# The Quantification Of The Effects Of Changes In Population Parameters On The Herd Immunity Threshold

N Georgette

###### Keywords

herd immunity threshold, immunization, mathematical modeling, measles

###### Citation

N Georgette. *The Quantification Of The Effects Of Changes In Population Parameters On The Herd Immunity Threshold*. The Internet Journal of Epidemiology. 2006 Volume 5 Number 1.

###### Abstract

The purpose was to develop a novel population parameter based (PPB) equation for the herd immunity threshold that incorporates the effects of population dynamics and immunization on the infectiousness of a disease and to analyze these effects. Previous research has not attempted this specific method. The researcher sought to improve cost effectiveness of outbreak response in resource-poor areas. This was achieved by solving for a PPB equation for the basic reproductive number and developing the threshold equation. The researcher applied this equation to three actual measles outbreaks.

The PPB equation demonstrates that, using data from the 2003 Marshall Islands measles outbreak, gradual immunization decreases both the effective and the basic reproductive numbers when compared to pulse immunization (from 3.48 to 3.05 and 18.38 to 17.47 respectively). This decreases the potency of the outbreak, thus reducing the associated morbidities, mortalities, and costs.

### 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, _{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 (_{0}
_{1}) include: 1) Directly calculating the number of secondary infections per infectious person; 2) Calculating R_{0} based on final prevalence of the disease; 3) Estimating R_{0} 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 _{1}):

Where _{o}

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

Dividing the first equation by the second yields:

Separation of variables and integration yields:

Because when

Solving for λ:

Because (λ/δ) = R_{0}, or the basic reproductive number (_{1}), this equation yields:

Equation 1(E1):

The effective reproductive number, or _{E}=S*R_{0}

Equation 2 (E2):

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

which yields (_{0}
_{o}

With the equation for _{0}

Equation 3 (E3):

Where _{T}
_{o}
_{R}

### Analysis Tools

These partial derivatives were computed under the flawed notion that _{o}
_{T}

Using the chain rule on the _{T}

Then, one computes the total derivative of _{T}

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):

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 _{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%.

With 76% recovering quickly, or

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

ρ = 0 and .00378 + .00287 + _{8}).

Total threshold (initial rate + V_{T}): .80+.105=.905

### 2000 Sri Lanka

All data gathered from: “Measles Outbreak in Sri Lanka, 1999-2000” from _{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):

(_{9}).

Because:

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 _{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 _{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.

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

Total threshold (initial rate + _{T}

Because _{o}
_{R}

### 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 _{o}

### 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, _{T}

##### Figure 26

The **ES4** is incorrect, and total derivatives should have been computed instead because _{o}
_{T}

Where _{0}
_{E}

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

The exact effects of gradual immunization on the herd immunity threshold are demonstrated in Table 3 (below). Table 3 demonstrates that gradual immunization decreases _{E}
_{0}

##### Figure 28

### 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 (R_{0}) is constant for a given outbreak.

Due to the assumption on the rate parameters, and thus on R_{0}, the observed change attributed to the gradual immunization must be an artifact of the hypothetical approximations for _{0} 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 R_{0} 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 R_{0} estimate. The author also notes that a singularity exists in the equation for R_{0} at the beginning of the epidemic (i.e. when

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, _{T}
_{R}

There are two major types of reproductive numbers: _{0}
_{E}
_{0}
_{11}). While gradual immunization obviously decreases the overall

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 _{0}
_{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 _{6}). However, Table 3 clearly demonstrates that gradual immunization is a more effective outbreak _{0}

Though .002 may not seem like a great deal of difference between the _{T}
_{8}), this amounts to 38,200 USD saved from gradually immunizing the populous alone. Also, extrapolating the decrease in _{0}
_{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 _{T}

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

As can be seen, a decrease in _{T}

As _{T}
_{o}

In contrast, an increase in _{T}
_{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 _{o}
_{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.