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PLP 6404

Epidemiology of Plant Diseases

Spring 2015

Ariena van Bruggen, modified from Katherine Stevenson and R.D. Berger

Lecture 11: Disease progress in time: simple models

Approaches to modeling disease progress

little use of much use of

mathematical ------------------------------- mathematical

models models

Three reasons for epidemic analysis:

investigation of factors affecting epidemic development

"comparative epidemiology", comparing different curves

evaluation of control strategies

Disease progress curve - graph of disease intensity over time (or degree-days)

Incorporates the influence of:

1. environmental conditions

2. pathogen characteristics

3. availability of susceptible host tissue

4. management practices

We consider three basic types of disease progress curves: linear, monomolecular and logisitc

[Note: the above curves were drawn in Word and are only approximately similar to the logistic

and monomolecular curves]

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Phases of a disease progress curve:

The equations of Vanderplank

Landmark publication: Plant Diseases: Epidemics and Control (1963)

To understand the progress of disease, van der Plank used the analogy of the increase of money.

Hence, the two terms commonly used in the past: simple interest diseases and compound

interest diseases.

For money placed in a bank at simple interest, the principle (capital) does NOT change. Interest

is accumulated each year; importantly, the interest does NOT earn interest.

The equation to describe this simple interest increase is:

Kt = K0 + K0 * r * t (as we have seen before)

where:

K0 is the original deposit,

r is the rate of interest,

t is time (for money, usually in years), and

Kt is the amount of money after time t.

Example:

when K0 = $100; r = 0.10 (10%); and t = year:

For 1 yr : $110 = $100 + 100 * 0.10 * 1

For 2 yr : $120 = $100 + 100 * 0.10 * 2

For 10 yr : $200 = $100 + 100 * 0.10 * 10

Progressive phase Degressive phase

dY/dt>0 dY/dt<0

dY/dt=0

t

Y

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In an analogy, certain diseases increase only from the amount of inoculum at the beginning of

the crop season, with only ONE CYCLE of production of infective inoculum. These diseases are

properly called monocyclic diseases (“simple interest” diseases).

In principle, we have a linear increase in this case (assuming healthy tissue is not limiting).

Absolute rate of disease increase (dy/dt = first derivative of y, slope of curve, rate of change) is

constant in this case

Summary Linear relation

integrated form: y = y0 + r t

differential form: dy/dt = r

constant rate of disease increase

Year

t

Y

Y0

0

0

t

dY/dt

r

0

0

1

0.3

Year

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In the case of compound interest, the interest on the capital is accumulated daily (or monthly)

and the interest earns interest in succeeding time periods.

The equation to describe the increase of money at compound interest is:

Kt = K0 * exp (r * t).

The “exp” in the equation is for “exponentiation”; the inverse is “natural logarithms” (to the

base “e”; i.e., 2.71828); expressed as loge; or commonly as “ln”.

Example of increase of money at compound interest for 10 yr:

For K0 = $100, r = 0.10, and t = 10:

Kt = $100 * exp (0.10 * 10) = $271.83

(untransformed) Kt = K0 * exp (r * t)

$271.83 = $100 * exp (0.1*10)

(log-transformed) ln (Kt) = ln (K0) + r * t;

therefore: ln (271.83) = ln (100) + 0.1 * 10

and: 5.60517 = 4.60517 + 1.0

Sometimes we know the beginning amount of deposit and the balance in the account at a later

date, but we may not know the exact value of “r”.

Solve the equation for r:

r = [ln (Kt) - ln (K0)] / t

Substitute the values from the previous example:

r = [ln (271.83) - ln (100)] / 10

r = [5.60517 - 4.60517] / 10

r = 0.10

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In the analogy, compound interest diseases have MANY CYCLES of infection and inoculum

production during the crop season. These diseases are properly called polycyclic diseases.

When we describe disease progress instead of increase in capital, we use y0 instead of K0 and yt

instead of Kt.

Disease progress of polycyclic diseases is described by the exponential model equation:

yt = y0 exp (r t)

The exponential rate equation (the derivative of the exponential model equation):

dyt/dt = r yt

Summary Exponential (called logarithmic by Vanderplank)

integrated form: y = y0 e rt (e is base of natural log)

linearized form: ln(y) = ln(y0) + r t

differential form: dy/dt = r y

unlimited increase in disease with time

absolute rate of disease increase is proportional to the amount of disease

dY/dt

t

0

0

t

Y

0

0

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The analogy between money and disease is faulty in several ways (Kranz, 1974):

Infections occur intermittently, not continuously (because of the influence of

environment, host changes, etc.

There is a limit to the amount of host tissue available for infection (no limit for money)

Newly infected host tissue is NOT immediately infective

Diseases often occur in foci.

Limiting host tissues available for infection

In monocyclic plant pathosystems, the disease increases usually without successive

generations of inoculum and infection within a crop season. The inoculum does not increase

very much nor does it move about very much except by the assistance of man. The increase in

the number of wilting tomato plants is NOT from the fungus moving from plant to plant, but

from the different times of infection and different levels of inoculum for the individual plants.

The number of healthy plants available for new infections declines over time. This can be

described by the monomolecular equation:

Y t = 1 - exp (-r t) = 1 – e –rt

where e= 2.718281828 (base of natural log)

In this equation, the rate r is called the monomolecular rate.

To obtain a straight line when plotted against time, you need to transform Yt:

Yt = ln [1/ (1-y)]

The rate (derivative) equation for the monomolecular curve is:

dyt/dt = r (1- yt)

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In natural epidemics of “simple interest diseases”, the epidemic rates are usually rather slow;

i.e., rm < 0.05, and commonly about rm 0.01.

Practice to calculate the values of y1, y2, rm, and t using a calculator.

E.g., to find the level of disease at time2, y2:

ln [1/(1-y2)] = ln [1/(1 - y1)] + rm t

when time2=20 days and time1=10 days, i.e. t=20-10=10;

rm=0.05 and y1=0.39, then ln [1/(1 - y1)] = 0.494 and ln [1/(1-y2)] =0.494 + 0.5 = 0.994

1/(1-y2)=exp(0.994)= 2.702821 and 1-y2= 0.369984 and y2=0.63

or to calculate rm : (do this yourself)

rm = {ln [1/(1-y2)] - ln [1/(1-y1)]} / t

or to calculate t : (do this yourself)

t = {ln [1/(1-y2)] - ln [1/(1-y1)]} / r

Summary Monomolecular (negative exponential)

integrated form: y = 1- (1-y0) e –rt

[note: if y0=0 then equation changes into the familiar form]

differential form: dy/dt = r(1-y)

linearized form: ln[1/(1-y)] = ln[1/(1-y0)] + r t

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The absolute rate of disease increase is proportional to the amount of healthy tissue

Also when the initial disease progress is exponential, the healthy area may become limited

until the carrying capacity is reached and disease intensity does not increase any further (similar

to the monomolecular situation).

If we add the correction factor (1-y) for the limit of available host tissue to the exponential rate

equation then we have the famous logistic rate equation:

dyt/dt = r yt (1 - yt)

for example, if yt = 0.5 and r = 0.1, how much new disease (dy/dt) appeared today?

Answer: input values into:

dyt/dt = r yt (1-yt)

dyt/dt = 0.1 * 0.5 * (1 - 0.5)

dyt/dt = 0.025

and if yt = 0.1 and r = 0.4, how much new disease would appear today?

Answer: dyt/dt = r yt (1-yt)

dyt/dt = 0.4 * 0.1 * (1 - 0.1)

dyt/dt = 0.036

If we accumulate all the dyt/dt’s over time, we have an S-shaped curve, symmetrical around the

“point of inflection” at y = 0.5. This S-shaped curve is the logistic curve described by:

dY/dt

t

0

0

t

Y

0

0

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The logistic model equation:

yt = 1/ [1 + B exp (-r t)];

where B = (1-y0)/ y0

Van der Plank suggested that the logistic transformation be used to analyze and compare these

curves.

The logistic transformation equation:

Yt = ln [yt/(1-yt)].

The values of Yt are called “logits”. These logit values can be obtained by:

i, substitution of the proper values into the equation above; or

ii, look up the values in a “Table of logits”.

When the values of yt are small (yt < 0.01), the correction factor (1-yt) is very close to 1.0, and

there is very little correction in the transformation.

When the logit transformation is applied to the y-values along a symmetrical- sigmoidal curve,

the logits linearize the curve as:

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The line through the logit-transformed values is fit by linear regression and is called “the logit

line”; the slope of which is “r” (the epidemic rate).

The logistic transformation equation is used to:

1) calculate epidemic rates (r): r = [logit (y2) - logit (y1)] /(t2 - t1)

2) predict future disease (y2): logit (y2) = logit (y1) + r (t2 - t1)

3) estimate initial disease (y1): logit (y1) = logit (y2)- r (t2 - t1)

4) determine the time interval (t) between disease levels : (t2 - t1) = [logit (y2) - logit (y1)] / r

Van der Plank called the value of r the “average apparent infection rate”.

Why “average”?

Answer: because it was the average rate through all logit values.

Why “apparent infection”?

Answer: because the rate was calculated from disease, not infections.

The value of “r” has been used in many epidemiological studies to compare:

1) different epochs of a pathosystem

2) different pathogens on the same host

3) different pathosystems (hosts & pathogens)

4) different treatments

5) different cultivars

6) different races of a pathogen

7) different environments

8) different forecasting systems

9) different cropping systems

10) different levels of parasitic fitness

11) different amounts of sanitation during the epidemic

12) different levels of y0.

An important point about the epidemic rate is: “in what units is the rate expressed (kph, mph,

eg.)?” In his 1963 book, van der Plank said the rate was in “per unit per day”, a relative rate.

Many researchers have published epidemic rates without defining in what units the rates were;

this is not correct!

Some people describe the epidemic curve as three phases:

1) The early exponential phase (van der Plank, 1963 called this the logarithmic phase).

This phase begins at onset and continues to about y = 0.03. Some people also call this the “lag”

phase (after the growth of bacteria), which is incorrect because the epidemic is increasing many

fold during this period.

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2) The logistic phase begins at the end of the exponential phase and continues to ca. y = 0.5.

3) The terminal phase begins at the end of the logistic phase and continues until finality. The

terminal phase is probably the least important to be distinguished for most pathosystems. If a

grower has a crop with y = 0.65 or 0.85, the difference is unimportant because both intensities

mean essentially total crop loss.

The greatest proportional increase in disease occurs in the exponential phase, often before the

disease is detected. From y = 0.0001 to y = 0.05 is a 200-fold increase. During this phase, control

measures are the most effective. Commonly, growers begin their control program in the logistic

phase which results in rather poor management of the epidemic.

Actual crop damage occurs in the mid-logistic to terminal phase. During this period, the disease

“merely” doubles in intensity!!

Van der Plank said that there is a limit to how fast an epidemic can go, and the latent period “p”

sets the limit. When p is short, r is usually fast. When p is long, r is usually slow. The product

of p* r, van der Plank called the “explosiveness” of the epidemic. The limits of p * r are in the

range of 0.0 to 6.0; values above 4.0 have rarely been found.

Examples: p r p* r Result .

Late blight: 4 0.4 1.6 somewhat explosive

Bean rust 10 0.4 4.0 very explosive

Leaf spots 10 0.1 1.0 Not explosive

Wheat rust 10 0.6 6.0 Extremely explosive .

Perhaps van der Plank placed too much emphasis on p * r, because late blight of potatoes would

have 2 1/2 infection cycles for each cycle of bean or cereal rust. Nevertheless, an explosive

epidemic (high p * r) is very difficult to control.

High p * r for a pathosystem with long p is unusually difficult to control because of the many

latent infections.

An important thing to remember about the logistic model equation is that it generates a

symmetrical-sigmoidal curve and the point of inflection is at y = 0.5. Another important point is

that there are symmetrical-sigmoidal curves that are NOT logistic even though they may start at

the same y0, reach the inflection point y = 0.5 at the same time, and finish at the same ymax.

J. C. Zadoks called these other curve shapes as “over-topped” and “under-topped” curves.

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Summary Logistic

integrated form:

linearized form: ln[y/(1-y)] = ln[(y0/(1-y0)] + rt

differential form: dy/dt = ry (1-y)

"r" is known as the ‘slope’ or ‘apparent infection rate’ or ‘the intrinsic rate of growth of

populations’

[note: the above graphs were drawn in Word and are only approximately similar to the logistic

curve]

t

dY/dt

Y=0.5

t

Y

Y=0.5

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Often, the logit-transformed proportions are NOT straight. In other words, the logistic curve does

not fit the data. What kind of distribution do the daily values of dy/dt’s have in that case? Most

commonly for polycyclic epidemics, the distribution of dy/dt’s is skewed to the left. This means

that the curve for the accumulating disease may still be sigmoidal, but asymmetrically

sigmoidal. The disease epidemic goes faster in the early stages (Berger, 1975).

There are several functions that can be fit to such distributions. S. Analytis (remember him from

the equation for the optimum curve?) suggested the Gompertz, von Bertalanffy, or

Mitscherlich functions. For now, we will only discuss the Gompertz function.

The Gompertz equation for increasing growth is:

yt = exp [-B * exp(-r * t)].

The B-value in the Gompertz equation is a position parameter; i.e., it positions the origin of the

transformed line at time t = 0 as: B = -ln(y0). The point of inflection is y = 1/e; where e =

2.718; that is y = 0.367879 (0.37).

The rate equation is:

dy/dt = rG * y * [-ln(y)].

The transformation equation to linearize the Gompertz function is Y = -ln [-ln(y)].

The integrated curve is sigmoidal, but it is asymmetrically sigmoidal about its point of

inflection at y = 0.37.

The area under the curve for a plot of the derivative is noticeably skewed to the left.

The curve for the Gompertz equation is much like the logistic, so you need to compare the

derivatives to see the difference.

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Epidemic rates for the Gompertz function are calculated with gompits, just like with logits.

With logits:

rl = [logit (y2) – logit (y1)] / (t2 – t1).

With gompits:

Rg = [gompit (y2) – gompit (y1)] / (t2 – t1).

Berger (1981) examined over 100 disease progress curves for polycyclic pathosystems with both

the logistic and Gompertz models. In all cases, the Gompertz model provided the better, closer fit

and with a random pattern of residuals.

Summary Gompertz (sigmoid but not symmetric)

integrated form:

linearized form: -ln[-ln(y)] = -ln[-ln(y0)] +rt

differential form: dy/dt = ry [-ln(y)]

Similar in shape but often better than logistic for describing epidemics

t

Y

t

dY/dt

Y=0.37 =1/e