scholarly journals Modelling infectious diseases with herd immunity in a randomly mixed population

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Kian Boon Law ◽  
Kalaiarasu M. Peariasamy ◽  
Hishamshah Mohd Ibrahim ◽  
Noor Hisham Abdullah
Keyword(s):  
Infectious Disease ◽  
Infectious Diseases ◽  
Transmission Rate ◽  
Herd Immunity ◽  
The United States ◽  
Sir Model ◽  
Mixed Population ◽  
Transmission Dynamics ◽  
Vaccination Rates ◽  
Reduced Risk

AbstractThe conventional susceptible-infectious-recovered (SIR) model tends to magnify the transmission dynamics of infectious diseases, and thus the estimated total infections and immunized population may be higher than the threshold required for infection control and eradication. The study developed a new SIR framework that allows the transmission rate of infectious diseases to decline along with the reduced risk of contact infection to overcome the limitations of the conventional SIR model. Two new SIR models were formulated to mimic the declining transmission rate of infectious diseases at different stages of transmission. Model A utilized the declining transmission rate along with the reduced risk of contact infection following infection, while Model B incorporated the declining transmission rate following recovery. Both new models and the conventional SIR model were then used to simulate an infectious disease with a basic reproduction number (r0) of 3.0 and a herd immunity threshold (HIT) of 0.667 with and without vaccination. Outcomes of simulations were assessed at the time when the total immunized population reached the level predicted by the HIT, and at the end of simulations. Further, all three models were used to simulate the transmission dynamics of seasonal influenza in the United States and disease burdens were projected and compared with estimates from the Centers for Disease Control and Prevention. For the simulated infectious disease, in the initial phase of the outbreak, all three models performed expectedly when the sizes of infectious and recovered populations were relatively small. As the infectious population increased, the conventional SIR model appeared to overestimate the infections even when the HIT was achieved in all scenarios with and without vaccination. For the same scenario, Model A appeared to attain the level predicted by the HIT and in comparison, Model B projected the infectious disease to be controlled at the level predicted by the HIT only at high vaccination rates. For infectious diseases with high r0, and at low vaccination rates, the level at which the infectious disease was controlled cannot be accurately predicted by the current theorem. Transmission dynamics of infectious diseases with herd immunity can be accurately modelled by allowing the transmission rate of infectious diseases to decline along with the reduction of contact infection risk after recovery or vaccination. Model B provides a credible framework for modelling infectious diseases with herd immunity in a randomly mixed population.

scholarly journals Modelling infectious diseases with herd immunity in a randomly mixed population

2021 ◽  
Author(s):  
Kian Boon Law ◽  
Kalaiarasu M. Peariasamy ◽  
Hishamshah Mohd. Ibrahim ◽  
Noor Hisham Abdullah
Keyword(s):  
Infectious Disease ◽  
Infectious Diseases ◽  
Transmission Rate ◽  
Herd Immunity ◽  
Sir Model ◽  
Mixed Population ◽  
Transmission Dynamics ◽  
Transmission Model ◽  
Infection Model ◽  
Vaccination Rates

Abstract Background The conventional susceptible-infectious-recovered (SIR) model tends to overestimate the transmission dynamics of infectious diseases and ends up with total infections and total immunized population exceeding the threshold required for control and eradication of infectious diseases. The study aims to overcome the limitation by allowing the transmission rate of infectious disease to decline along with the reducing risk of contact infection. Methods Two new SIR models were developed to mimic the declining transmission rate of infectious diseases at different stages of transmission. Model A mimicked the declining transmission rate along with the reducing risk of transmission following infection, while Model B mimicked the declining transmission rate following recovery. Then, the conventional SIR model, Model A and Model B were used to simulate an infectious disease with a basic reproduction number (r0) of 3.0 and a herd immunity threshold (HIT) of 0.667 with and without vaccination. The infectious disease was expected to be controlled or eradicated when the total immunized population either through infection or vaccination reached the level predicted by the HIT. Outcomes of simulations were assessed at the time when the total immunized population reached the level predicted by the HIT, and at the end of simulations. Findings All three models performed likewise at the beginning of the transmission when sizes of infectious and recovered were relatively small as compared with the population size. The infectious disease modelled using the conventional SIR model appeared completely out of control even when the HIT was achieved in all scenarios with and without vaccination. The infectious disease modelled using Model A appeared to be controlled at the level predicted by the HIT in all scenarios with and without vaccination. Model B projected the infectious disease to be controlled at the level predicted by the HIT only at high vaccination rates. At lower vaccination rates or without vaccination, the level at which the infectious disease was controlled cannot be accurately predicted by the HIT. Conclusion Transmission dynamics of infectious diseases with herd immunity can accurately be modelled by allowing the transmission rate of infectious disease to decline along with the combined risk of contact infection. Model B provides a more credible framework for modelling infectious diseases with herd immunity in a randomly mixed population.

scholarly journals Modelling infectious diseases with herd immunity in a randomly mixed population

2021 ◽  
Author(s):  
Kian Boon Law ◽  
Kalaiarasu M Peariasamy ◽  
Hishamshah Ibrahim ◽  
Noor Hisham Abdullah
Keyword(s):  
Infectious Disease ◽  
Infectious Diseases ◽  
Transmission Rate ◽  
Herd Immunity ◽  
Sir Model ◽  
Mixed Population ◽  
Transmission Dynamics ◽  
Transmission Model ◽  
Infection Model ◽  
Vaccination Rates

Abstract Background The conventional susceptible-infectious-recovered (SIR) model tends to overestimate transmission dynamics of infectious diseases and ends up with total infections exceeding the threshold required for control and eradication of infectious diseases. The study aims to overcome the limitation by allowing the transmission rate of infectious disease to decline along with the reducing risk of contact infection. MethodsTwo new SIR models were developed to mimic the declining transmission rate of infectious diseases at different stages of transmission. Model A mimicked the declining transmission rate along with the reducing risk of transmission following infection, while Model B mimicked the declining transmission rate following recovery. Then, the conventional SIR model, Model A and Model B were used to simulate an infectious disease with a basic reproduction number (r0) of 3.0 and a herd immunity threshold (HIT) of 0.667 with and without vaccination. The infectious disease was expected to be controlled or eradicated when the total immunized population either through infection or vaccination reached the level predicted by the HIT. Outcomes of simulations were assessed at the time when the total immunized population reached the level predicted by the HIT, and at the end of simulations.Findings All three models performed likewise at the beginning of transmission when sizes of infectious and recovered were relatively small as compared with the population size. The infectious disease modelled using the conventional SIR model appeared completely out of control even when the HIT was achieved in all scenarios with and without vaccination. The infectious disease modelled using Model A appeared to be controlled at the level predicted by the HIT in all scenarios with and without vaccination. Model B projected the infectious disease to be controlled at the level predicted by the HIT only at high vaccination rates. At lower vaccination rates or without vaccination, the level at which the infectious disease was controlled cannot be accurately predicted by the HIT. ConclusionTransmission dynamics of infectious diseases with herd immunity can accurately be modelled by allowing the transmission rate of infectious disease to decline along with the combined risk of contact infection. Model B provides a more realistic framework for modelling infectious diseases with herd immunity in a randomly mixed population.

scholarly journals Modelling infectious diseases with herd immunity in a randomly mixed population

2021 ◽  
Author(s):  
Kian Boon Law ◽  
Kalaiarasu M. Peariasamy ◽  
Hishamshah Mohd. Ibrahim ◽  
Noor Hisham Abdullah
Keyword(s):  
Infectious Disease ◽  
Infectious Diseases ◽  
Transmission Rate ◽  
Herd Immunity ◽  
Sir Model ◽  
Mixed Population ◽  
Transmission Dynamics ◽  
Transmission Model ◽  
Infection Model ◽  
Vaccination Rates

Abstract Background The conventional susceptible-infectious-recovered (SIR) model tends to overestimate the transmission dynamics of infectious diseases and ends up with total infections and total immunized population exceeding the threshold required for control and eradication of infectious diseases. The study aims to overcome the limitation by allowing the transmission rate of infectious disease to decline along with the reducing risk of contact infection. Methods Two new SIR models were developed to mimic the declining transmission rate of infectious diseases at different stages of transmission. Model A mimicked the declining transmission rate along with the reducing risk of transmission following infection, while Model B mimicked the declining transmission rate following recovery. Then, the conventional SIR model, Model A and Model B were used to simulate an infectious disease with a basic reproduction number (r0) of 3.0 and a herd immunity threshold (HIT) of 0.667 with and without vaccination. The infectious disease was expected to be controlled or eradicated when the total immunized population either through infection or vaccination reached the level predicted by the HIT. Outcomes of simulations were assessed at the time when the total immunized population reached the level predicted by the HIT, and at the end of simulations.Findings All three models performed likewise at the beginning of the transmission when sizes of infectious and recovered were relatively small as compared with the population size. The infectious disease modelled using the conventional SIR model appeared completely out of control even when the HIT was achieved in all scenarios with and without vaccination. The infectious disease modelled using Model A appeared to be controlled at the level predicted by the HIT in all scenarios with and without vaccination. Model B projected the infectious disease to be controlled at the level predicted by the HIT only at high vaccination rates. At lower vaccination rates or without vaccination, the level at which the infectious disease was controlled cannot be accurately predicted by the HIT. Conclusion Transmission dynamics of infectious diseases with herd immunity can accurately be modelled by allowing the transmission rate of infectious disease to decline along with the combined risk of contact infection. Model B provides a more credible framework for modelling infectious diseases with herd immunity in a randomly mixed population.

scholarly journals The attainment of herd immunity with reduced risk of contact infection in modelling

2021 ◽  
Author(s):  
Kian Boon Law ◽  
Kalaiarasu M Peariasamy ◽  
Hishamshah Ibrahim ◽  
Noor Hisham Abdullah
Keyword(s):  
Infectious Disease ◽  
Herd Immunity ◽  
Sir Model ◽  
Mixed Population ◽  
Reproduction Ratio ◽  
Force Of Infection ◽  
Population Immunity ◽  
Modelling Framework ◽  
And Control ◽  
Reduced Risk

Abstract The risk of contact infection among susceptible individuals in a randomly mixed population can be reduced by the presence of immune individuals and this concept is referred to as herd immunity1–3. The conventional susceptible-infectious-recovered (SIR) model does not feature a reduced risk of susceptible individuals in the transmission dynamics of infectious disease, therefore violates the fundamental of herd immunity4. Here we show that the reduced risk of contact infection among susceptible individuals in the SIR model can be attained by incorporating the proportion of susceptible individuals (model A) or the inverse of proportion of recovered individuals (model B) in the force of infection of the SIR model. We simulated the conventional SIR model and both new SIR models under the exact condition with a basic reproduction ratio of 3.0 and an expected herd immunity threshold of 0.667 (66.7%). All three models performed likewise at the initial stage of an epidemic. In the conventional SIR model, the epidemic continued until 94.0 % of the population had been infected and recovered, way above the threshold for eradication and control of the epidemic. Both models A and B simulated the epidemic waning when 66.7% and 75.6% of the population had been infected, as a result of the herd effect. As a result, model A provides a better framework for modelling vaccine-induced herd or population immunity, while model B provides a better framework for modelling infection-induced herd or population immunity. Our results demonstrate how the new modelling framework overcomes the drawback of the conventional SIR model and attain the effect of herd immunity in modelling outputs, which is important for modelling infectious disease in a randomly mixed population, especially for the COVID-19 pandemic.

scholarly journals The attainment of herd immunity with a reducing risk of contact infection in modelling 

2021 ◽  
Author(s):  
Kian Boon Law ◽  
Kalaiarasu M Peariasamy ◽  
Hishamshah Ibrahim ◽  
Noor Hisham Abdullah
Keyword(s):  
Infectious Disease ◽  
Disease Transmission ◽  
Herd Immunity ◽  
Sir Model ◽  
Mixed Population ◽  
Transmission Model ◽  
Force Of Infection ◽  
Initial Stage ◽  
And Control ◽  
Model Features

Abstract The risk of contact infection among susceptible individuals in a randomly mixed population can be reduced by the presence of immune individuals and this principle forms the fundamental of herd immunity. The conventional susceptible-infectious-recovered (SIR) model features an infection-induced herd immunity model, but does not include the reducing risk of contact infection among susceptible individuals in the transmission model, therefore tends to overestimate the transmission dynamics of infectious diseases. Here we show that the reducing risk of contact infection among susceptible individuals can be achieved by incorporating the proportion of susceptible individuals (model A) or the inverse of proportion of recovered individuals (model B) in the force of infection of the SIR model. We numerically simulated the conventional SIR model and both new SIR models A and B under the exact condition with a basic reproduction number of 3·0. Prior to the numerical simulation, the threshold for the eradication of infectious disease through herd immunity was expected to be 0·667 (66·7%) for all three models. All three models performed likewise at the initial stage of disease transmission. In the conventional SIR model, the infectious disease subsided when 94·0 % of the population had been infected and recovered, way above the expected threshold for eradication and control of the infectious disease. Both models A and B simulated the infectious disease to diminish when 66·7% and 75·6% of the population had been infected, showing herd immunity might protect more susceptible individuals from the infectious disease as compared to the projection generated by the conventional SIR. Our study shows that model A provides a better framework for modelling herd immunity through vaccination, while model B provides a better framework for modelling herd immunity through infection. Both models overcome the insufficiency of the conventional SIR model in attaining the effect of herd immunity in modelling outputs, which is important and relevant for modelling infectious disease, such as the COVID-19 in a randomly mixed population.

scholarly journals The attainment of herd immunity with a restructured risk of contact infection in compartmental models

2021 ◽  
Author(s):  
Kian Boon Law ◽  
Kalaiarasu M Peariasamy ◽  
Hishamshah Ibrahim ◽  
Noor Hisham Abdullah
Keyword(s):  
Infectious Disease ◽  
Herd Immunity ◽  
Approximate Solutions ◽  
Sir Model ◽  
Compartmental Models ◽  
Mixed Population ◽  
Reproduction Ratio ◽  
Force Of Infection ◽  
Infection Dynamics ◽  
The Difference

Abstract Introduction: The risk of contact infection among susceptible individuals in a randomly mixed population can be reduced by the presence of immune individuals and this concept is referred to as herd immunity. Although herd immunity is observed in vaccinated populations for some infectious diseases, it has never been truly attained in compartmental models such as the susceptible-infectious-recovered (SIR) model. This paper introduces a new SIR framework to overcome the limitation of the conventional SIR model in attaining herd immunity.Methods: Two SIR models were newly developed based on the reduced risk of contact infection. The first model A assumes that the risk of contact infection reduces as soon as susceptible individuals are infected and move from class S(t) to I(t), therefore incorporating prevalence of both infectious and susceptible individuals into its force of infection. The second model B assumes the risk of contact infection would reduce after infected individuals have recovered from infection and move from class I(t) to R(t), therefore incorporating the prevalence of infectious and the inverse of prevalence of recovered individuals into its force of infection. Then, numerical simulations were applied to obtain approximate solutions for all three conventional SIR model, new SIR model A and model B for comparison under exact and arbitrary conditions with β = 0.3 and σ = 0.1 to mimic the infection dynamics with basic reproduction ratio (r0) of 3.0 and herd immunity threshold (HIT) of 0.667 (66.7%).Results and discussion: All three models performed likewise at the initial stage of the epidemic. The conventional SIR model simulated the epidemic diminishing when 94.0% of the population had been infected and recovered, way above its HIT. Model A simulated the epidemic waning when 66.7% of the population had been infected and recovered, in line with its HIT. However, the model conceptualized herd immunity incorrectly. Model B simulated the epidemic waning at 75.6%, slightly above its HIT and was more in line with the fundamental of herd immunity. The difference between model A and model B can be attributed to the proportion of infectious individuals, and this would increase in infectious disease with high transmissibility. The threshold theorem derived based on r0 may not be sufficient for optimal control and eradication of infectious disease with high transmissibility like the COVID-19.Conclusion: The newly developed SIR model that includes the inverse of proportion of recovered individuals into its force of infection is more accurate and credible for modelling infection with high transmissibility or vaccine-induced herd immunity in a randomly mixed population, especially in COVID-19.

scholarly journals The attainment of herd immunity with a restructured risk of contact infection in compartmental models

2021 ◽  
Author(s):  
Kian Boon Law ◽  
Kalaiarasu M Peariasamy ◽  
Hishamshah Ibrahim ◽  
Noor Hisham Abdullah
Keyword(s):  
Infectious Disease ◽  
Herd Immunity ◽  
Approximate Solutions ◽  
Sir Model ◽  
Compartmental Models ◽  
Mixed Population ◽  
Reproduction Ratio ◽  
Force Of Infection ◽  
Infection Dynamics ◽  
The Difference

Abstract Introduction: The risk of contact infection among susceptible individuals in a randomly mixed population can be reduced by the presence of immune individuals and this concept is referred as herd immunity. Although herd immunity is observed in vaccinated population for some infectious diseases, it has never been truly attained in compartmental models such as the susceptible-infectious-recovered (SIR) model. This paper introduces a new SIR framework to overcome the limitation of the conventional SIR model in attaining herd immunity.Methods: Two SIR models were newly developed based on the reduced risk of contact infection. The first model A assumes that the risk of contact infection reduces as soon as susceptible individuals are infected and move from class S(t) to I(t), therefore incorporating prevalence of both infectious and susceptible individuals into its force of infection. The second model B assumes the risk of contact infection would reduce after infected individuals have recovered from infection and move from class I(t) to R(t), therefore incorporating the prevalence of infectious and the inverse of prevalence of recovered individuals into its force of infection. Then, numerical simulations were applied to obtain approximate solutions for all three conventional SIR model, new SIR model A and model B for comparison under exact and arbitrary conditions with β = 0.3 and σ = 0.1 to mimic the infection dynamics with basic reproduction ratio (r0) of 3.0 and herd immunity threshold (HIT) of 0.667 (66.7%).Results and discussion: All three models performed likewise at the initial stage of epidemic. The conventional SIR model simulated the epidemic diminishing when 94.0% of the population had been infected and recovered, way above its HIT. Model A simulated the epidemic waning when 66.7% of the population had been infected and recovered, in line with its HIT, however, the model conceptualized the herd immunity incorrectly. Model B simulated the epidemic waning at 75.6%, slightly above its HIT and was in line with the fundamental of herd immunity. The difference between model A and model B can be attributed to the proportion of infectious individuals, and this would increase in infectious disease with high transmissibility. The threshold theorem derived based on r0 may not be sufficient for optimal control and eradication of infectious disease with high transmissibility like the COVID-19.Conclusion: The newly developed SIR model that includes the inverse of proportion of recovered individuals into its force of infection is more accurate and credible for modelling infection with high transmissibility or vaccine-induced herd immunity in a randomly mixed population, especially in COVID-19.

scholarly journals The Effect of Vaccination Rates on the Infection of COVID-19 under the Vaccination Rate below the Herd Immunity Threshold

2021 ◽  
Vol 18 (14) ◽  
pp. 7491
Author(s):  
Yi-Tui Chen
Keyword(s):  
Infection Rate ◽  
Herd Immunity ◽  
The United States ◽  
Vaccination Rate ◽  
Infection Rates ◽  
Vaccination Rates ◽  
The United Kingdom ◽  
The World ◽  
Almost All ◽  
The Impact

Although vaccination is carried out worldwide, the vaccination rate varies greatly. As of 24 May 2021, in some countries, the proportion of the population fully vaccinated against COVID-19 has exceeded 50%, but in many countries, this proportion is still very low, less than 1%. This article aims to explore the impact of vaccination on the spread of the COVID-19 pandemic. As the herd immunity of almost all countries in the world has not been reached, several countries were selected as sample cases by employing the following criteria: more than 60 vaccine doses per 100 people and a population of more than one million people. In the end, a total of eight countries/regions were selected, including Israel, the UAE, Chile, the United Kingdom, the United States, Hungary, and Qatar. The results find that vaccination has a major impact on reducing infection rates in all countries. However, the infection rate after vaccination showed two trends. One is an inverted U-shaped trend, and the other is an L-shaped trend. For those countries with an inverted U-shaped trend, the infection rate begins to decline when the vaccination rate reaches 1.46–50.91 doses per 100 people.

scholarly journals Estimating SARS-CoV-2 infections from deaths, confirmed cases, tests, and random surveys

2021 ◽  
Vol 118 (31) ◽  
pp. e2103272118
Author(s):  
Nicholas J. Irons ◽  
Adrian E. Raftery
Keyword(s):  
Herd Immunity ◽  
The United States ◽  
Sir Model ◽  
Mitigation Strategies ◽  
Time Varying ◽  
Fatality Rate ◽  
Multiple Sources ◽  
Population Prevalence ◽  
Sample Testing ◽  
True Infection

There are multiple sources of data giving information about the number of SARS-CoV-2 infections in the population, but all have major drawbacks, including biases and delayed reporting. For example, the number of confirmed cases largely underestimates the number of infections, and deaths lag infections substantially, while test positivity rates tend to greatly overestimate prevalence. Representative random prevalence surveys, the only putatively unbiased source, are sparse in time and space, and the results can come with big delays. Reliable estimates of population prevalence are necessary for understanding the spread of the virus and the effectiveness of mitigation strategies. We develop a simple Bayesian framework to estimate viral prevalence by combining several of the main available data sources. It is based on a discrete-time Susceptible–Infected–Removed (SIR) model with time-varying reproductive parameter. Our model includes likelihood components that incorporate data on deaths due to the virus, confirmed cases, and the number of tests administered on each day. We anchor our inference with data from random-sample testing surveys in Indiana and Ohio. We use the results from these two states to calibrate the model on positive test counts and proceed to estimate the infection fatality rate and the number of new infections on each day in each state in the United States. We estimate the extent to which reported COVID cases have underestimated true infection counts, which was large, especially in the first months of the pandemic. We explore the implications of our results for progress toward herd immunity.

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