A Model of Dieting
Ronald E. Mickens; Denise N. Brewley; Matasha L. Russell
SIAM Review, Vol. 40, No. 3. (Sep., 1998), pp. 667-672.
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SIAM R E V . @ 1998 Society f o ~lridustrinl a ~ l dAppiieii hlntili,~natics 

Vol. 40, No. 3 . pp 667-672. September 1998 008 

CLASSROOM NOTES
EDITEDBY JACK hIACKI
Thzs sectzon contazns brzef notes of up lo five typed pagci Arthelei dlustrate noLtl ~ d e u iur~d
znszghts relatecl to ezther the teachzng of inathernatzcs for applzcatzons or thr teachzng oj the upp17-
catrons
A MODEL OF DIETING"
RONALD E. ~ I I C K E N S ~ ,DENISE N. BREWLEYJ, AND MATASHA L. RUSSELL~
Abstract. SVe construct a mathematical model of the dieting process and investigate the properties
of its solutions. SVhile the general solution cannot be explicitly obtained, we show that all the
important features can be determined by use of geometrical techlliques and linear stability analysis.
The results of this study are consistent with all accepted facts relitting to the dieting procrss.
Key words. dieting, mathematical modeling, differential equations
AMS subject classifications. 34A26, 34D99, 92B05
PII. SO036144596318740
1. Introduction. Dieting as a process and social phenoinenon is a n~ultibillion
dollar industry in the United States. This industry deals wit11 the following products
and services: (i) diet foods and nutritional suppleincnts: (ii) sports and dieting
clothing and equipment: (iii) exercise tapes. videos, and television programs: and
(iv) articles, books, and other publications on how to achieve weight reduction. However
long-term weight loss and its inaintenance is very difficult for most individuals
[I,41. A variety of complex personal and pl~ysiologicalfactors are the basis for this
situation [I,4, 51.
The time line for most individuals who lose weight, i.e., who through dieting
attempt to achieve their desired weight level, goes as follows:
(a) A decision is made to reduce body weight by a certain anlourit.
(b) A diet and physical exercise plan is selected.
(c) Initially, there is rapid weight loss.
(d) Eventually, the desired weight is attained.
(e) The weight loss is maintained for a (generally) short period of time.
(f) 	The individual returns to esseiltially the sanie .'eating pattern" as before the
weight was lost.
(g) Rather quicltly, the weight increases wit11 often a final body weight that is
larger than the initial (prediet) weight.
Our main purpose is to construct a rather straiglitforward rilathematical niodel
of the dieting process and investigate the properties of its solutions. The niodel
"Received by the editors December 27. 1996; accepted for publication March 7, 1997. This research
was supported by a grant froni NIH-NIGMS-h/IBRS. This work was performed by an employee of the
U.S. Government or under U.S. Governme~itcontract. Tlie LI.8.Govenirllrx~ltretains a nonexclusive.
royalty-free license to publish or reproduce the published form of this contribution, or allow others
to do so, for U.S. Government purposes. Copyright is owned by SIAM to the extent not limited by
these rights.
htt~p://www.siam.org/journals/sirev/40-3/31874.html 

?Department of Physics, Clark Atlanta University, Atlanta, GA 30314.
$Department of Mathematics, Clark Atlanta University, Atlanta, GA 30314.
§~epartnieritof Chemistry, Spelnlan College, Atlanta, GA 50314.
668 CLASSROOM NOTES
is given in terms of a nonlinear, first-order, ordinary differential equation. While
this equation cannot be solved exactly, in terms of elementary functions, we show
that all the important features of its solutions can be determined by use of phasespace
(geometrical) techniques and linear stability analysis [3]. A second and equally
important goal is to show that mathematical modeling techniques can be used to
provide important insight and guidance for a biomedical problem of great interest
to the general public. A third feature of this paper is to show that the value of a
nlathematical model may not come from giving exact numerical and/or analytical
results, but by providing qualitative knowledge of the possible solutions of relevance
to the system.
2. The model. Let W(t) denote the body weight of an individual at time t. Our
main assumption is that the rate of change of W(t), with respect to time, is given by
the relation
where F(t) is a food intake function, E(t) is a function that provides a measure of the
physical activity done by the individual, and M ( W ) is a body metabolism function
which depends on W. Note that F(t) is a nonnegative function which measures the
caloric intake of the dieter. E(t) is also a nonnegative function and is a measure of the
nonmetabolic energy needs of the body. The dominant component to this function is
physical exercise.
The function M ( W ) is nonnegative. All available data are consistent with it
having the functional form [2]
For our purposes, the following function is selected:
where p is a positive parameter. A more complex expression for M ( W ) that is consistent
with (2) is
where A and B are positive parameters. Use of (4) does not change any of the critical
conclusions of the paper; however, the details of the analysis become complex.
The substitution of (3) into (1)gives the differential equation
This equation provides a mathematical model of the dieting process. The central
question is what are the functions F(t) and E(t)?
3. Dieting strategy. In general, a diet consists of selecting a strategy or plan
for the determination of the amount of food (calories) to be eaten during a given
interval of time and the determination of the number of calories to be expended in
directed physical activity during this same interval of time. The simplest strategy
is to let both the food intake and exercise regiment be constant over a convenient
669CLASSROOM NOTES
and meaningful time interval. For example, the caloric intake can be selected to be
F ca!ories/day from food with E calories/day "burned" in exercise. Both F and E
are assumed to be constant. Note that the reference time interval is one day. This
particular diet strategy leads to the result
where the notation means that the left side of (6) is "averaged" over one day. In
general, the parameter X is nonnegative, i.e.,it is difficult to imagine any physiological
state of a normal human for which X would be negative.
Substituting the diet strategy given by (6) into (5) leads to our model equation
This is a nonlinear, first-order differential equation that depends on two parameters,
X and p.
4. Analysis of the model. The long-time behavior of the solutions to (7) is
determined by the fixed-points or constant solutions. This equation has a single
fixed-point at the value
An application of linear stability analysis [3]will allow the determination of the behavior
of the solutions in a small neighborhood of the fixed-point. To proceed, let
where t(0) is a small perturbation from the fixed-point. Then substituting (9) into
(7) and retaining only linear terms in t(t) gives
Since R > 0, it follows that
Lim t(t) = 0
t--tm
and the fixed-point W(t) = w is linearly stable. Consequently, for fixed X and P,
small perturbations in the body weight will die out.
The global behavior of the solutions for positive initial data, i.e., W(0) > 0, can
be found by examining (7). Now,
Hence, it can be concluded that all solutions monotonically approach the fixed-point
with increase in time 131, i.e.,
Lim W(t) = W,
t-03
This means that the fixed-point is globally stable; see Figure 1.
CLASSROOM NOTES
FIG. 1. Typical solutions of (7)for positive initial conditions.
5. Discussion. Under normal conditions, as stated in section 3, the parameter
X > 0. The situation where X <0 is clearly pathological in the sense that all solutions
to (7) tend to zero, i.e.,
Lim W(t) = 0, X 0.
t-m
This case corresponds to starvation: F = 0 and E > 0. (For a real person W(t)
cannot be negative. Thus, for X 5 0, if for some t = tl, W(tl) = 0, then W(t) = 0
for t > tl.) For a given individual P is constant and once a particular diet strategy
is selected (by specifying the value of A) the equilibrium weight W is determined.
Note, from (8), that W is proportional to X4I3and consequently increases faster than
a linear behavior on A.
Consider the situation where weight loss is desired. Let the initial weight be
Wi and assume that a smaller weight Wf is the goal of the dieter. To achieve this
reduction the parameter X must change from Xi to Xf, where
Figure 2 presents a graphic depiction of this case. The individual maintains weight
Wi until the time t = to. At that time the diet begins and the individual's weight
decreases monotonically to the new, smaller value Wf.
If the individual wishes to gain weight, then the reverse of the above procedure
is done, i.e., Xi must increase to Xf; see Figure 3.
In summary, we have constructed a mathematical model of the dieting process.
The model is formulated in terms of a nonlinear, first-order ordinary differential equation.
The model depends on two parameters, /3 and A. For a given individual P is
constant. The dieting strategy is determined by selecting a (positive) value for A.
Each value of X corresponds to a unique final body weight as given in (8). Since
W increases faster with X than a linear function, significant changes in the dieter's
"life-style" can lead to even larger changes in the value of the final body weight.
Note that for a given value of X there are many combinations of F and E which give
X = F - E = constant. Thus, a given diet strategy can be achieved by various combinations
of caloric (food) restriction and physical exercise. A detailed examination
CLASSROOM NOTES
FIG.2. Loss of weight diagram. The initial weight is W % and wfis the desired weight, Wi > wf.
FIG.3. Gain of weight diagram. The initial weight is ~i and wfis the desired weight, W,< wf.
of the linear stability analysis results indicates that the greatest decrease of weight
per unit time occurs at the start of the diet. Likewise, the weight loss per unit time
is smallest when the weight goal is nearly accomplished. An important conclusion is
that to stay at the desired weight Wf,the original diet strategy that brought about
the weight loss must be maintained "forever"! Finally, it should be indicated that the
results of this model are all consistent with current data on dieting and related issues
[I,41.
REFERENCES
[I] R. L. LEIBEL,M. ROSENBAUM, J. HIRSCH,Changes in energy e~penditureresulting fromAND
altered body weight, The New England Journal of Medicine, 332 (1995), pp. 621-628.
672 	 CLASSROOM NOTES
[2] T. A. MCMAHONE O n Size and Life, Scientific American Library, New York,AND J. T. BONNER, 

1983, pp. 64-66. 

[3] R. E. MICKENS,Oscillations i n Planar Dynamic Systems, World Scientific Publishers, River
Edge, NY, Appendix I.
[4] B. 	F. MILLER,The Complete Medical Guide, 4th ed., Simon and Schuster, New York, 1978,
Chapter 3.
[5] RESEARCHNEWSSECTION,Obese protein slims mice, Science, 269 (1995), pp. 475-476.