Abstract
A mathematical study of the size of a population ofdiabetes mellitus patients is carried out in this paper. The study also monitors the number of patients with complications. By appropriate definition of a parameter, the mathematical model may be classified as linear or non-linear. The non-linear case is discussed and the critical values of the population are analysed for stability. Numerical methods are developed for solving the model equations and the results of numerical simulations are reported.
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Abbreviations
- t :
-
time
- l :
-
increment int (the time step)
- C(t) :
-
number of diabetics with complications
- D(t) :
-
number of diabetics without complications
- N(t) :
-
number of diabetics (N = C + D)
- I :
-
incidence ofdiabetes mellitus
- J :
-
Jacobian
- μ :
-
natural mortality rate
- λ :
-
probability of developing a complication
- γ :
-
rate at which complications are cured
- ν :
-
rate at which patients with complications become severely disabled
- δ :
-
mortality rate due to complications
- Θ :
-
μ + δ + γ + ν
- β :
-
parameter used in definition of λ (non-linear model)
- χ1, χ2 :
-
eigenvalues
- C 0,N 0 :
-
initial values ofC andN
- C*, N* :
-
critical-point values ofC andN (continuous system)
- C +,N + :
-
fixed-point values ofC andN (discrete system)
- L :
-
local truncation error
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Dedicated to David M. Bartlett and Wiam Boutayeb
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Boutayeb, A., Chetouani, A., Achouyab, A. et al. A non-linear population model of diabetes mellitus. J. Appl. Math. Comput. 21, 127–139 (2006). https://doi.org/10.1007/BF02896393
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DOI: https://doi.org/10.1007/BF02896393