Introduction

The herd immunity threshold is the proportion of a population that must be immunized in order to cease an epidemic and impart indirect protection to those without personal immunity, thereby preventing the spread of a disease (₁). This concept has long been understood. In addition, the effects of the herd immunity threshold on diseases have been researched (₂).

The purpose of this experimentation was to develop an equation to dynamically approximate the herd immunity threshold via the incorporation of the dynamic changes in the population parameters (the fractions of the population susceptible, S; infectious, I; Recovered, R – also referred to as population dynamics in this paper) and gradual immunization as opposed to probability of infection or some other method previously developed (₁) (₃). This population parameter based (PPB) equation will quantify the effects of the population parameters and gradual immunization on the herd immunity threshold. This equation will then be tested on multiple past epidemics to determine its efficiency and identify possible trends related to the effects of the population parameters and gradual immunization on the threshold.

The researcher hypothesized this equation would effectively incorporate the effects of population dynamics and gradual immunization through the manipulation of an altered SIR model, and that gradual immunization would be an effective measure for reducing the infectiousness of a disease.

Though the spread of disease has been thoroughly analyzed (₁,₃), the incorporation of this spread into the calculation of the herd immunity has not been based on the dynamics of population parameters and gradual immunization analysis alone (₄). The four previously implemented techniques for determining the basic reproductive number (R₀ – a quantification of the infectiousness of the disease which is critical in the calculation of the herd immunity threshold) (₁) include: 1) Directly calculating the number of secondary infections per infectious person; 2) Calculating R₀ based on final prevalence of the disease; 3) Estimating R₀ based on transmission chains; 4) Approximation based on the initial increase in the force of infection (₄). A PPB approach allows for the more accessible understanding of how the population dynamics and gradual immunization affect the herd immunity threshold and the basic reproductive number, allowing improved analysis and approximation.

The comparison of pulse and gradual immunization has been an ongoing debate. Two papers have advocated the implementation of pulse immunization as an effective public health measure (₅,₆). They postulate that cyclical disease outbreak patterns warrant cyclical immunization patterns in pulses (₅,₆).

The World Health Organization (WHO) currently employs a blanket target, fixed at 95% (₇), for the herd immunity threshold of measles outbreaks, however this fixed method is not very effective at conserving resources. Measles primarily affects poor nations (₈), and therefore limiting the cost of outbreak control is paramount in order to make public health safety affordable and effective simultaneously. Reducing the cost and time of achieving the herd immunity threshold is the primary goal of this research. This goal can be reached through a two-pronged approach: analysis and approximation. Analysis tools will improve the understanding of exactly how various factors affect the threshold. This understanding allows the control of these factors, thereby limiting the herd immunity threshold. The second prong is the approximation of this now reduced threshold using the equation for the herd immunity threshold.

Methods

Formulation of Equations:

Primary Equation

The rate of immunization of susceptible is assumed to be proportional to the fraction susceptible.

A researcher-modified version of the SIR equation (with the un-modified version found in Infectious Diseases of Humans by Anderson and May) (₁):

Figure 1

Where S=fraction of population susceptible to the disease, I=fraction of population infectious, R=fraction of population recovered from disease, V_o =fraction of population successfully vaccinated during the outbreak.

λ =transmission rate, ρ=the proportion of susceptible population immunized per unit time, δ=recovery rate. The natural (non-disease-related) birth and death rates are assumed equal. Disease-related deaths are categorized as R.

Dividing the first equation by the second yields:

Figure 2

Separation of variables and integration yields:

Figure 3

Because when S=P (initial fraction of population susceptible), I=N (fraction of population initially infectious, or 1/Total Population).

Figure 4

Solving for λ:

Figure 5

Because (λ/δ) = R₀, or the basic reproductive number (₁), this equation yields:

Equation 1(E1):

Figure 6

The effective reproductive number, or R_E=S*R₀ can now be calculated:

Equation 2 (E2):

Figure 7

Because when (dI / dT ) < 0 the herd immunity threshold has been reached (₁), the following is calculated:

Figure 8

which yields (R₀ )(S)<1 and because P=S + I + R + V_o then:

Figure 9

With the equation for R₀ found above, the following equation is found:

Equation 3 (E3):

Figure 10

Where V_T is the herd immunity threshold value; the fraction of the population that must be immunized during the outbreak to achieve herd immunity, as opposed to V_o , the fraction of the population already successfully immunized during the outbreak (not necessarily fulfilling the herd immunity threshold). The fraction of the population that still must be immunized to achieve herd immunity (V_R ) can be calculated as follows:

Figure 11

Analysis Tools

These partial derivatives were computed under the flawed notion that S, I, and R are independent variables. The author now recognizes that these variables, as well as V_o and V_T , all depend on time t. (Please see the Discussion section for other issues arising from this oversight.) Thus, it is not meaningful to compute those partial derivatives in this system; please disregard those incorrect equations. In contrast, the total derivatives take the dependence on time into account, and these are computed below.

Using the chain rule on the V_T equation, one finds:

Figure 12

Then, one computes the total derivative of V_T with respect to S, I, and R by the following equations:

Figure 13

Please note that these equations demonstrate the effects of the change in time during which the change in the population parameter occurs.

Partial derivatives of functions are important in understanding the individual effects each parameter has on the overall threshold:

Equation Set 4 (ES4):

Figure 14

These partial derivatives elucidate the individual effects of each of the population parameters on the herd immunity threshold. Obviously, these partial derivatives should be optimized so that they are negative, thus decreasing the herd immunity threshold.

Outbreak Application

2006 Fiji

All data gathered from: “Measles outbreak and response in Fiji, February-May 2006.” In the Weekly epidemiological record (₇).

Background: A measles outbreak strikes the Pacific island nation of Fiji, and the WHO responds with massive immunization efforts among children under 5 years. The WHO supports a 95% immunization rate as a general guideline to stop the measles outbreak (₇).

Dealing with the 6-11 month old subgroup and with outbreak totals (₇):

Most recent data shows an immunization rate at beginning of the outbreak at 80%.

P= 1 - .8 = .20 and 33 cases out of 8,722 children in age range yields:

Figure 15

With 76% recovering quickly, or R = .00378(.76) = .00287

Figure 16

Assuming no additional immunizations during the outbreak (in order to calculate the maximum threshold):

ρ = 0 and .00378 + .00287 + S = .20 therefore S = .19335. Also, δ =1/8=.125 (as approximately 1/8 infectious persons recover per day) (₈).

Figure 17

Total threshold (initial rate + V_T): .80+.105=.905

2000 Sri Lanka

All data gathered from: “Measles Outbreak in Sri Lanka, 1999-2000” from The Journal of Infectious Diseases (₉).

Background: A measles outbreak erupted in the slums of Sri Lanka in late 1999, leading to a larger outbreak during early 2000, with the peak number of infected occurring at week 6, 2000. The nation had a 93% immunization rate at the commencement of the outbreak. The Sri Lanka health ministry responded with various public health measures (₉).

(All week #s of 2000):

Figure 18

(₉).

Because:

S + I + R = P therefore S + .00298 + .0001 = .07 which yields S = .069602

Figure 19

Total threshold (initial rate + V_T): .93+.017=.947

2003 Marshall Islands

All data gathered from: “Measles outbreak in the Republic of the Marshall Islands, 2003” from the International Journal of Epidemiology (₁₀).

Background: The Marshall Islands (henceforth to be referred to as RMI) had a population of 50,480 by last estimate previous to the outbreak(₁₀). The measles outbreak commenced on July 13th, 2003 and the nation had a 80% immunization rate with a one-dose MMR vaccine as of 2002 (this figure will be used to determine P). The outbreak ended on November 7th, 2003. During the outbreak, the health ministry implemented a massive gradual immunization campaign and public health program including transportation limits and school closings. During the outbreak, a 93% immunization rate with the MMR one-dose vaccine was achieved (₁₀).

This article provided day-by-day breakdowns of rash onset numbers by date. The greatest number of infected persons existed on 08/20/2003.

Figure 20

800 people were immunized from August 1st to August 20th, or 20 days (₁₀); therefore:

Figure 21

Total threshold (initial rate + V_T ): .80 + .133 = .933

Because V_o =.0158, V_R =.1172

Effects of Gradual Immunization

The outbreak control strategy in RMI implemented gradual immunization (ρ = 0.004) The researcher set up a hypothetical pulse immunization to compare immunization strategies. By setting ρ = 0 to model pulse immunization that has yet to occur, the effects of gradual versus pulse immunization can be compared.

If ρ = 0 , (for future pulse immunization) then V_o =0 as opposed to .0158. Assuming a proportional increase in I and R (for estimation purposes) the herd immunity threshold can be computed as follows:

Figure 22

Results

Below are a series of graphs demonstrating the individual effects of the three studied population parameters on the herd immunity threshold:

Figures 1-3 demonstrate clear and consistent trends in the effects of population parameters on the herd immunity threshold. It is important to note that the dependent variable, V_T , is the overall herd immunity threshold as opposed to the remaining proportion required to achieve the threshold.

Figure 23

Figure 24

Figure 25

Figure 26

Table 1: List of the equations analyzing effects of population dynamics and immunization on the herd immunity threshold created during this research.

The ES4 is incorrect, and total derivatives should have been computed instead because S,I, R, V_o , and V_T are each entirely dependent on time. Please see the “Analysis Tools” section above for the proper approach.

Where S=fraction of population susceptible to the disease, I=fraction of population infectious, R=fraction of population recovered from disease, V=fraction of population successfully vaccinated during the outbreak. λ=transmission rate, ρ=the proportion of susceptible population immunized per unit time, δ=recovery rate, P=initial fraction of the population susceptible, N=1/Total Population, R₀ =basic reproductive number, and R_E =effective reproductive number.

The equations of Table 1 (above) are those developed by the researcher during the course of this work. These equations revealed several important aspects of the herd immunity threshold that can be capitalized upon in order to reduce the threshold, and thereby the cost of attainment.

Figure 27

Table 2: Summary of threshold approximations (=the fraction of a population that must be immunized during the outbreak to achieve herd immunity.

The exact effects of gradual immunization on the herd immunity threshold are demonstrated in Table 3 (below). Table 3 demonstrates that gradual immunization decreases both R_E and R₀ . It also evinces that gradually immunizing a population decreases the herd immunity threshold.

Figure 28

Table 3: Demonstration of the effects of gradual immunization on the herd immunity threshold using data from the Marshall Islands Outbreak (where ? = the proportion of the susceptible population immunized per unit time, =the basic reproductive number; =the effective reproductive number, or *; =the fraction of a population that must be immunized during the outbreak to achieve herd immunity; =the fraction of the population remaining that needs to be immunized to achieve herd immunity)

Discussion

With the benefit of further mathematical training, the author revisited this work and would like to clarify certain aspects of it. First, the integration step in the model derivation implicitly assumed that all rate parameters (those of transmission λ, recovery δ, and immunization ρ) are constant with respect to time. Thus, the model assumes that the basic reproduction number (R₀) is constant for a given outbreak.

Due to the assumption on the rate parameters, and thus on R₀, the observed change attributed to the gradual immunization must be an artifact of the hypothetical approximations for I and R. Because the observed decrease in R₀ was used to support gradual over delayed pulse immunization, the author acknowledges that this interpretation is flawed. However, the intuition behind deploying vaccines as early as possible is strong: infections are driven by availability of susceptible hosts, which are removed via vaccination.

In light of the assumptions, the proper scope of the model from this paper is to estimate the constant R₀ for an outbreak, and use this to compute V_T. Ideally, the model should capture some of the environmental and societal structure of the population, thus improving the estimate of R0 compared to a universal one. The assumptions also mean that any changes imposed on the transmission, recovery, or immunization rates during the outbreak (i.e. due to school closures or quarantines) cannot be accurately captured. However, the model remains useful in estimating an outbreak-specific reproduction number and thus the herd immunity threshold.

Regression methods applied to data from several time points may improve the accuracy of the model’s R₀ estimate. The author also notes that a singularity exists in the equation for R₀ at the beginning of the epidemic (i.e. when S = P> and I = N). Numerical simulation of the underlying system of differential equations indicates that the model underestimates R0 near the beginning of the outbreak, and that accuracy improves as the epidemic progresses.

The author apologizes to the editors and readers of the Internet Journal of Epidemiology for the delay in correcting and clarifying these issues.

The minimization of the herd immunity threshold is an excellent method for conserving resources and reducing costs of outbreak control. This paper demonstrates several ways to achieve this.

Due to the fact that measles is a disease that primarily affects poor nations (₈), resource conservation is paramount. Table 3 demonstrates clearly that gradual immunization reduces the overall herd immunity threshold, V_T , not just the fraction remaining that needs to be immunized, V_R , at no extra cost.

There are two major types of reproductive numbers: R₀ , the basic reproductive number and R_E =(S)(R₀ ) the effective reproductive number (₁₁). While gradual immunization obviously decreases the overall effective reproductive number (by lowering S), it also decreases the basic reproductive number, as seen in Table 3.

Gradually immunizing the susceptible populous is an extremely cost-effective method for reducing the herd immunity threshold and thereby reducing expenses. In fact, gradual immunization not only saves money, it also inhibits the infectiousness of the disease by lowering the R₀ . This results in less infections occurring, reducing both morbidities and mortalities associated with the disease. However, pulse immunization, in which a large segment of the population is immunized nearly simultaneously, has been supported (₅,₆) as an effective outbreak control measure. The 1993 paper focused upon immunization in Israel, a nation with greater resources at its disposal than the nations discussed here, such as Fiji and Sri Lanka (₅). Even so, as demonstrated in Table 3, the more cost efficient method is gradual immunization, because it actually decreases the herd immunity threshold and protects the populous better as persons are constantly being immunized. Such data may support pulse immunization as an outbreak prevention tool, as it responds to cyclical disease occurrence (₆). However, Table 3 clearly demonstrates that gradual immunization is a more effective outbreak control tool, as it reduces the dynamic infectiousness of the disease (as shown by the decrease in R₀ ), thereby reducing morbidities, mortalities, and cost associated with the outbreak.

Though .002 may not seem like a great deal of difference between the V_T values in the RMI outbreak (Table 3), its importance becomes more apparent if this same data is hypothetically applied to the Sri Lanka outbreak. In Sri Lanka, with a population of 19.1 million, a .2 of a percentage point is 38,200 persons. At an average total cost of around 1 USD per immunization (₈), this amounts to 38,200 USD saved from gradually immunizing the populous alone. Also, extrapolating the decrease in R₀ to a proportional decrease in number of infections (for approximation purposes) in the Sri Lanka outbreak, an analysis of the effects can be obtained. Similar gradual immunization could have prevented 95 infections by week 6, 2000. With the 5 % fatality rate of measles infections in developing countries (₈) applied, such gradual immunization could have prevented about 5 fatalities by week 6, 2000.

Based on these data figures, it is clear that gradual immunization can reduce the ability of a disease to spread through a population. Such gradual immunization measures could have significant impacts reducing morbidities, mortalities, and costs associated with the outbreak.

These equations may also be applied in the United States. In the age of increased bioterrorism risk, the possibility of an attack increases. The beneficial effects of gradual immunization can be applied to bioterrorist-initiated outbreaks, especially when the vaccine is in limited supply, in production, or still in development. Gradually immunizing the population reduces the overall amount of vaccine needed, expediting outbreak control. It also reduces the infectiousness of the disease, reducing the number of outbreaks and related morbidities and mortalities. Also, E3 (see Table 1) may be used to approximate the herd immunity threshold dynamically in order to gauge progress and vaccine production requirements.

The ES4 (see Table 1) imparts important understanding of how exactly population dynamics affect the herd immunity threshold. Although outbreak control agencies exercise little control over the actual population dynamics, these equations maintain their usefulness. For example, with weekly counts in hand, the outbreak control agency can calculate the partial derivatives in order to better understand the effects their programs are having on the threshold itself. If there is a large positive derivative of V_T , programs are not effectively managing the outbreak.

When analyzed individually, the partial derivatives and Figures 1-3 demonstrate important aspects of population dynamics.

As can be seen, a decrease in S leads to a decrease in V_T . This decrease in S can be achieved in two ways: immunization or infection. Immunization is the obvious choice.

As I increases, V_T increases as well. Therefore, the limiting of infections is important in reducing the herd immunity threshold. This can be achieved by increasing the V_o and R compartments through immunization and recovery programs.

In contrast, an increase in R results in a decrease in V_T . This increase in R may be expedited by ensuring proper nutrition and the consumption of one Vitamin A supplement every 24 hours (₈). This not only limits mortality rates(₈), it also cost-effectively lowers the herd immunity threshold.

Finally, E3 (see Table 1) can be used primarily to approximate the herd immunity threshold. By approximating the herd immunity threshold dynamically based on weekly counts as opposed to implementing a fixed approximation of 95% like the WHO (₇), a more accurate threshold level, which was below 95% in all three examples (Table 2), can be approximated and achieved. This reduces the cost of outbreak control, of paramount importance in resource poor settings.

While this experimentation yielded informative results, with any and all scientific research projects, some issues remain. The equations formulated during this experimentation are based on the values S (fraction susceptible), R (fraction recovered), I (fraction infectious), and V_o (fraction immunized during outbreak). Because of this, the equations do not take into account the incubation period of the measles virus during which the person is infected but not infectious (₈). However, this factor is largely mitigated by categorizing these persons as “susceptible” for they are not yet infectious. Also, epidemiological surveys cannot calculate the number of exposed based on symptoms, and therefore these persons should be categorized as susceptible. This equation would therefore prove even more effective for diseases with shorter incubation periods, such as influenza (₁₂).

Future work could improve upon this aspect by developing a threshold equation that incorporates the “exposed” fraction. Such an improvement would be a good theoretical tool to analyze the effects of the exposed populous on the herd immunity threshold. However, it would have little practical purpose as the “exposed” fraction would be nigh impossible to calculate in real-time, which would be required by the equation developed in this paper.

Another less obvious benefit of the reduction of the herd immunity threshold is that a lower threshold is more easily attained when there is some popular resistance to immunization. When this popular resistance occurs, the lower the threshold is, the more likely it can be achieved.

In conclusion, this novel research provides the tools with which the herd immunity threshold can be better understood with respect to the effects of population dynamics and gradual immunization. With both analysis and approximation tools at hand, the cost of obtaining the herd immunity threshold can be lowered significantly, conserving vital resources in the hardest-hit resource-poor settings (₈). The overall cost of a measles immunization is approximately 1 USD, making a few percentage points difference relatively enormous given the GDP (Gross Domestic Product) of many of the nations worst affected (₈). These equations also demonstrate the effectiveness of gradual immunization at inhibiting the spread of an infectious disease through the population as compared to pulse immunization. Data presented in the paper suggests that such gradual immunization would reduce the number of infections, lowering the number of morbidities and mortalities caused by the outbreak.