Improving the cost effectiveness of vaccine preventable disease outbreak response through novel mathematical methods

Mathematics

Nathan T. Georgette

Abstract

Vaccine preventable diseases (VPDs) primarily affect the poorest nations around the

world. In order to help reduce the costs associated with countering these VPDs in resource-poor

settings, I set out to develop novel mathematical methods intended to improve the cost

effectiveness of VPD outbreak control.

In order to do so, I focus on the effects of outbreak response immunization on the

dynamics of the VPD spread with a reformulation of a classic model. Based upon this approach,

I developed a model that can determine the minimum number of vaccines needed to guarantee

the end of the VPD outbreak in real time. Building upon this model, I formulated a recursive

algorithm for predicting this minimum number of vaccines several days in advance. Finally, I

generated an equation relating the changes in the vaccination coverage in a population with

changes in the individual reproduction number (a quantification of the infectiousness of the

VPD).

The mathematical methods developed herein represent a novel approach to improving the

cost effectiveness of VPD outbreak control. The models for the minimum number of vaccines

needed to stop the outbreak can allow outbreak control agencies to expend the minimum amount

of resources required. These preserved funds may then be diverted to other areas of dire medical

need.

Table of Contents

Title Page………………………………………………………………………………………….1

Abstract……………………………………………………………………………………………2

Table of Contents………………………………………………………………………………….3

Introduction………………………………………………………………………………………..4

Methods……………………………………………………………………………………………6

Results……………………………………………………………………………………………16

Discussion………………………………………………………………………………………..18

References………………………………………………………………………………………..21

Introduction

Vaccines provide the opportunity to greatly reduce the disease burdens of nations around

the world [1]. Unfortunately, vaccination programs are costly [2] and thus place extreme

financial burdens on the health systems of impoverished nations, which are most affected by

Vaccine Preventable Diseases (VPDs) [3]. In recognition of this problem, this research seeks to

improve the cost effectiveness of VPD outbreak control with the goal that funds preserved

through increased cost effectiveness can be applied to other areas of dire medical need.

The primary aspect of outbreak control discussed in this paper is vaccination. There are

two windows of time during which a vaccination program may be implemented: Pre-Outbreak

Immunization (POI) and Outbreak Response Immunization (ORI – occurs during the outbreak).

This paper seeks to optimize the cost effectiveness of the ORI approach through novel

mathematical methods.

The herd immunity threshold is the minimum vaccination rate at which the mean number

of infections per infectious individual is reduced to less than one [4]. The Outbreak Response

Immunization Threshold (ORIT) is the minimum fraction of the population that must be

immunized during the outbreak in order to reduce the mean number of infections per infectious

individual to less than one. In other words, the herd immunity threshold = Pre-Outbreak

Immunization (POI) rate + ORIT. With the ability to determine and predict the ORIT during a

VPD outbreak, outbreak control agencies may expend the minimum amount of resources

purchasing and deploying vaccines while still accomplishing the ultimate goal of guaranteeing

the end of the outbreak at that point. Those measles (MMR) vaccines not used, with shelf lives

of around 2 years [5], may be employed at a later date.

The herd immunity threshold (and by extension, the ORIT) is most often calculated based

upon the infectiousness of the disease [4]. One of the most common quantifications of a

disease’s infectiousness is the basic reproduction number [6], which is defined as the mean

number of secondary infections per infected individual in a completely susceptible population,

and usually referred to as R0 [7]. Another quantification of infectiousness is the individual

reproduction number, Ri, which is the mean number of secondary infections per infected

individual during the outbreak, making this number highly variable. Several mathematical

models for both of these values have been developed [8-12].

Other approaches to determining the minimum vaccination deployment required to end

an outbreak have been developed. One notable perspective involves the “firefighter problem,”

first introduced in [13], in which the minimum number of firefighters required to control a fire

spreading through a grid is calculated. In the epidemiological application of this problem, the

fire is a VPD and the firefighters are vaccines, protecting the vertices at which they are located.

In other words, the minimum number of vaccines needed to contain the outbreak may be solved

for through an iterative algorithm solution to the firefighter problem. Significant advances have

been made on this front, including the examination of grids with three or more dimensions [14],

and even generalization for grids of several dimensions [15].

This paper presents novel mathematical methods through which the cost effectiveness of

vaccination programs may be improved, taking into account the limitations of the two primary

methods discussed above. To this end, I focus upon the effects of ORI on the individual

reproduction number (and by extension, the ORIT itself). Basing the mathematical methods

presented in this paper upon this relationship represents a novel approach to improving the cost

effectiveness of outbreak control.

Methods

Model for disease spread

The development of the model for disease spread was the first step in this investigation.

The basis of this model was the SIR (Susceptible-Infectious-Recovered) model first developed

by Kermack and McKendrick [16]. This model uses a system of differential equations in order

to state the relationship among the three compartments into which each member of the

population falls: susceptible (without the disease, but can fall ill), infectious (with the disease,

and can spread it), and recovered (without the disease, and now immune).

Kermack-McKenrick SIR Model [16]:

(i)

(ii)

(iii)

Where S is the fraction of the population susceptible, I is the fraction of the population

infectious, R is the fraction of the population recovered, t is time, ? is the infection rate, and d is

the recovery rate.

The Kermack-McKendrick SIR Model fails to consider the effects of ORI on disease

dynamics. In order to take these effects into account, I incorporated another variable, V, which is

the fraction of the population immunized during the outbreak (it is assumed that these newly

immunized persons were susceptible prior to vaccination). I postulated that the fraction of the

population immunized per day is proportional to the fraction of the population susceptible. In

other words, the immunization rate will be higher earlier in the outbreak and then will reduce as

the outbreak comes under control. Incorporating this approach into the Kermack-McKendrick

SIR Model resulted in the novel model for disease spread shown below:

SIRV Model:

(iv)

(v)

(vi)

(vii)

The variables for the SIRV Model remain the same as for the SIR model, with the addition of ?,

the immunization rate, and V, the fraction of the population immunized during the outbreak. The

immunization rate can be calculated as follows:

(viii)

Where P is the fraction of the population initially susceptible.

This novel SIRV Model assumes homogenous mixing of the population, the mass action

principle, and does not incorporate the effects of outbreak control strategies other than

immunization (i.e. school closings, transportation limits, quarantines, etc.). These limitations are

due to the focus of this research on immunization specifically.

Development of survey data-based model for ORIT

The primary goal for the approach to improving cost effectiveness of ORI was to develop

a formula with which the ORIT could be approximated in real time using daily survey data. This

novel ORIT model was based on the SIRV Model reproduced below:

(1)

(2)

(3)

(4)

Dividing (3) by (4) yielded:

(5)

Separation of variables and integration of (7) resulted in:

(6)

(7)

P was defined as the initial fraction of the population that was susceptible, and N was defined as

the fraction of the population initially infectious. These are the initial conditions with which C

was solved for using (8):

(8)

I then solved for ?:

(9)

Due to the fact that (?/d)=R0, or the basic reproduction number, I found:

(10)

The basic reproduction number and the individual reproduction number (Ri) are related by the

following equation:

(11)

Therefore, equations (10,11) lead to an individual reproduction number model that is as follows:

(12)

Because the ORIT is defined as the immunization rate which results in (dI/dt) < 0, the following

was calculated:

(13)

(14)

(15)

(16)

(17)

Given the fact that P = S + I + R + V, I introduced the variable O, the value for V that satisfies

the herd immunity threshold. In other words, O is the ORIT.

(18)

Where O is the ORIT. Inserting (10) into (18) and solving for O resulted in:

ORIT Model:

(19)

With this model for the ORIT, an outbreak control agency may insert raw field survey

data (values for S, I, and R) to determine in real time an approximation for the ORIT. I tested

this ORIT Model against two actual measles outbreaks in order to gauge its validity: the 2003

Republic of the Marshall Islands (RMI) measles outbreak [17] and the 2006 Fiji measles

outbreak [18].

Development of recursive algorithm to predict ORIT several days in advance

Granting the outbreak control agency advanced knowledge of the ORIT value several

days in advance would allow it to properly coordinate an immunization campaign utilizing the

fewest resources.

Therefore, I developed a predictive algorithm with which the outbreak control agency can

predict the ORIT several days in advance. In order to develop the recursive predictive algorithm,

I first graphed the ORIT Model’s estimations for the ORIT over the duration of the 2003 RMI

measles outbreak. This graph is presented in Figure 1.

ORIT Model approximated ORIT (RED)

Fraction of population immunized during outbreak (BLUE)

Figure 1. Changes in ORIT through the 2003 RMI measles outbreak.

10 20 30 40 50 600

0.05

0.1

0.15

0.2

Day Number

Upon analysis of the graph, a strong relationship between the changes in the ORIT and

the rate of immunization became evident. With this in mind, I posited the following inverse,

direct, and constant relationships:

(20.1)

(20.2)

(20.3)

Where k is an arbitrary constant of proportionality and t is time, in days. As ? is a fraction, the

4th root of it was taken in order to increase its magnitude. This higher value reflects the high

impact of ? on changes in the ORIT.

I first calculated the change in ORIT between two days for several instances during the

2003 RMI measles outbreak and found significant variation in this slope among the tested

instances. This variability in the slope of the ORIT line (an approximation for dO/dt) eliminated

the possibility that (dO/dt) is constant, thus eliminating relationship (20.3). A comparison of the

slopes at each of these points with their respective immunization rates (? ) demonstrated that the

slope and immunization rate were close to inversely proportional. Based upon this analysis, I

concluded that the inverse relationship (20.1) was the most reasonable possibility and found that

it seemed to only apply when dO/dt < 0.

In order to generate a recursive model based on this function, the value of k must

constantly update in a recursive fashion. Based on this necessity, I applied a modified form of

Euler’s method with secant, rather than tangent, lines. First, the slope of the secant line (defined

as m) over the two previous days was found to be:

(21)

Using this equation, the value for k was calculated based on the relationship between m and ρ in

the following equation:

(22)

I set ε as an approximation for dO/dt based on the previous constant of proportionality

calculated in (22). The value for ε was defined as follows:

(23)

Therefore, the basic recursive formula for the value of On was generated as follows based on the

modified version of Euler’s method discussed above:

(24)

Inserting (21) into (22), then (22) into (23), and finally (23) into (24) yielded the final recursive

definition with which the ORIT may be predicted.

Recursive ORIT Model:

(25)

The final recursive definition presented in (25) predicts O several days in advance by first

predicting O for the very next day. Using this prediction, the ORIT for the following day is

predicted and this procedure is repeated until the final value is determined.

In order to expedite the application of the recursive predictor, I developed an algorithm

for the prediction of the ORIT using the recursive formula in (25) for the Java programming

language, which I called the Recursive Prediction of the ORIT Algorithm (RPORITA).

The application of the RPORITA consists of the following steps:

1. Enter the background information of the outbreak, including total population, vaccination

coverage at beginning of outbreak, and recovery rate (defined as 1/8 for measles [3] in

this paper).

2. Enter the survey data for two consecutive days, including number of persons infectious

and recovered for each day and total number of persons immunized during the outbreak

for each day.

3. Select the day number for which the ORIT should be predicted and enter the number of

immunizations that will occur on each day until the day of prediction.

4. The RPORITA then returns the predicted ORIT for the selected day of prediction.

Development of model relating delta(V) to delta(Ri)

The ability to determine the effects of additional vaccination on the individual reproduction

number would provide insight into outbreak response and improving cost effectiveness. For this

reason, I seek to develop a model for delta(Ri), the change in the individual reproduction number,

as a function of delta(V), the change in the vaccination coverage. In order to approximate this

relationship, I solve for dRi/dV.

I used the model for the individual reproduction number I developed concurrently with

the ORIT Model, reproduced from equation (12):

By introducing variables into equation (viii) from above, the formulation for the

immunization rate became:

With V=the fraction of the population immunized during the outbreak, c=the current day

number, and b=the day number on which vaccination began during the outbreak. The

vaccinations that would cause the are assumed to occur instantaneously for the purposes of

this model. Therefore, both c and I are held constant. However, S would decrease due to the

increase in V caused by the vaccinations, yielding the relationship below:

In order to simplify the Ri function, I decomposed it into functions of its numerator and

denominator, or y and z, respectively.

Taking the derivatives of the numerator and denominator individually with respect to V yielded:

By the quotient rule for differentiation:

(33)

Inserting (29,30,31,32) into (33) resulted in:

In order to determine the effects of changes in V on Ri at extremely low incremental values, the

following approximation was established:

(35)

By (35), can be determined with the function below:

(36)

Results

The ORIT Model was tested against two actual measles outbreaks in order to determine

its accuracy. The ORIT Model determined that the ORIT was achieved for the 2003 RMI

measles outbreak at around day 58 of the outbreak (see Figure 1). This day corresponds to the

significant and consistent decline in the number of rash onsets per day during the week of August

31 2003 [17]. The same analysis was conducted for the 2006 Fiji measles outbreak. The ORIT

determined by the ORIT Model was achieved on day 78 of the outbreak, which corresponds to

the noticeable and continued decline in rash onsets per day during the week of April 5 2006 [18].

These declines mark the achievement of the ORIT. While the actual ORIT is difficult to

determine exactly from published data, the high correlation between the ORIT Model-based and

actual ORIT approximations provides strong supporting evidence for the accuracy of the novel

ORIT Model developed in this paper.

In order to test the accuracy of the RPORITA at predicting the ORIT, I used survey data

from two measles outbreaks, including the 2006 Fiji measles outbreak [18] and the 2003 RMI

measles outbreak [17]. I determined the data values required (discussed in Methods ) from the

figures in each paper. I predicted the ORIT for an approximate 20 day period with an average of

5 day intervals. The period of the highest rate of immunization was chosen for the test period.

The graphical depictions of these applications can be seen in Figure 3.

The number of days between the second day of direct data input and the day of the

prediction can be defined as the Prediction Day Difference (PDD). I define the difference

between the day of prediction and the beginning of ORI as Days since Start of Immunization

(DSI). In order to better understand the specific factors affecting the level of prediction error of

the RPORITA, two relationships were examined: 1) between DSI and prediction error, and 2)

between PDD and prediction error. These relationships were examined using data from both

outbreaks, allowing several conclusions to be drawn.

Figure 3 elucidates several key aspects of the RPORITA, where (A) is data from the

2006 Fiji measles outbreak and (B) is from the 2003 RMI measles outbreak. First, all of the

prediction errors are within 0.009 of the ORIT Model based approximation using direct input of

data. It is important to note, however, that only four of the thirty predictions (4/30 = 13.3%)

have a prediction error of greater than 0.006. In addition, all of the mean prediction errors are

within 0.005. Given the multitude of factors that affect the ORIT and the randomness inherent to

all biological action, the low error values provide supporting evidence for the accuracy of the

RPORITA.

By analyzing the data graphically presented in Figure 3, the most accurate method for

applying the RPORITA was determined. An overarching trend could not be established between

either 1) the PDD and prediction error, or 2) DSI and prediction error. However, in both

outbreaks, the prediction error mean is the highest when six days separate the day of data input

55 60 65 70 75 800.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

Day Number

ORIT model approximated ORIT based on direct data input (RED)

RPORITA prediction with PDD = 4 (BLUE)

RPORITA prediction with PDD = 5 (GREEN)

RPORITA prediction with PDD = 6 (PURPLE)

Figure 3. RPORITA predictions compared to threshold approximation based on direct data input.

35 40 45 50 55 600.09

0.1

0.11

0.12

0.13

Day Number

A

B

and the day for prediction (PDD = 6). Overall, based on the data from the RPORITA

application, the two most accurate methods for applying the RPORITA may be established:

• Predict the ORIT five days in advance (with overall mean |prediction error| of 0.00340).

• Predict the ORIT four days in advance by averaging the predictions generated with PDDs

of 4, 5, and 6 (with overall mean |prediction error| of 0.00223).

Discussion

In this paper, I developed an ORIT Model than can approximate the ORIT in real time

using field survey data, the RPORITA, a recursive algorithm for predicting the ORIT several

days in advance, and a model relating changes in the vaccination coverage with changes in the

individual reproduction number.

Equations (19) and (25) represent a novel approach to approximating and predicting the

ORIT, respectively. Most previous attempts have focused on determining the individual

reproduction number in real time through a variety of means. However, the individual

reproduction number and ORIT have a complex relationship [6]. Perhaps the most basic

distinction is that the individual reproduction number is a largely abstracted and theoretical value

intended more to inform policy (whether or not certain outbreak control strategies are effective)

than to quantify the amount of resources needed for immunization programs [9].

In order to determine the ORIT using the individual reproduction number approach, the

immunization rate that results in an individual reproduction number of less than unity must be

solved for [8], involving intermediary steps. In contrast, the value for the ORIT generated using

either the RPORITA or ORIT model may be immediately applied to ORI without any of these

interceding steps. Therefore, direct prediction of the ORIT sidesteps any additional predictions

or mathematical methods required by the reproduction number prediction.

The development of the mathematical models presented herein also took into account the

limitations of the “firefighter problem” approach to vaccination threshold modeling. While this

method does incorporate the effects of ORI on disease dynamics, it offers a simplistic view of

disease spread. The basic reproduction number of the disease is limited to the fixed possibilities

of 2n, where n is the number of dimensions of the grid in question [14]. In contrast, the

mathematical models developed in this paper account for the complex disease spread dynamics

through incorporation of field survey data.

Some limitations of the RPORITA resulted from the ORIT Model (19) upon which it is

based. This model assumes homogenous mixing of the population and the mass action principle

and does not take into account the exposed portion of the population, thereby limiting its

accuracy. In this respect, several aspects of both the ORIT Model and the RPORITA may be

improved upon. The RPORITA should be applied to several more outbreaks, especially those of

other VPDs, before being deemed fully ready for field application. In addition, the incorporation

of more variables into the ORIT Model and RPORITA, such as vaccine efficacy, quarantine,

school closings, and heterogeneity of the population, would improve its accuracy. However, the

simultaneous difficulty in determining these variables in real time for use in the formulations

would limit their practical application.

The immunization strategy discussed in the ORI portions of this paper is not the only

option for outbreak control agencies. In fact, there are three possible courses of action with

regard to ORI. The outbreak could run its course, maximizing the number of infections at the

lowest cost. Or, the outbreak control agency could immunize as many persons as possible, thus

minimizing the number of infections and maximizing cost. However, there is middle road, the

one facilitated by the mathematical models developed in this research: the outbreak control

agency could immunize the minimum number of persons required to achieve the ORIT. This

approach strikes a balance between limiting both infections and cost, resulting in an optimal

strategy for impoverished nations.

Another important factor with regard to immunizing only the minimum number of

persons required to achieve the ORIT involves cultural resistance to immunization that is

prevalent around the world [19]. By only immunizing the minimum number of persons

necessary, an outbreak control agency may find the achievement of the ORIT possible without

expending critical time and manpower persuading those with personal and/or religious objections

to cooperate.

The model for the individual reproduction number in equation (36) can be applied to

outbreak control strategy in order to determine the effects of vaccination on the ability of the

VPD to spread through the population at different points in an outbreak. This equation

demonstrates three critical factors affecting the ability of vaccination to reduce the individual

reproduction number: the point in the outbreak at which these vaccinations would occur

(involves variables c, S, and I), the vaccination coverage before these vaccinations (which

determines V), and the number of additional vaccinations ( VΔ ). With this tool the time at which

vaccination has the optimum impact on the individual reproduction number may be determined.

Also, equation (36) may help answer the question: are these vaccinations worth their financial

cost?

The primary purpose of the RPORITA is to predict, several days in advance, the

minimum number of vaccines required to reduce the individual reproduction number to less than

unity at that point and thereby gradually stop an outbreak. This capability would allow

impoverished nations (which are most affected by VPDs [3]) to plan and coordinate an

immunization drive that implements the minimum amount of resources needed to end a VPD

outbreak at that point. Despite the moderate prediction error, considering the multitude of

variables and factors that affect the ORIT during an outbreak, the RPORITA proves to be

relatively accurate in its prediction of the ORIT. Overall, the novel mathematical methods strike

a delicate balance with their achievement of real- time applicability, simplicity, and accuracy.

[3,689 words]

References

1. Andre FE, Booy R, Bock HL, Clemens J, Datta SK, et al. (2008) Vaccination greatly reduces disease, disability, death and inequity worldwide. B World Health Organ 86(2): 140-146.

2. Robertson S, Markowitz L, Berry D, Dini E, Orenstein W (1992). A Million Dollar Measles Outbreak: Epidemiology, Risk Factors, and a Selective Revaccination Strategy. Public Health Rep 107(1): 24-31.

3. World Health Organization (2007) Measles Fact Sheet. Available: http://www.who.int/mediacentre/factsheets/fs286/en/. Accessed 2008 April 01.

4. Fine PE (1993) Herd Immunity: History, Theory, Practice. Epidemiol Rev 15(2): 265-302.

5. Centers for Disease Control and Prevention (2003) Vaccine Management: Handling and Storage Details for Vaccines. Available: http://www.cdc.gov/vaccines/programs/vfc/downloads/vacc_mgmt_guide_utah.pdf. Accessed 2008 May 12.

6. Anderson R, May R (1991) Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford Science Publications. 772 p.

7. Diekmann O, Heesterbeek J, Metz J (1998) On the definition and computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J Math Biol 28(4): 365-382.

8. Fraser C (2007) Estimating Individual and Household Reproduction Numbers in an Emerging Epidemic. PLoS ONE 2(8): e758. doi:10.1371/journal.pone.0000758

9. Bettencourt LMA, Ribeiro RM (2008) Real Time Bayesian Estimation of the Epidemic Potential of Emerging Infectious Diseases. PLoS ONE 3(5): e2185. doi:10.1371/journal.pone.0002185

10. Wallinga J, Lipsitch M (2007) How generation intervals shape the relationship between growth rates and reproductive numbers. Proc R Soc Ser B-Bio 274(1609): 599-604.

11. Cauchemez S, Boëlle P, Thomas G, Valleron A (2006) Estimating in real time the efficacy of measures to control emerging communicable diseases. Am J Epidemiol 164(6): 591-597.

12. Farrington C, Whitaker H (2003) Estimation of effective reproduction numbers for infectious diseases using serological survey data. Biostatistics 4(4): 621-632.

13. Hartnell B (1995). Firefighter! An Application of Domination, presentation, Twentieth Conference on Numerical Mathematics and Computing, University of Manitoba in Winnipeg, Canada.

14. Develin M, Hartke SG (2007). Fire Containment in Grids of Dimension Three and Higher. Discrete Applied Mathematics 155(17): 2257-2268.

15. Ng KL, Raff P (2008). A generalization of the firefighter problem on z x z. Discrete Applied Mathematics 156(5): 730-745.

16. Kermack WO, McKendrick AG (1927) A Contribution to the Mathematical Theory of Epidemics. Proc Roy Soc Lond A 115:700-721.

17. Hyde T, Dayan G, Langidrik J, Nandy R, Edwards R, et al. (2006) Measles outbreak in the Republic of the Marshall Islands, 2003. Int J Epidemiol 35(2): 299-306.

18. Samuela S, Tuiketei T, Duncan R, Kubo T, Kool J, et al. (2006) Measles Outbreak and Response --- Fiji, February--May 2006. MMWR 55(35): 963-966.

19. Streefland PH (2001) Public doubts about vaccination safety and resistance against vaccination. Health Policy 55(3): 159-172.