A MATHEMATICAL MODEL FOR ATTENUATING THE SPREAD OF DIABETES AND ITS MANAGEMENT IN A POPULATION

We study the dynamics of diabetes in a population based on the etiology of the disease. In carrying out the study, we proposed that; a population generate non-diabetic non – susceptible sub-population, and a non-diabetic susceptible sub-population, the non-diabetic susceptible sub-population can further generate a population of diabetics without complication, who can later transit to a population with diabetic complications. Based on the etiology dynamics, we proposed control measures at the point of transition from the population to non-diabetic susceptible population, and at the point of transition from diabetes without complications to diabetes with complications. For this study, we intend to look at the control measure. In this regard, we proposed a mathematical model for the dynamics of diabetes by incorporating a control parameter h, so as to investigate how to control diabetes in a population. The result of the study suggested that; we need to control the incidence of diabetes, I(t), and improve the control measure, h, for transition from diabetes without complication to diabetes with complication. Thus entailing going further in research to; Look into the dynamics of the genetics of transmission of the diabetic gene, to investigate how to reduce the spread (and hence the incidence I(t)) of diabetes, and to also look into the influence of the control factor h, on the dynamics of glucose metabolism, this will give an insight on how to manage diabetic patients. of developing complications from D(t). following schematic diagram of the natural history dynamics of diabetic sub-population.


INTRODUCTION
Diabetes mellitus is a recognized consequence of hereditary haemochromatosis, David et al (2003). Genomic wide scans for linkage have reported a number of chromosomal regions that may harbor genes involved in type II diabetes, with the most promising, replicating findings on chromosomes 1q21-q24, 2q37, 12q24 and chromosome 20; Florence et al; (2003). Type I diabetes develops in individuals who are genetically susceptible; Janne et al (2004). In genetic epidemiology, population-based disease registers are commonly used to collect incidence and/or genotype data or other risk factor information concerning affected subjects and their relatives or a whole population, Janne (2008).
The incidence and prevalence of diabetes are increasing all over the world; complications of diabetes constitute a burden for the individuals and the whole society. It is now commonly admitted that diabetes is sweeping the globe as a silent epidemic largely contributing to the growing burden of non-communicable diseases and mainly encouraged by decreasing levels of activity and increasing prevalence of obesity, Bouteyab et al (2004). This trend of incidence & prevalence in a population, despite medical intervention, is a case for serious concern.
Accordingly experts suggested that the dynamics of incidence & prevalence of diabetes in a population depends on; 1) The dynamics of the natural history of the disease in a population (Bouteyab et al 2004).
2) The dynamics of diabetes gene frequency in a population (Masatoshi Nei, 2006). To understand and model the above dynamics, we need to know the natural history dynamics of diabetes in a population.
Using the above idea of the dynamics of the natural history of a diabetic population, we want to model the dynamics of diabetes in a population by incorporating control parameters, so as to investigate how to manage diabetic patients, and to regulate the spread (and thus the incidence) of diabetes in a population. For this purpose, we decomposed a population into; susceptible sub-population and Non-susceptible sub-population, and introduced control measures at two stages of the dynamics of diabetes as follows:

1.
Control measure at the stage of diabetes without complication, to inhibit transition from diabetes without g as a result of; inhibitory control measure, recovery from complications, and incidence of diabetes without complications respectively.

State II:
The number of diabetics with complications C(t) depletes by as a result of natural death, death from complications, severe disability and recovery from complications, and increases by ) (t D l as a result of developing complications from D(t). This gives the following schematic diagram of the natural history dynamics of the diabetic sub-population.

Non-susceptible subpopulation:
A non susceptible person will be non-diabetic with the following dynamics In this work we are going to develop equations that describe the dynamics of diabetes in a population that incorporates the inhibitory control parameter, h, to investigate how it can play a role in retarding transition from diabetes without complications to diabetes with complications.

MODELING Methodology
Here we state assumptions, notations, parameters, and model development Assumptions v Underlying population is large and finite v Individuals are assumed to have no complications at the point of first diagnosis at any time interval from the start of the screening v Probability ( ) of a diabetic person developing complications is assumed to be constant.

Notations
The following notations are used;

Preliminary Result
Now looking at system (2) above, this describes (1) The dynamics of the diabetic population, from diabetes without complication to diabetes with complications & vice-versa in a diabetic population.
First, let us study system (2), we shall solve the system analytically so as to gain an insight into the dynamics of the evolution of the diabetic population from diabetes without complications, to diabetes with complications & vice-versa for the following reason; Analytical solutions give room for sensitivity analysis which will give more insight into the dynamics of the diabetic population.
Assuming a steady state f or I(t) i. e. I(t) = I(independent of time, t), and differentiating the first equation of system (2) with respect to t, we have; Using system (2) in equation (3), we have: From the first equation of system (2), we have The auxiliary equation for the homogeneous part of (6) is For the particular solution, we have, using method of undetermined coefficients.
Using (7) in (6), we have:   (9) constitute the solution of the system (2). Using the initial conditions we have; Therefore the solutions C(t) & D(t) to system (2) are:

Consider the following system of O.D.E
We obtain the critical values as follows; Assuming a steady state f or I(t) (i.e. I(t)=I(independent of time t)), then at critical points the above system reduces to; (15) & (16), we have: This implies that, the solutions C(t) & D(t) to the O.D.Es will revolve around the critical point values.

LIMITING CASE BEHEAVIOUR
Taking the limit of the solutions (13)    Where

DISCUSSION ON LIMITING CASE BEHEAVIOUR
The following discussion is based on the results of section 2.5 1.
From (i) of section 2.5; rate of developing complications becomes very high (which may result from lack of optimal glucose control for patients without complications), when this occurs, the number of diabetics without complications D(t) depletes to zero (0)  . This implies that, with varnishing rate of rec overy ( g ) from complications, C(t) becomes too large ( ¥ ), this is because of injection by l .
On the other hand, D(t) stabilizes at m l h h This translates to the need for regulating I and h to bring down D(t) and C(t) respectively. 8) From (viii) of section 2.5; Rate of mortality due to complications vanishes, when this occurs, the number diabetics without complications D(t) and the number of diabetics with complications C(t) approaches: This implies that, with very high recovery rate (which can be as a result of rigorous recovery programme for patients with complications), dereasning rate of developing complications and dereasinging mortality rate due to complications, C(t) drops to zero, while D(t) stabilizes at . This translates to the need f or regulating I and h to bring down D(t) and C(t) respectively.
From the above discussion, the results suggest that; to Contr ol diabetes, we need to reduce the incidence of diabetes I and improve the rate of retardati on of transition to diabetes with complication. This will, respectively reduce cases of diabetes incidence and manage sufferers to extinction.

CONCLUSSION
From the above discussion, we conclude as follows; The system of equations describing the dynamics of diabetes in the susceptible sub-population suggest that; we need to c ontrol the incidence I and improve the rate of inhibition, h for transition from diabetes without complication to diabetes with complication. This entails going further in research to; 1. Look into the dynamics of the genetics of transmission of the diabetic gene, to investigate how to reduce the spread(and hence the incidence I) of diabetes 2. Look into the effect of physical exercise and/or dieting on the dynamics of glucose metabolism, this will give an insight on how to manage diabetic patients.