedding dynamics. By applying our framework to real-world data, we found Reww for SARS-CoV-2 showed similar trend and values to clinically-based Re. Reww for IAV ranged from 0.66 to 1.52 with a clear peak in the winter season, which agrees with previously reported Re. We also succeeded in predicting wastewater concentration in a few weeks from available wastewater-based data. These results indicate that our framework potentially enables near real-time monitoring of approximated Re and prediction of infectious disease dynamics through wastewater surveillance, which limits the delay between infection and reporting. Our framework is useful especially for regions where reliable clinical surveillance is not available and notifiable surveillance is abolished, and can be expanded to multiple infectious diseases that have been detected from wastewater.

During the SARS-CoV-2 pandemic, the effective reproduction number (R-eff) has frequently been used to describe the course of the pandemic. Analytical properties of R-eff are rarely studied. We analytically examine how and under which conditions the conventional susceptible–infected–removed (SIR) model (without infection age) serves as an approximation to the infection-age-structured SIR model. Special emphasis is given to the role of R-eff, which is an implicit parameter in the infection-age-structured SIR model and an explicit parameter in the approximation. The analytical findings are illustrated by a simulation study about an hypothetical intervention during a SARS-CoV-2 outbreak and by historical data from an influenza outbreak in Prussian army camps in the region of Arnsberg (Germany), 1918–1919.

Key epidemiological parameters, including the effective reproduction number, R(t), and the instantaneous growth rate, r(t), generated from an ensemble of models, have been informing public health policy throughout the COVID-19 pandemic in the four nations of the United Kingdom of Great Britain and Northern Ireland (UK). However, estimation of these quantities became challenging with the scaling down of surveillance systems as part of the transition from the “emergency” to “endemic” phase of the pandemic.The Office for National Statistics (ONS) COVID-19 Infection Survey (CIS) provided an opportunity to continue estimating these parameters in the absence of other data streams. We used a penalised spline model fitted to the publicly-available ONS CIS test positivity estimates to produce a smoothed estimate of the prevalence of SARS-CoV-2 positivity over time. The resulting fitted curve was used to estimate the “ONS-based” R(t) and r(t) across the four nations of the UK. Estimates produced under this model are compared to government-published estimates with particular consideration given to the contribution that this single data stream can offer in the estimation of these parameters.Depending on the nation and parameter, we found that up to 77% of the variance in the government-published estimates can be explained by the ONS-based estimates, demonstrating the value of this singular data stream to track the epidemic in each of the four nations. We additionally find that the ONS-based estimates uncover epidemic trends earlier than the corresponding government-published estimates.Our work shows that the ONS CIS can be used to generate key COVID-19 epidemiological parameters across the four UK nations, further underlining the enormous value of such population-level studies of infection. This is not intended as an alternative to ensemble modelling, rather it is intended as a potential solution to the aforementioned challenge faced by public health officials in the UK in early 2022.

Background: During the COVID-19 pandemic, analytics and predictive models built on regional data provided timely, accurate monitoring of epidemiological behavior, informing critical planning and decision-making for health system leaders. At Atrium Health, a large, integrated healthcare system in the southeastern United States, a team of statisticians and physicians created a comprehensive forecast and monitoring program that leveraged an array of statistical methods. Methods: The program utilized the following methodological approaches: (i) exploratory graphics, including time plots of epidemiological metrics with smoothers; (ii) infection prevalence forecasting using a Bayesian epidemiological model with time-varying infection rate; (iii) doubling and halving times computed using changepoints in local linear trend; (iv) death monitoring using combination forecasting with an ensemble of models; (v) effective reproduction number estimation with a Bayesian approach; (vi) COVID-19 patients hospital census monitored via time series models; and (vii) quantified forecast performance. Results: A consolidated forecast and monitoring report was produced weekly and proved to be an effective, vital source of information and guidance as the healthcare system navigated the inherent uncertainty of the pandemic. Forecasts provided accurate and precise information that informed critical decisions on resource planning, bed capacity and staffing management, and infection prevention strategies. Conclusions: In this paper, we have presented the framework used in our epidemiological forecast and monitoring program at Atrium Health, as well as provided recommendations for implementation by other healthcare systems and institutions to facilitate use in future pandemics.

The effective reproduction number serves as a metric of population-wide, time-varying disease spread. During the early years of the COVID-19 pandemic, this metric was primarily derived from case data, which has varied in quality and representativeness due to changes in testing volume, test-seeking behavior, and resource constraints. Deriving nowcasting estimates from alternative data sources such as wastewater provides complementary information that could inform future public health responses. We estimated county-aggregated, sewershed-restricted wastewater-based SARS-CoV-2 effective reproduction numbers from May 1, 2022 to April 30, 2023 for five counties in California with heterogeneous population sizes, clinical testing rates, demographics, wastewater coverage, and sampling frequencies. We used two methods to produce sewershed-restricted effective reproduction numbers, both based on smoothed and deconvolved wastewater concentrations. We then population-weighted and aggregated these sewershed-level estimates to arrive at county-level effective reproduction numbers. Using mean absolute error (MAE), Spearman’s rank correlation (ρ), confusion matrix classification, and cross-correlation analyses, we compared the timing and trajectory of our two wastewater-based models to: (1) a publicly available, county-level ensemble of case-based estimates, and (2) county-aggregated, sewershed-restricted case-based estimates. Both wastewater models demonstrated high concordance with the traditional case-based estimates, as indicated by low mean absolute errors (MAE ≤ 0.09), significant positive Spearman correlation (ρ ≥ 0.66), and high confusion matrix classification accuracy (≥ 0.81). The relative timings of wastewater- and case-based estimates were less clear, with cross-correlation analyses suggesting strong associations for a wide range of temporal lags that varied by county and wastewater model type. This methodology provides a generalizable, robust, and operationalizable framework for estimating county-level wastewater-based effective reproduction numbers. Our retrospective evaluation supports the potential usage of real-time wastewater-based nowcasting as a complementary epidemiological tool for surveillance by public health agencies at the state and local levels. Based on this research, we produced publicly available wastewater-based nowcasts for the California Communicable diseases Assessment Tool (calcat.cdph.ca.gov).

We develop a mathematical model to investigate the effect of contact tracing on containing epidemic outbreaks and slowing down the spread of transmissible diseases. We propose a discrete-time epidemic model structured by disease-age which includes general features of contact tracing. The model is fitted to data reported for the early spread of COVID-19 in South Korea, Brazil, and Venezuela. The calibrated values for the contact tracing parameters reflect the order pattern observed in its performance intensity within the three countries. Using the fitted values, we estimate the effective reproduction number Re and investigate its responses to varied control scenarios of contact tracing. Alongside the positivity of solutions, and a stability analysis of the disease-free equilibrium are provided. [ABSTRACT FROM AUTHOR]

Key epidemiological parameters, including the effective reproduction number, RðtÞ, and the instantaneous growth rate, rðtÞ, generated from an ensemble of models, have been informing public health policy throughout the COVID-19 pandemic in the four nations of the United Kingdom of Great Britain and Northern Ireland (UK). However, estimation of these quantities became challenging with the scaling down of surveillance systems as part of the transition from the "emergency" to "endemic" phase of the pandemic. The Office for National Statistics (ONS) COVID-19 Infection Survey (CIS) provided an opportunity to continue estimating these parameters in the absence of other data streams. We used a penalised spline model fitted to the publicly-available ONS CIS test positivity estimates to produce a smoothed estimate of the prevalence of SARS-CoV-2 positivity over time. The resulting fitted curve was used to estimate the "ONS-based" RðtÞ and rðtÞ across the four nations of the UK. Estimates produced under this model are compared to government-published estimates with particular consideration given to the contribution that this single data stream can offer in the estimation of these parameters. Depending on the nation and parameter, we found that up to 77% of the variance in the government-published estimates can be explained by the ONS-based estimates, demonstrating the value of this singular data stream to track the epidemic in each of the four nations. We additionally find that the ONS-based estimates uncover epidemic trends earlier than the corresponding government-published estimates. Our work shows that the ONS CIS can be used to generate key COVID-19 epidemiological parameters across the four UK nations, further underlining the enormous value of such population-level studies of infection. This is not intended as an alternative to ensemble modelling, rather it is intended as a potential solution to the aforementioned challenge faced by public health officials in the UK in early 2022. [ABSTRACT FROM AUTHOR]

During the COVID-19 pandemic, analytics and predictive models built on regional data provided timely, accurate monitoring of epidemiological behavior, informing critical planning and decision-making for health system leaders. At Atrium Health, a large, integrated healthcare system in the southeastern United States, a team of statisticians and physicians created a comprehensive forecast and monitoring program that leveraged an array of statistical methods. The program utilized the following methodological approaches: (i) exploratory graphics, including time plots of epidemiological metrics with smoothers; (ii) infection prevalence forecasting using a Bayesian epidemiological model with time-varying infection rate; (iii) doubling and halving times computed using changepoints in local linear trend; (iv) death monitoring using combination forecasting with an ensemble of models; (v) effective reproduction number estimation with a Bayesian approach; (vi) COVID-19 patients hospital census monitored via time series models; and (vii) quantified forecast performance. A consolidated forecast and monitoring report was produced weekly and proved to be an effective, vital source of information and guidance as the healthcare system navigated the inherent uncertainty of the pandemic. Forecasts provided accurate and precise information that informed critical decisions on resource planning, bed capacity and staffing management, and infection prevention strategies. In this paper, we have presented the framework used in our epidemiological forecast and monitoring program at Atrium Health, as well as provided recommendations for implementation by other healthcare systems and institutions to facilitate use in future pandemics. [ABSTRACT FROM AUTHOR]

Monkey pox causes a rash which can be uncomfortable, itchy, and painful and its early detection is vital to every control mechanisms. Hence, the objective of this paper was the development and simulation of a mathematical model for monkey-pox transmission disease in Nigeria using Ordinary Differential Equations. The feasible region of the model was verified and solutions positivity was shown. We achieved the disease free equilibrium and computed effective reproduction number, of the model system. We show the global stability of disease free equilibrium and we found that the disease free equilibrium of the model system is globally asymptotically stable if Re < 1 and . The model system is considered mathematically and epidemiologically well posed. Furthermore, the simulations of the model shows that the average secondary cases of disease increases as exposed individual increases and rate of infection increases. Again, the effective reproduction number reduces as vaccination increases and it is observed that as exposed nonhuman transmits at low rate than symptomatic reduced, it reduces the secondary cases of the disease.

BackgroundPrior to the development of COVID-19 vaccines, policymakers instituted various non-pharmaceutical interventions (NPIs) to limit transmission. Prior studies have attempted to examine the extent to which these NPIs achieved their goals of containment, suppression, or mitigation of disease transmission. Existing evidence syntheses have found that numerous factors limit comparability across studies, and the evidence on NPI effectiveness during COVID-19 pandemic remains sparse and inconsistent. This study documents the magnitude and variation in NPI effectiveness in reducing COVID-19 transmission (i.e., reduction in effective reproduction rate [Reff] and daily contact rate) in Italy, the United States, the United Kingdom, and China.MethodsOur rapid review and narrative synthesis of existing research identified 126 studies meeting our screening criteria. We selected four contexts with >5 articles to facilitate a meaningful synthesis. This step yielded an analytic sample of 61 articles that used data from China, Italy, the United Kingdom, and the United States.ResultsWe found wide variation and substantial uncertainty around the effectiveness of NPIs at reducing disease transmission. Studies of a single intervention or NPIs that are the least stringent had estimated Reff reductions in the 10–50% range; those that examined so-called “lockdowns” were associated with greater Reff reductions that ranged from 40 to 90%, with many in the 70–80% range. While many studies reported on multiple NPIs, only six of the 61 studies explicitly used the framing of “stringency” or “mild versus strict” or “tiers” of NPIs, concepts that are highly relevant for decisionmakers.ConclusionExisting evidence suggests that NPIs reduce COVID-19 transmission by 40 to 90 percent. This paper documents the extent of the variation in NPI effectiveness estimates and highlights challenges presented by a lack of standardization in modeling approaches. Further research on NPI effectiveness at different stringency levels is needed to inform policy responses to future pandemics.

Acute bacterial infection of the intestine is caused by ingestion of food or water containing vibrio cholera. The symptoms include acute water diarrhea and vomiting which can result in severe dehydration or water loss. Sanitary conditions in the environment play an important role. Hence, the objective of this paper as to evaluate the effects of poor sanitation and public awareness in modeling bacterial infection amongst the students of a tertiary institution in Kaura Namoda, Zamfara State, Nigeria. We incorporated effectiveness of drug and awareness for proper hygiene and sanitation into our model. The disease free and endemic equilibrium were determined. The effective reproduction number Re was showed. Numerical results of the dynamics system of the transmission of bacterial infection were presented and we found that as the effective contact rate increases, the effective reproduction number increases. Also as the effectiveness of compliance of good hygiene increases, the effective reproduction number decreases by varying the contact rate. More so, as production rate of acute diarrhea bacteria increases, it increases the secondary cases of the infected individuals. [ABSTRACT FROM AUTHOR]

The ongoing COVID‐19 pandemic has posed a tremendous threat to the public and health authorities. Wuhan, as one of the cities experiencing the earliest COVID‐19 outbreak, has successfully tackled the epidemic finally. The main reason is the implementing of Fangcang shelter hospitals, which rapidly and massively scale the health system's capacity to treat COVID‐19 confirmed cases with mild symptoms. To give insights on what degree Fangcang shelter hospitals have contained COVID‐19 in Wuhan, we proposed a piecewise smooth model regarding the patient triage scheme and the bed capacities of Fangcang shelter hospitals and designated hospitals. We used data on the cumulative number of confirmed cases, recovered cases, deaths, and data on the number of hospitalized individuals in Fangcang shelter hospitals and designated hospitals in Wuhan to parameterize the targeted model. Our results showed that diminishing the bed capacity or delaying the opening time of Fangcang shelter hospitals, both would result in worsening the epidemic by increasing the total number of infectives and hospitalized individuals and the effective reproduction number Re(t)$$ {R}_e(t) $$. The findings demonstrated that Fangcang shelter hospitals avoided 17,013 critical infections and 17,823 total infections while it saved 7 days during the process of controlling the effective reproduction number Re(t)<1$$ {R}_e(t)<1 $$. Our study highlighted the critical role of Fangcang shelter hospitals in curbing and eventually stopping COVID‐19 outbreak in Wuhan, China. These findings may provide a valuable reference for decision‐makers in regarding ramping up the health system capacity to isolate groups of people with mild symptoms in areas of widespread infection. [ABSTRACT FROM AUTHOR]

The effective reproduction number ( R t ) is one of the most important epidemiological parameters, providing suggestions for monitoring the development trend of diseases and also for adjusting the prevention and control policies. However, a few studies have focused on the performance of some common computational methods for Rt. The purpose of this article is to compare the performance of three computational methods for Rt: the time-dependent (TD) method, the new time-varying (NT) method, and the sequential Bayesian (SB) method. Four evaluation methods—accuracy, correlation coefficient, similarity based on trend, and dynamic time warping distance—were used to compare the effectiveness of three computational methods for Rt under different time lags and time windows. The results showed that the NT method was a better choice for real-time monitoring and analysis of the epidemic in the middle and late stages of the infectious disease. The TD method could reflect the change of the number of cases stably and accurately, and was more suitable for monitoring the change of Rt during the whole process of the epidemic outbreak. When the data were relatively stable, the SB method could also provide a reliable estimate for Rt, while the error would increase when the fluctuation in the number of cases increased. The results would provide suggestions for selecting appropriate Rt estimation methods and making policy adjustments more timely and effectively according to the change of Rt. [ABSTRACT FROM AUTHOR]

Starting from May 31, 2023, the local transmission of monkeypox (Mpox) in mainland China began in Beijing. Till now, the transmission characteristics have not been explored. Based on the daily Mpox incidence data in the first 3 weeks of Beijing (from May 31 to June 21, 2023), we employed the instant‐individual heterogeneity transmission model to simultaneously calculate the effective reproduction number (Re) and the degree of heterogeneity (k) of the Beijing epidemic. We additionally simulated the monthly infection size in Beijing from July to November and compared with the reported data to project subsequent transmission dynamics. We estimated Re to be 1.68 (95% highest posterior density [HPD]: 1.12−2.41), and k to be 2.57 [95% HPD: 0.54−83.88], suggesting the transmission of Mpox in Beijing was supercritical and didn't have considerable transmission heterogeneity. We projected that Re fell in the range of 0.95−1.0 from July to November, highlighting more efforts needed to further reduce the Mpox transmissibility. Our findings revealed supercritical and homogeneous transmission of the Mpox epidemic in Beijing. Our results could serve as a reference for understanding and predicting the ongoing Mpox transmission in other regions of China and evaluating the effect of control measures. [ABSTRACT FROM AUTHOR]

This study presents a mathematical model to investigate the patterns of transmission in lymphatic filariasis. The model considers chronic, acute, and asymptomatic individuals and integrates key control strategies. Random synthetic data is generated robustly through numerical solutions to closely replicate real-world scenarios and encompass uncertainties. The synthetic data adheres to a Gaussian distribution to ensure validity and reliability. Following the derivation of the basic and effective reproduction number using the next generation matrix approach, Latin Hypercube Sampling (LHS) and the Partial Rank Correlation Coefficient (PRCC) algorithm is utilized to assess the parameters that significantly influence the model outputs. The study examine the trajectories of different population compartments through numerical simulations over time, with particular emphasis on the role played by asymptomatic individuals in the transmission of the disease. To assess the potential for disease elimination, the study introduces a range of strategies involving protective measures, treatment interventions, and mosquito control. These strategies are determined through sensitivity analysis. The findings demonstrate that the simultaneous implementation of all control measures has a noteworthy effect in managing lymphatic filariasis. In conclusion, the proposed model enhances understanding of lymphatic filariasis dynamics and informs effective control strategies.

BackgroundCountries across Europe have faced similar evolutions of SARS-CoV-2 variants of concern, including the Alpha, Delta, and Omicron variants.Materials and methodsWe used data from GISAID and applied a robust, automated mathematical substitution model to study the dynamics of COVID-19 variants in Europe over a period of more than 2 years, from late 2020 to early 2023. This model identifies variant substitution patterns and distinguishes between residual and dominant behavior. We used weekly sequencing data from 19 European countries to estimate the increase in transmissibility (Δβ) between consecutive SARS-CoV-2 variants. In addition, we focused on large countries with separate regional outbreaks and complex scenarios of multiple competing variants.ResultsOur model accurately reproduced the observed substitution patterns between the Alpha, Delta, and Omicron major variants. We estimated the daily variant prevalence and calculated Δβ between variants, revealing that: (i) Δβ increased progressively from the Alpha to the Omicron variant; (ii) Δβ showed a high degree of variability within Omicron variants; (iii) a higher Δβ was associated with a later emergence of the variant within a country; (iv) a higher degree of immunization of the population against previous variants was associated with a higher Δβ for the Delta variant; (v) larger countries exhibited smaller Δβ, suggesting regionally diverse outbreaks within the same country; and finally (vi) the model reliably captures the dynamics of competing variants, even in complex scenarios.ConclusionThe use of mathematical models allows for precise and reliable estimation of daily cases of each variant. By quantifying Δβ, we have tracked the spread of the different variants across Europe, highlighting a robust increase in transmissibility trend from Alpha to Omicron. Additionally, we have shown that the geographical characteristics of a country, as well as the timing of new variant entrances, can explain some of the observed differences in variant substitution dynamics across countries.

Accurately estimating the effective reproduction number is crucial for characterizing the transmissibility of infectious diseases to optimize interventions and responses during epidemic outbreaks. In this study, we improve the estimation of the effective reproduction number through two main approaches. First, we derive a discrete model to represent a time series of case counts and propose an estimation method based on this framework. We also conduct numerical experiments to demonstrate the effectiveness of the proposed discretization scheme. By doing so, we enhance the accuracy of approximating the underlying epidemic process compared to previous methods, even when the counting period is similar to the mean generation time of an infectious disease. Second, we employ a negative binomial distribution to model the variability of count data to accommodate overdispersion. Specifically, given that observed incidence counts follow a negative binomial distribution, the posterior distribution of secondary infections is obtained as a Dirichlet multinomial distribution. With this formulation, we establish posterior uncertainty bounds for the effective reproduction number. Finally, we demonstrate the effectiveness of the proposed method using incidence data from the COVID-19 pandemic.

Background: Monitoring the transmission of coronavirus disease 2019 (COVID-19) requires accurate estimation of the effective reproduction number (Rt). However, existing methods for calculating Rt may yield biased estimates if important real-world factors, such as delays in confirmation, pre-symptomatic transmissions, or imperfect data observation, are not considered. Method: To include real-world factors, we expanded the susceptible-exposed-infectious-recovered (SEIR) model by incorporating pre-symptomatic (P) and asymptomatic (A) states, creating the SEPIAR model. By utilizing both stochastic and deterministic versions of the model, and incorporating predetermined time series of Rt, we generated simulated datasets that simulate real-world challenges in estimating Rt. We then compared the performance of our proposed particle filtering method for estimating Rt with the existing EpiEstim approach based on renewal equations. Results: The particle filtering method accurately estimated Rt even in the presence of data with delays, pre-symptomatic transmission, and imperfect observation. When evaluating via the root mean square error (RMSE) metric, the performance of the particle filtering method was better in general and was comparable to the EpiEstim approach if perfectly deconvolved infection time series were provided, and substantially better when Rt exhibited short-term fluctuations and the data was right truncated. Conclusions: The SEPIAR model, in conjunction with the particle filtering method, offers a reliable tool for predicting the transmission trend of COVID-19 and assessing the impact of intervention strategies. This approach enables enhanced monitoring of COVID-19 transmission and can inform public health policies aimed at controlling the spread of the disease.

Objective: To investigate the effect of Enterovirus A71 (EV71) vaccination on the transmissibility of different enterovirus serotypes of hand, foot, and mouth disease (HFMD) in Zhejiang, China. Methods: Daily surveillance data of HFMD and EV71 vaccination from August 2016 to December 2019 were collected. Epidemic periods for each HFMD type were defined, and the time-varying effective reproduction number (Rt) was estimated, which could provide more direct evidence of disease epidemics than case number. General additive models (GAMs) were employed to analyze associations between EV71 vaccination quantity and rate and HFMD transmissibility. The epidemic prevention threshold, represented by required vaccination numbers and rates, was also estimated. Results: Vaccinating every 100,000 children ≤ 5 years could lead to a decrease in the Rt of EV71-associated HFMD by 14.44% (95%CI: 6.76%, 21.42%). Additionally, a positive correlation was observed between vaccinations among children ≤ 5 years old (per 100,000) and the increased transmissibility of other HFMD types (caused by enteroviruses other than EV71 and CA16) at 1.82% (95%CI: 0.80%, 2.84%). It was estimated that an additional 362,381 vaccinations, corresponding to increased vaccine coverage to 54.51% among children ≤ 5 years could effectively prevent EV71 epidemics in Zhejiang. Conclusions: Our findings highlight the importance of enhancing EV71 vaccine coverage for controlling the epidemic of EV71-HFMD and assisting government officials in developing strategies to prevent HFMD.

Acute bacterial infection of the intestine is caused by ingestion of food or water containing vibrio cholera. The symptoms include acute water diarrhea and vomiting which can result in severe dehydration or water loss. Sanitary conditions in the environment play an important role. Hence, the objective of this paper as to evaluate the effects of poor sanitation and public awareness in modeling bacterial infection amongst the students of a tertiary institution in Kaura Namoda, Zamfara State, Nigeria. We incorporated effectiveness of drug and awareness for proper hygiene and sanitation into our model. The disease free and endemic equilibrium were determined. The effective reproduction number was showed. Numerical results of the dynamics system of the transmission of bacterial infection were presented and we found that as the effective contact rate increases, the effective reproduction number increases. Also as the effectiveness of compliance of good hygiene increases, the effective reproduction number decreases by varying the contact rate. More so, as production rate of acute diarrhea bacteria increases, it increases the secondary cases of the infected individuals.

The search for better tools for interpreting and understanding wastewater surveillance has continued since the beginning of the coronavirus disease 2019 (COVID-19) pandemic. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has continued to mutate, thus complicating the interpretation of surveillance results. We assessed the Omicron variants (BA.1, BA.2, and BA.5) associated with wastewater-derived SARS-CoV-2 RNA trends by estimating the effective reproduction number (Reff) using an epidemic model that integrates explicitly the SARS-CoV-2 N2 gene concentration detected in wastewater through rt-qPCR quantitative analysis. The model inferred COVID-19 cases based on wastewater data and compared them with the ones reported by clinical surveillance. The variant of the SARS-CoV-2 associated with the wastewater-derived viral RNA was monitored through wastewater whole-genome sequencing. Three major waves between January and September 2022 were associated with the Omicron subvariants (BA.1, BA.2, and BA.5). This work showed that disease trends can be monitored using estimates of the effective reproduction number which is simple and easy to understand. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a SEIR-type COVID-19 model where the infected class is further divided into subclasses with individuals in intensive care (ICUs) and ventilation units. The model is calibrated with the symptomatic COVID-19 cases, deaths, and the number of patients in ICUs and ventilation units as reported by Republic of Turkey, Ministry of Health for the period 11 March 2020 through 30 May 2020 when the nationwide lockdown is in order. COVID-19 interventions in Turkey are incorporated into the model to detect the future trend of the outbreak accurately. We tested the effect of underreporting and we found that the peaks of the disease differ significantly depending on the rate of underreporting, however, the timing of the peaks remains constant. The lockdown is lifted on 1 June, and the model is modified to include a time-dependent transmission rate which is linked to the effective reproduction number ℛt through basic reproduction number ℛ0. The modified model captures the changing dynamics and peaks of the outbreak successfully. With the onset of vaccination on 13 January 2021, we augment the model with the vaccination class to investigate the impact of vaccination rate and efficacy. We observe that vaccination rate is a more critical parameter than the vaccine efficacy to eliminate the disease successfully. [ABSTRACT FROM AUTHOR]

Accepted by: Prof. M. Zied Babai This research empirically identifies the best-performing forecasting methods for the Effective Reproduction Number Rt of coronavirus disease 2019, the most used epidemiological parameter for policymaking during the pandemic. Furthermore, based on the most accurate forecasts for the United Kingdom, we model the excess exports and imports during the pandemic (using World Trade Organization data), whilst simultaneously controlling for governmental decisions, i.e. lockdown(s) and vaccination. We provide empirical evidence that the longer the lockdown lasts, the larger the cost to the economy is, predominantly for international trade. We show that imposing a lockdown leads to exports falling by 16.55% in the United Kingdom; without a lockdown, the respective decrease for the same period would be only 1.57%. On the other hand, efforts towards fast population vaccination improve the economy. We believe our results can help policymakers to make better decisions before and during future pandemics. [ABSTRACT FROM AUTHOR]

Rational allocation of limited vaccine resources is one of the key issues in the prevention and control of emerging infectious diseases. An age-structured infectious disease model with limited vaccine resources is proposed to explore the optimal vaccination ages. The effective reproduction number R e of the epidemic disease is computed. It is shown that the reproduction number is the threshold value for eradicating disease in the sense that the disease-free steady state is globally stable if R e < 1 and there exists a unique endemic equilibrium if R e > 1 . The effective reproduction number is used as an objective to minimize the disease spread risk. Using the epidemic data from the early spread of Wuhan, China and demographic data of Wuhan, we figure out the strategies to distribute the vaccine to the age groups to achieve the optimal vaccination effects. These analyses are helpful to the design of vaccination schedules for emerging infectious diseases. [ABSTRACT FROM AUTHOR]

We discuss the spread of a piece of news in a population. This is modeled by SIR model of epidemic spread. The model can be reduced to a nonlinear differential equation for the number of people affected by the news of interest. The differential equation has an exponential nonlinearity and it can be approximated by a sequence of nonlinear differential equations with polynomial nonlinearities. Exact solutions to these equations can be obtained by the Simple Equations Method (SEsM). Some of these exact solutions can be used to model a class of waves associated with the spread of the news in a population. The presence of exact solutions allow to study in detail the dependence of the amplitude and the time horizon of the news waves on the wave parameters, such as the size of the population, initial number of spreaders of the piece of the news, transmission rate, and recovery rate. This allows for recommendations about the change of wave parameters in order to achieve a large amplitude or appropriate time horizon of the news wave. We discuss five types of news waves on the basis of the values of the transmission rate and recovery rate—types A, B, C, D, and E of news waves. In addition, we discuss the possibility of building wavetrains by news waves. There are three possible kinds of wavetrains with respect of the amplitude of the wave: increasing wavetrain, decreasing wavetrain, and mixed wavetrain. The increasing wavetrain is especially interesting, as it is connected to an increasing amplitude of the news wave with respect to the amplitude of the previous wave of the wavetrain. It can find applications in advertising, propaganda, etc. [ABSTRACT FROM AUTHOR]

Describing viral outbreaks, such as the COVID-19 pandemic, often involves employing compartmental models composed of ordinary differential equation (ODE) systems. Estimating the parameter values for these ODE models is crucial and relies on accessible data. To accurately represent realistic pandemic scenarios with diverse situations, it is necessary to consider model parameters as time dependent. However, estimating such time-dependent parameters, like transition rates in compartmental models, is notoriously challenging due to the unknown function class of these parameters. In this study, we propose a novel approach by using an Augmented Kalman Smoother (AKS) combined with an Expectation-Maximization (EM) algorithm to simultaneously estimate all time-dependent parameters in an SIRD compartmental model. Our approach can be applied to general ODE systems with time-varying parameters, requiring no prior knowledge of model parameters or additional assumptions on the function class of the ODE time dependencies. A key advantage of our method compared to other methods is that it does not require assumptions about the parameterization of the serial interval distribution for estimating SIRD model parameters. Applying our approach to COVID-19 data in Germany, we adequately describe time-series data with strong fluctuations and multiple waves, obtaining non-parametric model-based time-course estimates for the effective reproduction number. [ABSTRACT FROM AUTHOR]

Influenza-Like Illness (ILI) constitutes a significant global health concern characterized by its high infection rates and widespread distribution worldwide. While influenza viruses, primarily types A and B, are primary contributors to ILI cases, other respiratory viruses also play a role in its prevalence. Jakarta, Indonesia's largest and densely populated city, has consistently reported a notable weekly number of ILI cases from 2016 to mid-2022. Intriguingly, this pattern of cases is irregular and does not exhibit a direct association with seasonal climate fluctuations. In response to this complex scenario, we have developed a SLIR mathematical model featuring a cumulative generating operator in the form of a multiple-terms sigmoid function, obtained from weekly cumulative data to derive model solutions. A total of 12 terms within the sigmoid function yielded a decent fit to the actual data spanning 339 weeks. Our correlation analysis unveiled distinct temporal relationships within the model, revealing an 8-week time lag between the dynamics of the infection rate and the latent compartment, along with a 2-week lag marking the incubation period between the latent and infected compartments. Furthermore, the effective reproduction number displayed recurrent fluctuations around a threshold of 1, indicating the endemic characteristics where infection persists within the population. This in-depth comprehension of ILI transmission dynamics and effective reproduction numbers plays a significant role in devising control measures and informed policy-making decisions. [ABSTRACT FROM AUTHOR]

Background: Monitoring the transmission of coronavirus disease 2019 (COVID-19) requires accurate estimation of the effective reproduction number (Rt). However, existing methods for calculating Rt may yield biased estimates if important real-world factors, such as delays in confirmation, pre-symptomatic transmissions, or imperfect data observation, are not considered. Method: To include real-world factors, we expanded the susceptible-exposed-infectiousrecovered (SEIR) model by incorporating pre-symptomatic (P) and asymptomatic (A) states, creating the SEPIAR model. By utilizing both stochastic and deterministic versions of the model, and incorporating predetermined time series of Rt, we generated simulated datasets that simulate real-world challenges in estimating Rt. We then compared the performance of our proposed particle filtering method for estimating Rt with the existing EpiEstim approach based on renewal equations. Results: The particle filtering method accurately estimated Rt even in the presence of data with delays, pre-symptomatic transmission, and imperfect observation. When evaluating via the root mean square error (RMSE) metric, the performance of the particle filtering method was better in general and was comparable to the EpiEstim approach if perfectly deconvolved infection time series were provided, and substantially better when Rt exhibited short-term fluctuations and the data was right truncated. Conclusions: The SEPIAR model, in conjunction with the particle filtering method, offers a reliable tool for predicting the transmission trend of COVID-19 and assessing the impact of intervention strategies. This approach enables enhanced monitoring of COVID-19 transmission and can inform public health policies aimed at controlling the spread of the disease. [ABSTRACT FROM AUTHOR]

In this study, we use a modified SEIR compartmental model to investigate the transmission dynamics of tuberculosis as well as its detection and treatment. The construction of this model is based on the hypothesis that the total population can be divided into seven compartments: susceptible, vaccinated, latently infected, diagnosed infected, undiagnosed infected, treated individuals, delayed treated individuals, and recovered individuals with a nonautonomous system. The effective reproduction number shows that the amplitude of the effective reproduction number decreases with the increasing vaccination rate and increases with the increase in the degree of seasonality of TB. The stability analyses of the model show that the value of the basic reproduction number R0 acts as a threshold between disease-free and endemic equilibrium. The model is found to be locally and globally asymptotically stable at the disease-free equilibrium when R01. The sensitivity of the parameters of the corresponding autonomous system is examined using the partial rank correlation coefficients (PRCC) analysis, which demonstrates that identification has a positive index and treatment has a negative index. The model is simulated using the RK-45 numerical method, and the parameter values for the model are taken from the available literature. Finally, the model outcome was compared with the real field data and found to be consistent.

Bovine cysticercosis and human taeniasis are neglected diseases caused by the beef tapeworm Taenia saginata. These diseases affect both human and animal health, rural livestock producers’ livelihoods, and the economies of the nations. Bovine cysticercosis makes beef unfit for human consumption, thus decreasing the cattle market value. In this study, a mathematical model for the dynamics of human taeniasis and bovine cysticercosis is examined in the context of diseases’ control efforts. The analysis of the basic model shows that both disease free and endemic equilibria exist. The basic reproduction number R0 is computed by the next generation method. The disease free equilibrium is globally asymptotically stable (GAS) when R01. To determine parameters that drive the diseases, the normalized forward sensitivity index method is adopted. The findings demonstrate that human and animal recruitment rates, the probability of humans to contract taeniasis, the rate at which humans with taeniasis defecate in the environment and T. saginata eggs’ natural mortality rate influence the diseases’ dynamics. The effects of several interventions including vaccination of cattle, treatment of infected humans and cattle, proper beef cooking, enhanced hygiene and sanitation, and the use of chemicals to kill T.saginata eggs in the environment are evaluated. When such interventions are administered, the model exhibits forward bifurcation, and secondary infections reduce significantly with time. Therefore, to control the diseases we recommend that more efforts be directed to treat humans with taeniasis and proper beef cooking and meat inspection be encouraged.

BackgroundChina managed coronavirus disease 2019 (COVID-19) with measures against Class B infectious diseases, instead of Class A infectious diseases, in a major shift of its epidemic response policies. We aimed to generate robust information on the transmission dynamics of novel coronavirus infection in Shanxi, a province located in northern China, after the implementation of the “Class B infectious disease Class B management” policy.MethodsWe consolidated infection data in Shanxi province from December 6, 2022 to January 14, 2023 through a network questionnaire survey and sentinel surveillance. A dynamics model of the SEIQHCVR was developed to track the infection curves and effective reproduction number (Rt).ResultsOur model was effective in estimating the trends of novel coronavirus infection, with the coefficient of determination (R2) above 90% in infections, inpatients, and critically ill patients. The number of infections in Shanxi province as well as in urban and rural areas peaked on December 20, 2022, with the peak of inpatients and critically ill patients occurring 2 to 3 weeks after the peak of infections. By the end of January 2023, 87.72% of the Shanxi residents were predicted to be infected, and the outbreak subsequently subsided. A small wave of COVID-19 infections may re-emerge at the end of April. In less than a month, the Rt values of positive infections, inpatients and critically ill patients were all below 1.0.ConclusionThe outbreak in Shanxi province is currently at a low prevalence level. In the face of possible future waves of infection, there is a strong need to strengthen surveillance and early warning.

This paper considers SEPIR, an extension of the well-known SEIR continuous simulation compartment model. Both models can be fitted to real data as they include parameters that can be estimated from the data. SEPIR deploys an additional presymptomatic infectious compartment, not modelled in SEIR but known to exist in COVID-19. This stage can also be fitted to data. We focus on how to fit SEPIR to a first wave of COVID. Both SEIR and SEPIR and the existing SEIR models assume a homogeneous mixing population with parameters fixed. Moreover, neither includes dynamically varying control strategies deployed against the virus. If either model is to represent more than just a single wave of the epidemic, then the parameters of the model would have to be time dependent. In view of this, we also show how reproduction numbers can be calculated to investigate the long-term overall outcome of an epidemic. [ABSTRACT FROM AUTHOR]

Background: Severe acute respiratory syndrome (SARS) is a form of atypical pneumonia which took hundreds of lives when it swept the world two decades ago. The pathogen of SARS was identified as SARS-coronavirus (SARS-CoV) and it was mainly transmitted in China during the SARS epidemic in 2002-2003. SARS-CoV and SARS-CoV-2 have emerged from the SARS metapopulation of viruses. However, they gave rise to two different disease dynamics, a limited epidemic, and an uncontrolled pandemic, respectively. The characteristics of its spread in China are particularly noteworthy. In this paper, the unique characteristics of time, space, population distribution and transmissibility of SARS for the epidemic were discussed in detail. Methods: We adopted sliding average method to process the number of reported cases per day. An SEIAR transmission dynamics model, which was the first to take asymptomatic group into consideration and applied indicators of R0, Reff, Rt to evaluate the transmissibility of SARS, and further illustrated the control effectiveness of interventions for SARS in 8 Chinese cities. Results: The R0 for SARS in descending order was: Tianjin city (R0 = 8.249), Inner Mongolia Autonomous Region, Shanxi Province, Hebei Province, Beijing City, Guangdong Province, Taiwan Province, and Hong Kong. R0 of the SARS epidemic was generally higher in Mainland China than in Hong Kong and Taiwan Province (Mainland China: R0 = 6.058 ± 1.703, Hong Kong: R0 = 2.159, Taiwan: R0 = 3.223). All cities included in this study controlled the epidemic successfully (Reff<1) with differences in duration. Rt in all regions showed a downward trend, but there were significant fluctuations in Guangdong Province, Hong Kong and Taiwan Province compared to other areas. Conclusion: The SARS epidemic in China showed a trend of spreading from south to north, i.e., Guangdong Province and Beijing City being the central regions, respectively, and from there to the surrounding areas. In contrast, the SARS epidemic in the central region did not stir a large-scale transmission. There were also significant differences in transmissibility among eight regions, with R0 significantly higher in the northern region than that in the southern region. Different regions were able to control the outbreak successfully in differences time. [ABSTRACT FROM AUTHOR]

The use of vaccines has always been controversial. Individuals in society may have different opinions about the benefits of vaccines. As a result, some people decide to get vaccinated, while others decide otherwise. The conflicting opinions about vaccinations have a significant impact on the spread of a disease and the dynamics of an epidemic. This study proposes a mathematical model of COVID-19 to understand the interactions of two populations: the low risk population and the high risk population, with two preventive measures. Unvaccinated individuals with chronic diseases are classified as high risk population while the rest are a low risk population. Preventive measures used by low risk group include vaccination (pharmaceutical way), while for the high risk population they include wearing masks, social distancing and regular hand washing (non-pharmaceutical ways). The susceptible and infected sub-populations in both the low risk and the high risk groups were studied in detail through calculations of the effective reproduction number, model analysis, and numerical simulations. Our results show that the introduction of vaccination in the low risk population will significantly reduce infections in both subgroups.

The effective reproduction number, $ R_t $, is a vital epidemic parameter utilized to judge whether an epidemic is shrinking, growing, or holding steady. The main goal of this paper is to estimate the combined $ R_t $ and time-dependent vaccination rate for COVID-19 in the USA and India after the vaccination campaign started. Accounting for the impact of vaccination into a discrete-time stochastic augmented SVEIR (Susceptible-Vaccinated-Exposed-Infectious-Recovered) model, we estimate the time-dependent effective reproduction number $ (R_t) $ and vaccination rate $ (\xi_t) $ for COVID-19 by using a low pass filter and the Extended Kalman Filter (EKF) approach for the period February 15, 2021 to August 22, 2022 in India and December 13, 2020 to August 16, 2022 in the USA. The estimated $ R_t $ and $ \xi_t $ show spikes and serrations with the data. Our forecasting scenario represents the situation by December 31, 2022 that the new daily cases and deaths are decreasing for the USA and India. We also noticed that for the current vaccination rate, $ R_t $ would remain greater than one by December 31, 2022. Our results are beneficial for the policymakers to track the status of the effective reproduction number, whether it is greater or less than one. As restrictions in these countries ease, it is still important to maintain safety and preventive measures.