Caveats on COVID-19 herd immunity threshold: the Spain case

AbstractAfter a year of living with the COVID-19 pandemic and its associated consequences, hope looms on the horizon thanks to vaccines. The question is what percentage of the population needs to be immune to reach herd immunity, that is to avoid future outbreaks. The answer depends on the basic reproductive number, R0, a key epidemiological parameter measuring the transmission capacity of a disease. In addition to the virus itself, R0 also depends on the characteristics of the population and their environment. Additionally, the estimate of R0 depends on the methodology used, the accuracy of data and the generation time distribution. This study aims to reflect on the difficulties surrounding R0 estimation, and provides Spain with a threshold for herd immunity, for which we considered the different combinations of all the factors that affect the R0 of the Spanish population. Estimates of R0 range from 1.39 to 3.10 for the ancestral SARS-CoV-2 variant, with the largest differences produced by the method chosen to estimate R0. With these values, the herd immunity threshold (HIT) ranges from 28.1 to 67.7%, which would have made 70% a realistic upper bound for Spain. However, the imposition of the delta variant (B.1.617.2 lineage) in late summer 2021 may have expanded the range of R0 to 4.02–8.96 and pushed the upper bound of the HIT to 90%.

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Caveats on COVID-19 Herd Immunity Threshold: the Spain case

10.21203/rs.3.rs-596691/v1 ◽

2021 ◽

Author(s):

David Garcia-Garcia ◽

Enrique Morales ◽

Eva S. Fonfría ◽

Isabel Vigo ◽

Cesar Bordehore

Keyword(s):

Upper Bound ◽

Generation Time ◽

Time Distribution ◽

Herd Immunity ◽

Transmission Capacity ◽

Spanish Population ◽

Basic Reproductive Number ◽

Reproductive Number ◽

Population Estimates ◽

Parameter Measuring

Abstract After a year of living with the COVID-19 pandemic and its associated consequences, hope looms on the horizon thanks to vaccines. The question is what percentage of the population needs to be immune to reach herd immunity, that is to avoid future outbreaks. The answer depends on the basic reproductive number, R0, a key epidemiological parameter measuring the transmission capacity of a disease. Besides the virus itself, R0 depends on the characteristics of the population and their environment. Additionally, the estimate of R0 depends on the methodology used, the accuracy of data, and the generation time distribution. The aim of this study is to provide a herd immunity threshold for Spain, for which we considered the different combinations of these elements to obtain the R0 for the Spanish population. Estimates of R0 range from 1.39 to 3.10, with the largest differences produced by the choice of the methodology to estimate R0. With these values, the herd immunity threshold ranges from 28.1–67.7%, which makes 70% a realistic upper bound for Spain.

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Epidemiology of the Zika virus outbreak in the Cabo Verde Islands, West Africa

10.1101/198952 ◽

2017 ◽

Author(s):

José Lourenço ◽

Maria de Lourdes Monteiro ◽

Tomás Valdez ◽

Júlio Monteiro Rodrigues ◽

Oliver G. Pybus ◽

...

Keyword(s):

West Africa ◽

Zika Virus ◽

Attack Rate ◽

Herd Immunity ◽

Basic Reproductive Number ◽

Reproductive Number ◽

Mathematical Framework ◽

Single Wave ◽

Cabo Verde ◽

Small Observation

AbstractIntroductionThe Zika virus (ZIKV) outbreak in the island nation of Cabo Verde was of unprecedented magnitude in Africa and the first to be associated with microcephaly in the continent.MethodsUsing a simple mathematical framework we present a first epidemiological assessment of attack and observation rates from 7,580 ZIKV notified cases and 18 microcephaly reports between July 2015 and May 2016.ResultsIn line with observations from the Americas and elsewhere, the single-wave Cabo Verdean ZIKV epidemic was characterized by a basic reproductive number of 1.85 (95% CI, 1.5 −2.2), with overall the attack rate of 51.1% (range 42.1 - 61.1) and observation rate of 2.7% (range 2.29 - 3.33).ConclusionCurrent herd-immunity may not be sufficient to prevent future small-to-medium epidemics in Cabo Verde. Together with a small observation rate, these results highlight the need for rapid and integrated epidemiological, molecular and genomic surveillance to tackle forthcoming outbreaks of ZIKV and other arboviruses.

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Estimating effective reproduction number using generation time versus serial interval, with application to COVID-19 in the Greater Toronto Area, Canada

10.1101/2020.05.24.20109215 ◽

2020 ◽

Cited By ~ 1

Author(s):

Jesse Knight ◽

Sharmistha Mishra

Keyword(s):

Time Series ◽

Generation Time ◽

Time Distribution ◽

Reproduction Number ◽

Reproductive Number ◽

Transmission Potential ◽

Greater Toronto Area ◽

Critical Measure ◽

Serial Interval ◽

Interval Distribution

AbstractBackgroundThe effective reproductive number Re(t) is a critical measure of epidemic potential. Re(t) can be calculated in near real time using an incidence time series and the generation time distribution—the time between infection events in an infector-infectee pair. In calculating Re(t), the generation time distribution is often approximated by the serial interval distribution—the time between symptom onset in an infector-infectee pair. However, while generation time must be positive by definition, serial interval can be negative if transmission can occur before symptoms, such as in covid-19, rendering such an approximation improper in some contexts.MethodsWe developed a method to infer the generation time distribution from parametric definitions of the serial interval and incubation period distributions. We then compared estimates of Re(t) for covid-19 in the Greater Toronto Area of Canada using: negative-permitting versus non-negative serial interval distributions, versus the inferred generation time distribution.ResultsWe estimated the generation time of covid-19 to be Gamma-distributed with mean 3.99 and standard deviation 2.96 days. Relative to the generation time distribution, non-negative serial interval distribution caused overestimation of Re(t) due to larger mean, while negative-permitting serial interval distribution caused underestimation of Re(t) due to larger variance.ImplicationsApproximation of the generation time distribution of covid-19 with non-negative or negative-permitting serial interval distributions when calculating Re(t) may result in over or underestimation of transmission potential, respectively.

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Reassessing the Human Immunodeficiency Virus Type 1 Life Cycle through Age-Structured Modeling: Life Span of Infected Cells, Viral Generation Time, and Basic Reproductive Number, R0

Journal of Virology ◽

10.1128/jvi.01799-08 ◽

2009 ◽

Vol 83 (15) ◽

pp. 7659-7667 ◽

Cited By ~ 37

Author(s):

Christian L. Althaus ◽

Anneke S. De Vos ◽

Rob J. De Boer

Keyword(s):

Generation Time ◽

Basic Reproductive Number ◽

Reproductive Number ◽

Viral Production ◽

Death Rates ◽

Immunodeficiency Virus ◽

Infected Cells ◽

Age Structured ◽

Hiv 1

ABSTRACT The rapid decay of the viral load after drug treatment in patients infected with human immunodeficiency virus type 1 (HIV-1) has been shown to result from the rapid loss of infected cells due to their high turnover, with a generation time of around 1 to 2 days. Traditionally, viral decay dynamics after drug treatment is investigated using models of differential equations in which both the death rate of infected cells and the viral production rate are assumed to be constant. Here, we describe age-structured models of the viral decay dynamics in which viral production rates and death rates depend on the age of the infected cells. In order to investigate the effects of age-dependent rates, we compared these models with earlier descriptions of the viral load decay and fitted them to previously published data. We have found no supporting evidence that infected-cell death rates increase, but cannot reject the possibility that viral production rates increase, with the age of the cells. In particular, we demonstrate that an exponential increase in viral production with infected-cell age is perfectly consistent with the data. Since an exponential increase in virus production can compensate for the exponential loss of infected cells, the death rates of HIV-1-infected cells may be higher than previously anticipated. We discuss the implications of these findings for the life span of infected cells, the viral generation time, and the basic reproductive number, R 0.

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Implications of Test Characteristics and Population Seroprevalence on “Immune Passport” Strategies

Clinical Infectious Diseases ◽

10.1093/cid/ciaa1019 ◽

2020 ◽

Cited By ~ 7

Author(s):

Daniel B Larremore ◽

Kate M Bubar ◽

Yonatan H Grad

Keyword(s):

False Positive ◽

Herd Immunity ◽

Basic Reproductive Number ◽

Reproductive Number ◽

Serological Tests ◽

Test Characteristics ◽

Simple Question ◽

Minimum Requirements ◽

Test Specificity ◽

Repeat Infection

Abstract Various forms of “immune passports” or “antibody certificates” are being considered in conversations around reopening economies after periods of social distancing. A critique of such programs focuses on the uncertainty around whether seropositivity means immunity from repeat infection. However, an additional important consideration is that the low positive predictive value of serological tests in the setting of low population seroprevalence and imperfect test specificity will lead to many false-positive passport holders. Here, we pose a simple question: how many false-positive passports could be issued while maintaining herd immunity in the workforce? Answering this question leads to a simple mathematical formula for the minimum requirements of serological tests for a passport program, which depend on the population prevalence and the value of the basic reproductive number, R0. Our work replaces speculation in the press with rigorous analysis, and will need to be considered in policy decisions that are based on individual and population serology results.

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Modelling the Transmission Dynamics of Tuberculosis in the Ashanti Region of Ghana

Interdisciplinary Perspectives on Infectious Diseases ◽

10.1155/2020/4513854 ◽

2020 ◽

Vol 2020 ◽

pp. 1-16

Author(s):

Felix Okoe Mettle ◽

Prince Osei Affi ◽

Clement Twumasi

Keyword(s):

Deterministic Model ◽

Equilibrium Points ◽

Herd Immunity ◽

Transmission Dynamics ◽

Basic Reproductive Number ◽

Reproductive Number ◽

Seir Model ◽

High Burden ◽

Ashanti Region ◽

Over Time

Mathematical models can aid in elucidating the spread of infectious disease dynamics within a given population over time. In an attempt to model tuberculosis (TB) dynamics among high-burden districts in the Ashanti Region of Ghana, the SEIR epidemic model with demography was employed within both deterministic and stochastic settings for comparison purposes. The deterministic model showed success in modelling TB infection in the region to the transmission dynamics of the stochastic SEIR model over time. It predicted tuberculosis dying out in ten of twelve high-burden districts in the Ashanti Region, but an outbreak in Obuasi municipal and Amansie West district. The effect of introducing treatment at the incubation stage of TB transmission was also investigated, and it was discovered that treatment introduced at the exposed stage decreased the spread of TB. Branching process approximation was used to derive explicit forms of relevant epidemiological quantities of the deterministic SEIR model for stability analysis of equilibrium points. Numerical simulations were performed to validate the overall infection rate, basic reproductive number, herd immunity threshold, and Malthusian parameter based on bootstrapping, jackknife, and Latin Hypercube sampling schemes. It was recommended that the Ghana Health Service should find a good mechanism to detect TB in the early stages of infection in the region. Public health attention must also be given to districts with a potentially higher risk of experiencing endemic TB even though the estimates of the overall epidemic thresholds from our SEIR model suggested that the Ashanti Region as a whole had herd immunity against TB infection.

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Estimating the reproductive number R0 of SARS-CoV-2 in the United States and eight European countries and implications for vaccination

10.1101/2020.07.31.20166298 ◽

2020 ◽

Cited By ~ 2

Author(s):

Ruian Ke ◽

Ethan Obie Romero-Severson ◽

Steven Sanche ◽

Nick Hengartner

Keyword(s):

United States ◽

Herd Immunity ◽

The United States ◽

Control Measures ◽

European Countries ◽

Basic Reproductive Number ◽

Reproductive Number ◽

Individual Level ◽

Epidemic Period ◽

Induced Immunity

SARS-CoV-2 rapidly spread from a regional outbreak to a global pandemic in just a few months. Global research efforts have focused on developing effective vaccines against SARS-CoV-2 and the disease it causes, COVID-19. However, some of the basic epidemiological parameters, such as the exponential epidemic growth rate and the basic reproductive number, R0, across geographic areas are still not well quantified. Here, we developed and fit a mathematical model to case and death count data collected from the United States and eight European countries during the early epidemic period before broad control measures were implemented. Results show that the early epidemic grew exponentially at rates between 0.19-0.29/day (epidemic doubling times between 2.4-3.6 days). We discuss the current estimates of the mean serial interval, and argue that existing evidence suggests that the interval is between 6-8 days in the absence of active isolation efforts. Using parameters consistent with this range, we estimated the median R0 value to be 5.8 (confidence interval: 4.7-7.3) in the United States and between 3.6 and 6.1 in the eight European countries. This translates to herd immunity thresholds needed to stop transmission to be between 73% and 84%. We further analyze how vaccination schedules depends on R0, the duration of vaccine-induced immunity to SARS-CoV-2, and show that individual-level heterogeneity in vaccine induced immunity can significantly affect vaccination schedules.

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COVID-19 PANDEMICS: HOW FAR ARE WE FROM HERD IMMUNITY?

10.1101/2020.12.19.20248571 ◽

2020 ◽

Author(s):

Carlos Hernandez-Suarez ◽

Efren Murillo-Zamora ◽

Francisco Espinoza Gómez

Keyword(s):

Infection Rate ◽

Study Design ◽

Removal Rate ◽

Herd Immunity ◽

Basic Reproductive Number ◽

Reproductive Number ◽

Fatality Risk

ABSTRACTObjectivesto estimate the current number of total infections in a region in order to measure the progress of the epidemic with the purpose of reopening activities and planning the deployment of vaccines.Study designWe recovered estimates of the basic reproductive number (R0) and the Infection Fatality Risk (IFR) as well as the number of confirmed cases and deaths in several countries.Methodsthis works presents an expression to estimate the number of remaining susceptible in a population using the observed number of SARS-CoV-2 related deaths and current estimates of R0 and IFR.Resultsthe epidemic will infect most of the population causing 2.5 deaths per thousand inhabitants on average, and herd immunity will be achieved when the number of deaths per thousand inhabitants is close to two. This work introduces an expression to provide estimates of the number of remaining susceptible in a region using the reported number of deaths.Conclusionsany region with fewer than 2.5 deaths per thousand individuals will continue accumulating deaths until this average is achieved, and the infection rate will exceed the removal rate until the number of deaths is about two deaths per thousand, when herd immunity is reached. Waves may occur in any region where the number of deaths is below the herd immunity level.

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A NOVEL APPLICATION OF A CLASSICAL METHOD FOR CALCULATING THE BASIC REPRODUCTIVE NUMBER, R0 FOR A GENDER AND RISK STRUCTURED TRANSMISSION DYNAMIC MODEL OF HUMAN PAPILLOMAVIRUS INFECTION

International Journal of Biomathematics ◽

10.1142/s1793524513500460 ◽

2013 ◽

Vol 06 (06) ◽

pp. 1350046 ◽

Cited By ~ 1

Author(s):

KATY TOBIN ◽

CATHERINE COMISKEY

Keyword(s):

Human Papillomavirus ◽

Dynamic Model ◽

Classical Method ◽

Control Strategies ◽

Basic Reproductive Number ◽

Reproductive Number ◽

Population Estimates ◽

Transmission Dynamic ◽

Papillomavirus Infection ◽

Parameter Values

Mathematical models are increasingly being used in the evaluation of control strategies for infectious disease such as the vaccination program for the Human Papillomavirus (HPV). Here, an ordinary differential equation (ODE) transmission dynamic model for HPV is presented and analyzed. Parameter values for a gender and risk structured model are estimated by calibrating the model around the known prevalence of infection. The effect on gender and risk sub-group prevalence induced by varying the epidemiological parameters are investigated. Finally, the outcomes of this model are applied using a classical mathematical method for calculating R0 in a heterogeneous mixing population. Estimates for R0 under various gender and mixing scenarios are presented.

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Two new compartmental epidemiological models and their equilibria

10.1101/2021.09.03.21263050 ◽

2021 ◽

Author(s):

Jonas Balisacan ◽

Monique Chyba ◽

Corey Shanbrom

Keyword(s):

Compartmental Model ◽

Herd Immunity ◽

Basic Reproductive Number ◽

Reproductive Number ◽

Mathematical Epidemiology ◽

Epidemiological Models ◽

Classical Models ◽

Asymptomatic Individuals ◽

The Stability ◽

Disease Free

Compartmental models have long served as important tools in mathematical epidemiology, with their usefulness highlighted by the recent COVID-19 pandemic. However, most of the classical models fail to account for certain features of this disease and others like it, such as the ability of exposed individuals to recover without becoming infectious, or the possibility that asymptomatic individuals can indeed transmit the disease but at a lesser rate than the symptomatic. Furthermore, the rise of new disease variants and the imperfection of vaccines suggest that concept of endemic equilibrium is perhaps more pertinent than that of herd immunity. Here we propose a new compartmental epidemiological model and study its equilibria, characterizing the stability of both the endemic and disease-free equilibria in terms of the basic reproductive number. Moreover, we introduce a second compartmental model, generalizing our first, which accounts for vaccinated individuals, and begin an analysis of its equilibria.

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