Dynamics of a disabled population in Morocco
 Abdesslam Boutayeb^{1}Email author and
 Abdelaziz Chetouani^{1}
DOI: 10.1186/1475925X22
© Boutayeb and Chetouani; licensee BioMed Central Ltd. 2003
Received: 18 November 2002
Accepted: 5 February 2003
Published: 5 February 2003
Abstract
Background
The disabled population constitutes a class of people needing special care and necessitating important economic and social effort.
Methods
In this paper, using specific parameter settings, partial differential equations are used to model the temporal change of the proportion of the disabled population in Morocco.
Results
Combining different forms and values of the parameters, a numerical method is proposed and three scenarios are considered. These forms and values are determined by data fitting and simulation.
Conclusions
The experiments show clearly the dynamical evolution of the disabled population with time and age according to each scenario.
Introduction
Although the definition of disability may vary from one country to another, the handicapped population represents a special category which appeals for specific needs and more attention. Around the world, it is estimated that more than 10% of the population can be classified as disabled. However, the numbers given by different countries vary. In 1987, 32 million people of the United States were considered disabled, accounting for 13.5% of the total population [1]. A survey in the United Kingdom indicated in 1988 that 6.5 million people representing 14% of the population, could be disabled [2]. In Morocco, 10% of the population suffer from physical disability [3], whereas in China nearly 52 million, constituting 5% of the whole population were considered disabled according to a sampling survey carried out in 1988 [4]. Zhou and Li [5] considered a matrix model of the disabled population with application to Chinese data. Boutayeb and Derouich [6] stressed the effect of a natural " accumulation" with age in chronical diseases.
In this paper we consider the disabled continuous model with age structure. The dynamics are modelled by hyperbolic partial differential equations with initial condition and a nonlocal boundary condition at the zero age. The presence of integral in the boundary condition is well known, accounting for the new born from the population. Numerical methods are proposed and simulation is carried out with different values of the parameters. Throughout this paper the terms disabled and handicapped are used interchangeably to refer to the inability or incapacity to meet certain standard, physical, social, occupational or economical responsibility.
Formulation of the model and numerical method
Notations
In our model, the total population is divided into two groups: the healthy and the disabled. The following notations will be used throughout this paper.
x (a, t): the number of the healthy population aged a at time t,
y(a,t): the number of the disabled population aged a at time t,
u(a,t) = x(a,t) + y(a,t) : the number of the total population aged a at time t,
s(a,t): the survival rate of healthy population aged a at time t,
d(a,t): the death rate of healthy population aged a at time t,
d'(a,t): the death rate of disabled population aged a at time t, (d' = d + δ),
q(a,t): the survival rate of disabled population aged a at time t,
β (a, t) : the fertility rate of the healthy women aged a at time t,
γ (a, t): the fertility rate of the disabled women aged a at time t,
c(a,t): the handicapping rate of the healthy population aged a at time t,
e(a,t) : the rehabilitating rate of the disabled population aged a at time t,
A : final age.
Formulation of the model
The handicapping rate c(0, t) is assumed to be the percentage of the congenital disability rate in year t. Assuming that the number of males is equal to the number of females, from these assumptions and population theory, we get the following model
Where β(a,t) = 0, γ(a,t) = 0, when a <A _{0} or a >A _{1} and [A _{0},A _{1}] is the child bearing age of women and J is the time interval [0,T].
By adding x(a,t) and y(a,t) and using the notations in subsection (2.1) we obtain :
where m(a, t) = d(a,t) + δ(a,t)r(a,t), b(a,t) = (β (a, t)(1  r(a,t)) + γ (a, t)), and ε (a, t) = (c(a, t)s(a, t) + e(a, t)q(a, t) + δ(a, t)), ω(a, t) = c(a,t)s(a,t). This leads to the following equation
Numerical approximation
The partial differential equation (2) is solved over a mesh of T years time and A years of age. The time and the age variables will be discretized, respectively, at the points t _{ n }= nl, n = 0,...,N and a _{ m }= mh, m = 0,...,M.
Hence, if a final age A is considered and the usual age categories of ten years observed every five years on a century scale then A = Mh and T = Nl so that for A = T = 100, l = 5 years and h = 10 years.
Discretizing equation (2) in time and age and using the LaxWendroff method [7] gives:
A standard von Neumann stability analysis [7] shows that this numerical scheme is stable and convergent provided that p ≤ 1.
The literature on theoretical and numerical solutions of structured population models is abundant [3, 8, 9]. In search of clarity and simplicity, we considered the percentage r(a,t) of the disabled population as given by equation (2) and easily implemented by the numerical scheme (3).
Parameters expressions
 1.
Natural death rate of healthy population : d(a,t)
With 50% of the population under the age of 20 and 90% less than 55 years old, the Moroccan population is young. Notwithstanding a decreasing tendency in fertility rate and an improvement of life expectancy, the age structure is not expected to change rapidly. It is assumed that this rate is linearly decreasing with age and slowly decreasing with time. The following explicit form is used
 2.
Death rate of disabled population : d' (a, t)
As stressed by many studies [3], this rate differs from the death rate of the healthy population by δ(a,t) so that
d'(a,t) = d(a,t) + δ(a,t),
where the expression of δ(a, t) is assumed to be given by
δ(a, t) = c _{3} + c _{4} a^{2}  c _{5} t
 3.
Rehabilitating rate : e(a, t)
Some disabilities may be cured, provided they are diagnosed at the early stage. It is assumed that e(a, t) has a form which is slowly increasing with time and decreasing with age, namely:
 4.
Handicapping rate : c(a, t)
As mentioned by the authors [3], several forms for the handicapping rate can be suggested. Throughout this paper three forms will be used for c(a, t):

Linear form for slow increase with age and decrease with time
c(a, t) = c _{7} + c _{8} a  c _{9} t.

Quadratic form for medium increase with age and decrease with time
c(a,t) = c _{10} × a × (c _{11}  a)  c _{12} × t.

Exponential form for rapid increase with age and decrease with time
c(a,t) = (c _{13} × a × (c _{14}  a)  c _{15} t) .
The parameters c _{1},...,c _{5} were determined by fitting the continuous curves d(a,t) and δ(a,t) to discrete data. Whereas the parameters c _{6},...,c _{16} were determined by simulations based on possible scenarios.
Parameters of the model
parameter  c _{1}  c _{2}  c _{3}  c _{4}  c _{5}  c _{6}  c _{8}  c _{8} 

value  0.009  0.001  0.00001  0.00001  0.001  0.00005  0.00004  0.00001 
parameter  c _{9}  c _{10}  c _{11}  c _{12}  c _{13}  c _{14}  c _{15}  c _{16} 
value  0.001  0.00001  0.6  0.0001  0.0001  0.6  0.00001  0.00001 
Results
Several scenarios may be considered by combining different forms and values of the parameters to simulate the dynamics of the handicapped population between age 0 and 100 years in a certain time interval. In the present study, we concentrate on three scenarios. Scenario (1) low congenital disability with linear form of handicapping rate. Scenario (2) medium congenital disability with quadratic form of handicapping rate. Scenario (3) high congenital disability with exponential form of handicapping rate. The numerical method (3) was implemented with different values of the parameters. The output gives the percentages of the disabled population for a range of time and age.
Temporal change of the disabled population with scenario (1)
time  5  10  15  20  25  30  35  40  45  50 

age 10  2.44  2.7  2.9  3.14  3.9  4.29  4.53  4.87  5.68  6.17 
age 20  2.62  3.17  3.54  4.22  5.19  6.77  7.18  8.26  9.45  7.31 
age 30  2.84  4.51  5.01  5.54  6.21  7.33  8.81  8.20  9.46  7.33 
age 40  2.98  5.32  6.71  7.25  8.75  9.13  9.87  10.21  14.43  12.67 
age 50  3.19  6.43  7.12  8.44  9.03  10.15  11.24  12.37  13.76  14.34 
age 60  3.45  7.21  8.18  9.51  10.33  11.7  12.89  15.51  16.96  20.34 
age 70  3.69  8.75  10.5  11.7  12.21  13.56  14.84  16.58  21.36  25.67 
Temporal change of the disabled population with scenario (2)
time  5  10  15  20  25  30  35  40  45  50 

age 10  2.9  3.06  4.17  5.21  6.68  8.41  10.81  11.5  11.87  12.28 
age 20  2.95  3.43  4.48  5.51  6.72  9.21  10.71  12.27  13.53  13.79 
age 30  3.05  5.12  4.91  6.19  7.33  10.08  11.81  13.74  14.31  15.10 
age 40  3.75  4.23  5.11  7.88  8.44  12.65  13.71  14.23  15.77  16.48 
age 50  3.98  5.71  7.33  9.12  10.14  13.42  14.98  17.23  19.13  20.74 
age 60  4.08  8.41  9.8  11.38  12.51  14.72  16.15  18.25  20.38  22.56 
age 70  4.25  11.33  13.15  16.13  19.11  22.42  23.55  25.75  28.38  29.86 
Temporal change of the disabled population with scenario (3)
time  5  10  15  20  25  30  35  40  45  50 

age 10  3.5  3.76  4.62  6.68  8.40  9.52  11.5  12.55  13.51  14.32 
age 20  3.68  6.54  11.76  16.92  21.68  22.56  25.64  28.3  29.5  29.9 
age 30  3.9  7.88  15.95  22.80  26.04  31.8  33.51  35.4  36.6  37.5 
age 40  4.85  8.76  16.25  24.16  30.21  33.3  37.91  38.64  41.12  41.16 
age 50  5.36  11.35  21.33  28.19  32.8  34.66  38.45  39.12  41.51  42.24 
age 60  5.42  12.02  21.81  31.4  33.5  35.71  38.9  39.49  41.66  42.84 
age 70  5.87  12.55  23.11  32.19  36.2  36.93  39.14  40.8  41.55  42.92 
Discussion
 1.
It is well known that some chronic diseases significantly contribute to the number of disabled people. For instance, diabetes is the commonest cause of blindness in people under the age of 65 in the UK, and has been reported to account for 46% of lowerlimb amputations carried by the NHS [13, 14]. Similar figures are reported in Europe and the United States, where the total annual economic cost of diabetes in 1997 was estimated to be US $98 billion of which US $54 billion for indirect costs attributed to disability and mortality. In Morocco and developing countries in general, the lack of medical care makes the situation even worse. Instead of struggling to find a way to look after people with blindness, amputations or kidney failure, why not try to avoid or at least to delay as much as possible the occurrence of these complicated situations?
 2.
Road accidents constitute another source of physical disability. In Morocco, more than 80000 accidents occur every year of which 9% lead to some kind of disability. There is urgency to find ways of alleviating this disaster and their consequences. It is important to identify causative and contributing factors(drivers, road conditions, traffic signals) in order to take efficient preventive measures.
 3.
Last, but not least, in spite of the many major achievements of modern medicine, the number of congenital disabilities is still, high especially in developing countries. Improvement of medical care and sensitization of the population should be combined to reduce handicaps at birth or developed in childhood.
The results presented in this paper indicate that a lowering of both the congenital disability and the handicapping rates will result in a reduction of the proportion of the disabled population at different levels of time and age. For instance, after 50 years time, the percentage of the disabled population of age 70 may be reduced from 42% (scenario (3)) to 25% (scenario (1)). Likewise, after 20 years time, the percentage of the disabled population of age 20 may be reduced from 17% (scenario (3)) to 5.5% (scenario (2)).
 (a)
To manage the needs of a given disabled population. The model predicts the age structure of the disabled population and consequently allows decisions to be made for each age category with regards to education, health, employment, and other needs.
 (b)
To control, as far as possible, the proportion of the disabled population in different age categories by making efficient decisions to help reduce the rate of congenital disability and other predictable controllable forms of disability.
Conclusion
The disabled population constitutes a class of people needing special care and necessitating important economic and social effort. Rather than dealing with the management of the disability per say, our purpose is to suggest efficient methods to prevent disability as much as possible. The mathematical model proposed illustrates clearly the dynamics of the disabled population in each age category and for different periods of time. In order to be able to compare different scenarios, a quantification of the handicapped population is obtained through numerical simulation with variable values of the parameters. Medical strategies may be considered on the basis of this model. For instance, congenital disability may and should be reduced by awareness of the ability to control pregnancy and childbirth, information and sensitization can reduce the excessive number of accidents leading to disability, and finally, diabetes education and other health care measures can reduce the incidence of disabilities throughout the entire age spectrum.
Tables 2, 3 and 4 illustrate clearly the evolution of the disabled population with time and age according to different scenarios. It can be seen that a reduction of the rate of congenital disability and/or the handicapping rate will induce a reduction of the number of disabled people in all age categories. The results of the simulations using our model can help healthcare planners predict the actual needs of each age category with regards to education, employment, housing, etc. Moreover, the proportion of the disabled population can certainly be reduced if health authorities adopt a strategy that helps lower the congenital disability and the handicapping rates.
Declarations
Acknowledgement
This work is partially supported by PARS MI 23.
Authors’ Affiliations
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