N3. Herd immunity

Epidemics have long influenced public health policies and research priorities, shaping global efforts to mitigate disease spread. In modern epidemiology, mathematical modeling has emerged as an essential tool for understanding disease dynamics and guiding intervention strategies. At the core of these models lies the basic reproduction number (R₀), a fundamental metric in infectious disease modeling.  R₀ is a crucial measure of a pathogen’s transmissibility and outbreak potential, making it one of the most frequently utilized metrics in the study of infectious disease dynamics.

The ability of a pathogen to establish and sustain transmission is directly linked to R₀, making it a cornerstone for understanding transmission dynamics. By quantifying the expected number of secondary infections caused by a single infected individual in a fully susceptible population, R₀ provides crucial insights into epidemic potential. Reducing R₀ below 1 is fundamental to epidemic prevention, achievable through widespread immunization, improved healthcare access, and public awareness campaigns.  Tracking R₀ enables public health officials to design targeted interventions, such as vaccination campaigns, social distancing measures, and quarantine protocols. Understanding the interplay between pathogen characteristics, host susceptibility, and environmental factors is vital for accurately estimating R₀ and developing effective control strategies. Integrating biological, environmental, and social factors into epidemiological models enhances the ability to predict and contain infectious disease outbreaks.

1.  Basic reproduction number (R₀)

R₀ quantifies the potential spread of an infectious disease within a fully susceptible population. It is defined as the expected number of secondary infections generated by a single infected individual in the absence of immunity or intervention measures.

Mathematically, R0 can be expressed as:

$$R_0 = SβL$$

where:

[latex]S[/latex] (Susceptible Population):  The number of susceptible individuals in a population

[latex]β[/latex] (Transmission Rate):  The probability of disease transmission between an infected and a susceptible individual.

[latex]L[/latex] (Infectious Period):  The average duration an individual remains contagious.

The interpretation of R₀ follows a clear epidemiological framework:

If [latex]R_0>1[/latex] → The disease is likely to spread exponentially, leading to an outbreak or epidemic.

If [latex]R_0=1[/latex] → The disease remains stable within the population, meaning each infected individual replaces themselves with exactly one new case.

If [latex]R_0<1[/latex] → The disease will gradually decline and eventually disappear, as each infected individual transmits the infection to fewer than one person on average.

R₀ values for different diseases

This equation highlights the key determinants of disease spread:  higher transmission efficiency, prolonged infectious periods, and frequent interactions within a population all contribute to increasing R₀. Consequently, public health interventions, such as reducing contact rates (social distancing), shortening the infectious period (early treatment), and lowering transmission probability (vaccination), aim to decrease R₀ and control disease outbreaks.The basic reproduction number varies significantly across infectious diseases, reflecting differences in their transmissibility and outbreak potential. For COVID-19, early pandemic variants the R₀ range between 2 – 3, and Omicron variant with R₀ of around 8.2. This variation demonstrates how viral mutations can dramatically change transmission potential.

Seasonal influenza, for instance, has an estimated R₀ ranging from 0.9 to 2.1, indicating moderate transmission rates. The 1918 flu pandemic had a slightly higher R₀ of 1.4 to 2.8, contributing to its widespread impact. In contrast, measles, one of the most highly contagious diseases, has an R₀ between 12 and 18, meaning a single infected person can spread the virus to a large number of susceptible individuals. Monkeypox (MPXV) presents a more variable transmission pattern. Earlier studies estimated an R₀ of 0.83, suggesting self-limiting outbreaks. However, recent modeling in non-endemic regions found R₀ values between 1.10 and 2.40, raising concerns about its potential for sustained transmission. The 2014 Ebola virus outbreak in West Africa is the largest outbreak of the genus Ebolavirus to date. The maximum likelihood estimates of the basic reproduction number are 1.51 for Guinea, 2.53 for Sierra Leone and 1.59 for Liberia.

These variations highlight the necessity for disease-specific control measures, emphasizing the importance of tailored public health strategies to mitigate transmission risks effectively. Estimating R₀ comes with inherent uncertainties, particularly during the early stages of an outbreak. Most modeling simulations produce a range of R₀ values, reflecting difficulties in accurately determining key parameters. One major challenge is identifying the true number of cases, as many mild or asymptomatic infections may go undetected, contributing to unobserved transmission. These uncertainties underscore the importance of continuous epidemiological surveillance and refinement of disease models to improve public health response strategies.

Factors influencing R₀

Several factors influence R₀, extending beyond the intrinsic properties of a pathogen to include environmental and social determinants. Pathogen characteristics such as virulence, which affects disease severity and infectious duration, play a crucial role – higher virulence often leads to increased R₀. The mode of transmission also impacts R₀, with airborne diseases like measles exhibiting higher values compared to contact-based infections. Additionally, the incubation period influences transmission dynamics, as shorter incubation times accelerate infection cycles. Antigenic variation, observed in viruses like influenza, allows pathogens to evade host immunity, prolonging outbreaks and sustaining transmission.

Beyond pathogen biology, environmental and social factors significantly affect R₀. Population density facilitates frequent interactions, increasing disease spread, whereas behavioral practices such as improved hygiene, sanitation, and social distancing can effectively reduce contact rates and lower R₀. The efficiency of the healthcare system further modulates R₀, as rapid diagnosis, effective treatment, and robust containment strategies help suppress transmission and mitigate outbreak severity. Together, these factors shape disease dynamics, highlighting the complexity of controlling infectious disease spread and the necessity of integrated public health strategies.

2.  Effective reproductive number (R_t)

The effective reproduction number (R_t) is a theoretical indicator of the course of an infectious disease that allows policymakers to evaluate whether current or previous control efforts have been successful or whether additional interventions are necessary. This metric, however, cannot be directly observed and must be inferred from available data. One approach to obtaining such estimates is fitting SIR models to disease incidence data.

In the early days of the SARS-CoV-2 pandemic, given the absence of vaccines and the lack of effective therapeutics, governments primarily relied on non-pharmaceutical interventions (NPIs) to reduce the transmission of SARS-CoV-2, thereby lowering the death toll. Although effective in preventing deaths, NPIs such as mobility restrictions and stay-at-home orders impose a burden on society with economic and psychological costs. In addition to this, the effectiveness of these interventions wanes over time as compliance progressively diminishes. Following these considerations, policymakers strive to find an adequate balance between the interventions’ severity and acceptable transmission levels.

In this decision-making process, the effective reproduction number plays a crucial role. Briefly, the effective reproduction number, R_t, is the time-varying average number of secondary cases caused by a primary case at a calendar time [latex]t[/latex], and it is a theoretical indicator of the course of an infectious process. The interpretation of R_t is the same as for R₀: above the epidemic threshold [latex]R_t > 1[/latex], each infectious person leads to more than one secondary infectious person, and the disease is (re)emerging; below that threshold, there is limited secondary transmission.

Generally speaking, R_t is the result of a combination of intrinsic (decline in susceptible individuals) and extrinsic (change in contact patterns due to the implementation of control measures) factors, for which there are no readily available measurements. One, therefore, must resort to statistical methods to obtain an approximation of this epidemic indicator from compartmental models. However, R_t estimates from compartmental models are sensitive to data availability and assumptions in the model structure.

In practice, policymakers can use R_t in two ways. First, as a guide to assess in near real-time whether the interventions are succeeding [latex](R_t < 1)[/latex] or whether it is required to increment the response’s strength. Second, in retrospective analyses to assess how policy decisions, population immunity, and other factors have impacted transmission at specific points in time.

3.  Critical vaccination threshold (p_c)

Herd immunity occurs when a sufficient proportion of the population becomes immune – either through vaccination or natural infection – thereby reducing disease transmission. This protects both immunized individuals and those who cannot be vaccinated, such as people with contraindications or compromised immune systems. Vaccination directly lowers R₀ by decreasing the proportion of susceptible individuals. The critical vaccination threshold (p_c) required to achieve herd immunity is calculated using:

$$p_c = 1−\frac{1}{R_0}$$

For example, diseases with higher R₀ values require a larger proportion of the population to be immunized. Measles, with an R₀ of approximately 12 – 18, necessitates immunizing 92% – 95% of the population, whereas seasonal influenza, with an R₀ of about 1.3, requires only around 23% immunity to prevent outbreaks.

This relationship follows from the idea that if [latex]R_0−1[/latex] out of the [latex]R_0[/latex] individuals to whom an infected person might have transmitted the disease are vaccinated, each infected individual will generate fewer than one secondary infection. Thus, in general, p_c can be expressed as:

$$p_c =\frac{R_0−1}{R_0}$$

Vaccination strategies aim to lower R₀ by reducing susceptibility through immunization or by minimizing contact rates via social distancing and other public health interventions. For highly contagious diseases, targeted vaccination of high-transmission groups (e.g., healthcare workers and essential personnel) can be especially effective.

Vaccine efficacy

When designing vaccination programs, the effectiveness of the vaccine is a pivotal factor in determining the coverage needed to achieve herd immunity and effectively control the spread of infectious diseases. Vaccine efficacy refers to the percentage reduction in disease incidence among vaccinated individuals compared to those who are unvaccinated, typically measured under controlled conditions such as clinical trials. The success of a vaccination program hinges not only on achieving the critical vaccination threshold (p_c) but also on accounting for the effectiveness of the vaccine. This adjustment ensures that enough individuals in the population are protected to reduce transmission. The relationship between vaccine efficacy (E_v) and the actual proportion of the population that must be vaccinated (V_c) can be expressed as:

$$V_c = \frac{p_c}{E_v}$$

For example, if the critical vaccination threshold p_c is 80% (0.8) and the vaccine efficacy E_v is 90% (0.9), then:

$$V_c = \frac{0.8}{0.9} ≈ 89\%$$

This means nearly 89% of the population must be vaccinated to achieve herd immunity. If vaccine efficacy is lower – say 70% – the required coverage rises to approximately 114%, indicating that additional interventions or booster doses would be necessary.

Vaccine efficacy plays a crucial role in shaping vaccination programs and determining the required coverage to achieve herd immunity. When vaccines have high efficacy, a smaller proportion of the population needs to be immunized, making it easier to control disease spread. Conversely, for vaccines with moderate efficacy, a larger portion of the population must be vaccinated, which can present logistical and resource challenges. Additionally, the efficacy observed in clinical trials may not always translate directly to real-world effectiveness due to factors such as improper storage, delays in dosing, or individual variations in immune response. To account for these discrepancies, vaccination programs must incorporate a buffer in coverage calculations to maintain adequate protection. Furthermore, population heterogeneity – including differences in susceptibility, contact patterns, and vaccine access – necessitates tailored strategies to ensure that adjusted coverage goals are met across diverse demographics and regions.

Limitations of the herd immunity threshold in real-world epidemics

Herd immunity is often perceived as a definitive threshold beyond which disease transmission ceases, but this oversimplified view does not align with real-world epidemiology. The commonly used formula for the herd immunity threshold, [latex]1−1/R_0[/latex], assumes homogeneous mixing, stable population dynamics, and lifelong immunity. For instance, if R₀ is 3, the threshold is estimated at 66%. However, real-world populations exhibit heterogeneous mixing, where individuals with high contact rates acquiring immunity may reduce transmission, but susceptible subgroups can sustain localized outbreaks. Additionally, population turnover through births and migration continuously replenishes the susceptible pool, explaining why diseases like measles persist despite strong lifelong immunity and high vaccination coverage.

Moreover, waning immunity and viral evolution further complicate herd immunity dynamics.  Many respiratory viruses, including influenza, RSV, and coronaviruses, allow reinfections due to immune waning or antigenic drift. The COVID-19 pandemic demonstrated that vaccines, while effective in reducing severe illness, provide only temporary protection and do not entirely block transmission. The emergence of immune-evasive variants, such as Omicron, has raised the effective reproduction number (R_t), increased herd immunity thresholds, and contributed to breakthrough infections.

Furthermore, vaccine hesitancy remains a critical challenge, as misinformation, distrust in healthcare systems, and logistical barriers hinder widespread uptake, preventing communities from reaching theoretical immunity levels. These complexities highlight that herd immunity is not a fixed endpoint but a dynamic process influenced by epidemiological, immunological, and behavioral factors. As a result, infectious diseases rarely disappear entirely but often establish endemicity, necessitating ongoing public health interventions to manage their impact.

Licenses and Attributions

“Herd immunity” by Michelle McCully is adapted from “Understanding the Basic Reproduction Number (R₀): Calculation, Applications, and Limitations in Epidemiology” in the Open Journal of Epidemiology 15 (2):272-295, 2025 by Hamid H. Hussien, Khalid Rhamtallah Genawi, Nuha Hassan Hagabdulla, and Khalda M.Y. Ahmed and used under CC BY 4.0, and “Inferring the effective reproductive number from deterministic and semi-deterministic compartmental models using incidence and mobility data” in PLoS Computation Biology 18(6): e1010206, 2022 by Jair Andrade and Jim Duggan and used under CC BY 4.0.  “Herd immunity” is licensed under CC BY-NC 4.0.