Final Epidemic Size Research Articles

Effects of changes in temperature on Zika dynamics and control.

When a rare pathogen emerges to cause a pandemic, it is critical to understand its dynamics and the impact of mitigation measures. We use experimental data to parametrize a temperature-dependent model of Zika virus (ZIKV) transmission dynamics and analyse the effects of temperature variability and control-related parameters on the basic reproduction number (R0) and the final epidemic size of ZIKV. Sensitivity analyses show that these two metrics are largely driven by different parameters, with the exception of temperature, which is the dominant driver of epidemic dynamics in the models. Our R0 estimate has a single optimum temperature (≈30°C), comparable to other published results (≈29°C). However, the final epidemic size is maximized across a wider temperature range, from 24 to 36°C. The models indicate that ZIKV is highly sensitive to seasonal temperature variation. For example, although the model predicts that ZIKV transmission cannot occur at a constant temperature below 23°C (≈ average annual temperature of Rio de Janeiro, Brazil), the model predicts substantial epidemics for areas with a mean temperature of 20°C if there is seasonal variation of 10°C (≈ average annual temperature of Tampa, Florida). This suggests that the geographical range of ZIKV is wider than indicated from static R0 models, underscoring the importance of climate dynamics and variation in the context of broader climate change on emerging infectious diseases.

COVID-19 Global Humanitarian Response Plan: An optimal distribution model for high-priority countries

Background:The 2019 novel coronavirus disease (COVID-19) has spread rapidly worldwide, and the outbreak of the disease was designated a global pandemic by the World Health Organization. Such outbreaks would certainly be catastrophic for some of the best-ranked health systems and would be more catastrophic in countries with more fragile health systems. Accordingly, the World Health Organization and other organizations have been appealing to donor countries to support a rapid response plan. The primary objectives of this response plan are to appeal for funds from donors and to distribute these funds to the most affected countries based on the requirements. Methods:In this study, we developed a mathematical model to provide initial insights into the efficient and equitable distribution of urgent funds to high-priority countries. Three phases were proposed for the construction of this mathematical model. In the first phase, the final epidemic sizes in all the target countries were predicted by using three epidemiological models. In the second phase, the urgent requirements for each country were estimated in parallel with the estimates issued by the humanitarian response plan, based on the size of the epidemic and several other factors. In the third and final phase, a multi-objective optimization model was proposed. The first objective was to maximize the funds from donors to cover all the requirements. The second objective was to minimize the unmet demands by ensuring a fair distribution of the urgent funds based on the requirements of the target countries. Results:Predictions of the basic reproduction numbers and the final epidemic sizes were calculated for all target countries. The urgent requirements were estimated, and the requirements issued by the humanitarian response plan for all target countries were also considered. Moreover, a proposed response plan for the distribution network was demonstrated. Donors must provide urgent funds exceeding US$ 2,608,084,209 to cover at least 40 % of each target country’s requirements. Overall, results demonstrate the importance of an urgent and fair distribution of funds to the target countries to overcome the outbreak of COVID-19. Conclusions:Rapid responses by donor countries to humanitarian appeals will facilitate the immediate and fair distribution of relief supplies to the poorest countries. This distribution may help to support health systems, restrain the spread of COVID-19, and prevent an unlimited catastrophe.

Dynamics analysis of typhoid fever with public health education programs and final epidemic size relation

In this paper, an epidemic model is develop and use to investigate the transmission dynamics of the typhoid fever epidemic (TF, a bacterial infection caused by Salmonella serotype Typhi bacteria). The model assesses the impact of public health education programs (PHEP) on reducing the pathogenesis of TF which can cause large outbreaks especially in resource-poor settings. The model is fitted well to the data for TF cases for Taiwan, China. Results from our mathematical analysis reveal that the disease-free equilibrium (DFE) of the model is globally asymptotically stable (GAS) when the basic reproduction number (R0) is below or equal to unity, and unstable when it is above unity. Further analysis also shows that the endemic equilibrium (EE) of the model is GAS whenever the R0 is above unity with some certain conditions , indicating the potential for the TF to spread and cause outbreaks in the community. We obtain a final size relation with consideration of human-to-human transmission route that could be used to report the actual size of the outbreaks over the cause of the epidemic period. Furthermore, sensitivity analysis results reveal the most sensitive parameters that are vital to combat the TF epidemic in Taiwan. Also, a wavelets analysis is performed to explore significant periodicities of the TF outbreaks in Taiwan, China.

Free ticket, discount ticket or intermediate of the best of two worlds – Which subsidy policy is socially optimal to suppress the disease spreading?

With the aid of the evolutionary vaccination game on a scale-free network, we design a new subsidy policy, named degree dependent subsidy, where cooperative agents get incentives according to their connectivity or degree. That is, agents, having a greater degree, receive a higher incentive, and vice versa. Here we presume that vaccinators are cooperative agents. The new scheme can be said to an intermediate policy between two previously studies policies, namely free ticket and flat discount policies. The former policy distributes free tickets to cooperative hub agents as a priority, whereas the latter dispenses a fixed discount to every cooperator. We compare the efficiency of each policy in terms of having a less infectious state with a minimum social cost. While investigating the performance of the three policies in terms of average social payoff–which takes into account the cost of vaccination as well as infection–the free ticket scheme is found to be the most appealing policies among the three when the budget for subsidy is quite low. The degree dependent subsidy policy outperforms others for a moderate budget size, while the flat discount policy requires a higher budget to effectively suppress the disease. We further estimate threshold levels of the subsidy budget for each policy beyond which subsidizing results in excessive use of vaccination. As a whole, concerning vaccination coverage and final epidemic size, the degree-dependent subsidy scheme outperforms the flat discount scheme, but is dominated by the free ticket policy.

Optimal Immunity Control and Final Size Minimization by Social Distancing for the SIR Epidemic Model

The aim of this article is to understand how to apply partial or total containment to SIR epidemic model during a given finite time interval in order to minimize the epidemic final size, that is the cumulative number of cases infected during the complete course of an epidemic. The existence and uniqueness of an optimal strategy are proved for this infinite-horizon problem, and a full characterization of the solution is provided. The best policy consists in applying the maximal allowed social distancing effort until the end of the interval, starting at a date that is not always the closest date and may be found by a simple algorithm. Both theoretical results and numerical simulations demonstrate that it leads to a significant decrease in the epidemic final size. We show that in any case the optimal intervention has to begin before the number of susceptible cases has crossed the herd immunity level, and that its limit is always smaller than this threshold. This problem is also shown to be equivalent to the minimum containment time necessary to stop at a given distance after this threshold value.

Comment on “Poorly known aspects of flattening the curve of COVID-19″

This short research note describes and summarizes several recent peer-reviewed and non-peer-reviewed studies on the concept of flattening-the-curve (FTC) in the context of the COVID-19 pandemic. This note also highlights contradictory findings of these studies in terms of the effect of FTC on the total number of infections (the final epidemic size), and poses a research problem for future studies.

The Basic Reproduction Number as a Loop Gain Matrix.

The COVID-19 pandemic and the disordered reactions of most governments made the importance of mathematical modelling and model-based predictions evident, even outside the scientific community. The basic reproduction number n}{}mathcal {R}_{0}n quickly entered the common jargon, as a concise but effective tool to communicate the spreading power of a disease and estimate, at least roughly, the possible outcomes of the epidemic. However, while n}{}mathcal {R}_{0}n is easily defined for simple models, its proper definition is more subtle for larger, state-of-the-art models. Here we show that it is nothing else than the spectral radius of the gain matrix of a linear system, and that this matrix generalizes n}{}mathcal {R}_{0}n in the computation of the vector-valued final epidemic size and epidemic threshold, in a large class of finite-dimensional SIR-like models.

Mobility Restrictions and the Control of COVID-19

A recent study on the impact of mobility controls on the final size of epidemics by Espinoza, Castillo-Chavez, and Perrings (2020) found that mobility restrictions between areas experiencing different levels of disease risk and with different public health infrastructures do not always reduce the final epidemic size. Indeed, restrictions on the mobility of people from high-risk to low-risk areas can increase, not reduce, the total number of infections. Since the first response of many countries to the COVID-19 pandemic was to implement mobility restrictions, it is worth bearing in mind the implications of the Espinoza result when considering the effectiveness of such restrictions.

Epidemic spreading in a weighted pig trade network

The inclusion of edge weights can add valuable insights in the spreading processes within trade networks and may identify factors influencing the final epidemic size. The aim of the study was to evaluate the effect of different network versions on the outcome of an epidemiological model. The weighted network versions included the number of trade contacts (A), the sum of delivered livestock (B) and the mean number of delivered livestock per trade contact (C). Furthermore, other factors, e.g. transmission probability and farm type of primary outbreak, were tested for their impact on the final epidemic size. From 2013–2014, data from a pig trade network in Northern Germany was recorded containing 678 farms connected by 1,018 directed edges. An epidemiological model was implemented considering a higher probability of disease spread for edges with a higher weight for each of the combinations between network version and transmission probability. Only transmission routes following the network structure were considered for disease transmission. The outcome of the epidemiological model (number of infected farms) was tested with a generalized linear mixed model including the fixed effects network version (unweighted, A, B, C), transmission probability and farm type of primary outbreak (breeding farm, farrowing farm, finishing farm, farrow-to-finishing farm, unknown) as well as all twofold interactions. The results revealed that all fixed effects as well as all twofold interactions were significant (p ≤ 0.05), i.e. in the following only the impact of the interactions on the number of infected farms can be interpreted. Network versions B and C showed in all combinations the highest number of infected farms independent of the underlying transmission probability. The unweighted network and network version A showed a significant increase of infected farms with increasing transmission probability. All interactions including the farm type of primary outbreak revealed a significant higher number of infected farms for farm types located at the beginning of the production chain, e.g. breeding farms. These farm types reached also more other farms in 1–4 steps compared to farm types located near to the end of the production chain. The inclusion of edge weights has a significant effect on the outcome of epidemiological models and dependent on the chosen edge weight the results need to be interpreted accordingly.

COVID-19 outbreak, social distancing and mass testing in Kenya-insights from a mathematical model

As reported by the World Health Organization (WHO), the world is currently facing a devastating pandemic of a novel coronavirus (COVID-19), which started as an outbreak of pneumonia of unknown cause in the Wuhan city of China in December 2019. Since then, the respiratory disease has exponentially spread to over 210 countries. By the end of April, COVID-19 had caused over three million confirmed cases of infections and over 200,000 fatalities globally. The trend poses a huge threat to global public health. Understanding the early transmission dynamics of the infection and evaluating the effectiveness of control measures is crucial for assessing the potential for sustained transmission to occur in new areas. We employed a SEIHQRD delay differential mathematical transmission model with reported Kenyan data on cases of COVID-19 to estimate how transmission varies over time and which population to target for mass testing. The model is concise in structure, and successfully captures the course of the COVID-19 outbreak, and thus sheds light on understanding the trends of the outbreak and the vulnerable populations. The results show that, the government should target population in the informal settlement for mass testing and provide affordable sanitizers and clean water to this population. The model results also indicate that people with pre-existing non-communicable diseases (NCDs) should be identified and given special medical care. Given the absence of vaccine at the moment, non-pharmaceutical intervention is needed to effectively reduce the final epidemic size.

Implementation of a real-time, data-driven online Epidemic Calculator for tracking the spread of COVID-19 in Singapore and other countries.

While there are many online data dashboards on COVID-19, there are few analytics available to the public and non-epidemiologists to help them gain a deeper insight into the COVID-19 pandemic and evaluate the effectiveness of social intervention measures. To address the issue, this study describes the methods underlying the development of a real-time, data-driven online Epidemic Calculator for tracking COVID-19 growth parameters. From publicly available infection case and death data, the calculator is used to estimate the effective reproduction number, final epidemic size, and death toll. As a case study, we analyzed the results for Singapore during the "Circuit Breaker” period from April 7, 2020 to the end of May 2020. The calculator shows that the stringent measures imposed have an immediate effect of rapidly slowing down the spread of the coronavirus. After about two weeks, the effective reproduction number reduced to about 1.0. Since then, the number has been fluctuating around 1.0 for more than a month.The COVID-19 Epidemic Calculator is available in the form of an online Google Sheet and the results are presented as Tableau Public dashboards at www.cv19.one. By making the calculator readily accessible online, the public can have a tool to assess the effectiveness of measures to control the pandemic meaningfully.

Assessing the impact of adherence to Non-pharmaceutical interventions and indirect transmission on the dynamics of COVID-19: a mathematical modelling study.

Adherence to public health policies such as the non-pharmaceutical interventions implemented against COVID-19 plays a major role in reducing infections and controlling the spread of the diseases. In addition, understanding the transmission dynamics of the disease is also important in order to make and implement efficient public health policies. In this paper, we developed an SEIR-type compartmental model to assess the impact of adherence to COVID-19 non-pharmaceutical interventions and indirect transmission on the dynamics of the disease. Our model considers both direct and indirect transmission routes and stratifies the population into two groups: those that adhere to COVID-19 non-pharmaceutical interventions (NPIs) and those that do not adhere to the NPIs. We compute the control reproduction number and the final epidemic size relation for our model and study the effect of different parameters of the model on these quantities. Our results show that there is a significant benefit in adhering to the COVID-19 NPIs.

Short-term forecast in the early stage of the COVID-19 outbreak in Italy. Application of a weighted and cumulative average daily growth rate to an exponential decay model.

To estimate the size of the novel coronavirus (COVID-19) outbreak in the early stage in Italy, this paper introduces the cumulated and weighted average daily growth rate (WR) to evaluate an epidemic curve. On the basis of an exponential decay model (EDM), we provide estimations of the WR in four-time intervals from February 27 to April 07, 2020. By calibrating the parameters of the EDM to the reported data in Hubei Province of China, we also attempt to forecast the evolution of the outbreak. We compare the EDM applied to WR and the Gompertz model, which is based on exponential decay and is often used to estimate cumulative events. Specifically, we assess the performance of each model to short-term forecast of the epidemic, and to predict the final epidemic size.Based on the official counts for confirmed cases, the model applied to data from February 27 until the 17th of March estimate that the cumulative number of infected in Italy could reach 131,280 (with a credibility interval 71,415-263,501) by April 25 (credibility interval April 12 to May 3). With the data available until the 24st of March the peak date should be reached on May 3 (April 23 to May 23) with 197,179 cumulative infections expected (130,033–315,269); with data available until the 31st of March the peak should be reached on May 4 (April 25 to May 18) with 202,210 cumulative infections expected (155.235–270,737); with data available until the 07st of April the peak should be reached on May 3 (April 26 to May 11) with 191,586 (160,861-232,023) cumulative infections expected. Based on the average mean absolute percentage error (MAPE), cumulated infections forecasts provided by the EDM applied to WR performed better across all scenarios than the Gompertz model.An exponential decay model applied to the cumulated and weighted average daily growth rate appears to be useful in estimating the number of cases and peak of the COVID-19 outbreak in Italy and the model was more reliable in the exponential growth phase.

Network structure-based interventions on spatial spread of epidemics in metapopulation networks.

Mathematical modeling of epidemics is fundamental to understand the mechanism of the disease outbreak and provides helpful indications for effectiveness of interventions for policy makers. The metapopulation network model has been used in the analysis of epidemic dynamics by taking individual migration between patches into account. However, so far, most of the previous studies unrealistically assume that transmission rates within patches are the same, neglecting the nonuniformity of intervention measures in hindering epidemics. Here, based on the assumption that interventions deployed in a patch depend on its population size or economic level, which have shown a positive correlation with the patch's degree in networks, we propose a metapopulation network model to explore a network structure-based intervention strategy, aiming at understanding the interplay between intervention strategy and other factors including mobility patterns, initial population, as well as the network structure. Our results demonstrate that interventions to patches with different intensity are able to suppress the epidemic spreading in terms of both the epidemic threshold and the final epidemic size. Specifically, the intervention strategy targeting the patches with high degree is able to efficiently suppress epidemics. In addition, a detrimental effect is also observed depending on the interplay between the intervention measures and the initial population distribution. Our study opens a path for understanding epidemic dynamics and provides helpful insights into the implementation of countermeasures for the control of epidemics in reality.

How best can finite-time social distancing reduce epidemic final size?

Given maximal social distancing duration and intensity, how can one minimize the epidemic final size, or equivalently the total number of individuals infected during the outbreak? A complete answer to this question is provided and demonstrated here for the SIR epidemic model. In this simplified setting, the optimal solution consists in enforcing the highest confinement level during the longest allowed period, beginning at a time instant that is the unique solution to certain 1D optimization problem. Based on this result, we present numerical essays showing the best possible performance for a large set of basic reproduction numbers and lockdown durations and intensities.

Phenomenological Modelling of COVID-19 Epidemics in Sri Lanka, Italy, the United States, and Hebei Province of China.

The COVID-19 pandemic has resulted in increasing number of infections and deaths every day. Lack of specialized treatments for the disease demands preventive measures based on statistical/mathematical models. The analysis of epidemiological curve fitting, on number of daily infections across affected countries, provides useful insights on the characteristics of the epidemic. A variety of phenomenological models are available to capture the dynamics of disease spread and growth. The number of daily new infections and cumulative number of infections in COVID-19 over four selected countries, namely, Sri Lanka, Italy, the United States, and Hebei province of China, from the first day of appearance of cases to 2nd July 2020 were used in the study. Gompertz, logistic, Weibull, and exponential growth curves were fitted on the cumulative number of infections across countries. AIC, BIC, RMSE, and R2 were used to determine the best fitting curve for each country. Results revealed that the most appropriate growth curves for Sri Lanka, Italy, the United States, and China (Hebei) are the logistic, Gompertz, Weibull, and Gompertz curves, respectively. Country-wise, overall growth rate, final epidemic size, and short-term forecasts were evaluated using the selected model. Daily log incidences in each country were regressed before and after the identified peak time of the respective outbreak of epidemic. Hence, doubling time/halving time together with daily growth rates and predictions was estimated. Findings and relevant interpretations demonstrate that the outbreak seems to be extinct in Hebei, China, whereas further transmissions are possible in the United States. In Italy and Sri Lanka, current outbreaks transmit in a decreasing rate.

INVESTIGATING THE ROLE OF WITHIN- AND BETWEEN-PATCH MOVEMENT IN A DYNAMIC MODEL OF DISEASE SPREAD

The impact of human mobility on the spreading of disease in a metapopulation is emphasized on interconnecting between patches, whereas the current volume of movement within the local population is usually neglected. Here, the role of internal commuters is taken into account by two means, a local transmission rate and the volume of internal commuters. Dynamic model of human mobility in the metapopulation with gravity coupling is presented. In conjunction with the disease spreading, the impact on invasion threshold and epidemic final size are analyzed. For two-patch model, we show that under fixing parameters in gravity model, the existence of invasion threshold depends on the difference of local transmission rates and the proportion of internal commuters between two patches. For a fully connected network with an identical transmission rate, the difference in patch final sizes is driven by patch distribution of internal commuters. By neglecting the effect of spatial variation in a simple core–satellite model, we show that the heterogeneity of internal commuters and gravity coupling induce a complex pattern of threshold, which depend mostly on the exponent in gravity model, and are responsible for the differences among local epidemic sizes.

Simulation models of dengue transmission in Funchal, Madeira Island: Influence of seasonality.

The recent emergence and established presence of Aedes aegypti in the Autonomous Region of Madeira, Portugal, was responsible for the first autochthonous outbreak of dengue in Europe. The island has not reported any dengue cases since the outbreak in 2012. However, there is a high risk that an introduction of the virus would result in another autochthonous outbreak given the presence of the vector and permissive environmental conditions. Understanding the dynamics of a potential epidemic is critical for targeted local control strategies.Here, we adopt a deterministic model for the transmission of dengue in Aedes aegypti mosquitoes. The model integrates empirical and mechanistic parameters for virus transmission, under seasonally varying temperatures for Funchal, Madeira Island. We examine the epidemic dynamics as triggered by the arrival date of an infectious individual; the influence of seasonal temperature mean and variation on the epidemic dynamics; and performed a sensitivity analysis on the following quantities of interest: the epidemic peak size, time to peak, and the final epidemic size.Our results demonstrate the potential for summer and autumn season transmission of dengue, with the arrival date significantly affecting the distribution of the timing and peak size of the epidemic. Late-summer arrivals were more likely to produce large epidemics within a short peak time. Epidemics within this favorable period had an average of 11% of the susceptible population infected at the peak, at an average peak time of 95 days. We also demonstrated that seasonal temperature variation dramatically affects the epidemic dynamics, with warmer starting temperatures producing large epidemics with a short peak time and vice versa. Overall, our quantities of interest were most sensitive to variance in the date of arrival, seasonal temperature, transmission rates, mortality rate, and the mosquito population; the magnitude of sensitivity differs across quantities.Our model could serve as a useful guide in the development of effective local control and mitigation strategies for dengue fever in Madeira Island.

Vaccinating SIS epidemics under evolving perception in heterogeneous networks.

Vaccination is an effective intervention against epidemics. Previous work has demonstrated that psychological cognition affects individual behavior. However, perceptual differences between individuals, as well as the dynamics of perceptual evolution, are not taken into account. In order to explore how these realistic characteristics of psychological cognition influence collective vaccination behavior, we propose a prospect theory based evolutionary vaccination game model, where the evolution of reference points is used to characterize changes in perception. We compare the fractions of vaccinated individuals and infected individuals under variable reference points with those under the expected utility theory and the fixed reference point, and highlight the role of evolving perception in promoting vaccination and contributing to epidemic control. We find that the epidemic size under variable reference point is always less than that under the expected utility theory. Finding that there exists a vaccination cost threshold for the cognitive effect, we develop a novel mixed-reference-point mechanism by combining individual psychological characteristics with network topological feature. The effectiveness of this mechanism in controlling the network epidemics is verified with numerical simulations. Compared with pure reference points, the mixed-reference-point mechanism can effectively reduce the final epidemic size, especially at a large vaccination cost.Graphical abstract

Epidemiological Modeling of COVID-19 in Saudi Arabia: Spread Projection, Awareness, and Impact of Treatment

The first case of COVID-19 originated in Wuhan, China, after which it spread across more than 200 countries. By 21 July 2020, the rapid global spread of this disease had led to more than 15 million cases of infection, with a mortality rate of more than 4.0% of the total number of confirmed cases. This study aimed to predict the prevalence of COVID-19 and to investigate the effect of awareness and the impact of treatment in Saudi Arabia. In this paper, COVID-19 data were sourced from the Saudi Ministry of Health, covering the period from 31 March 2020 to 21 July 2020. The spread of COVID-19 was predicted using four different epidemiological models, namely the susceptible–infectious–recovered (SIR), generalized logistic, Richards, and Gompertz models. The assessment of models’ fit was performed and compared using four statistical indices (root-mean-square error (RMSE), R squared (R2), adjusted R2 ( Radj2), and Akaike’s information criterion (AIC)) in order to select the most appropriate model. Modified versions of the SIR model were utilized to assess the influence of awareness and treatment on the prevalence of COVID-19. Based on the statistical indices, the SIR model showed a good fit to reported data compared with the other models (RMSE = 2790.69, R2 = 99.88%, Radj2 = 99.98%, and AIC = 1796.05). The SIR model predicted that the cumulative number of infected cases would reach 359,794 and that the pandemic would end by early September 2020. Additionally, the modified version of the SIR model with social distancing revealed that there would be a reduction in the final cumulative epidemic size by 9.1% and 168.2% if social distancing were applied over the short and long term, respectively. Furthermore, different treatment scenarios were simulated, starting on 8 July 2020, using another modified version of the SIR model. Epidemiological modeling can help to predict the cumulative number of cases of infection and to understand the impact of social distancing and pharmaceutical intervention on the prevalence of COVID-19. The findings from this study can provide valuable information for governmental policymakers trying to control the spread of this pandemic.