Development and Analysis of a Mathematical Model for the Population Dynamics of Diabetes Mellitus During Pregnancy

Abstract

Diabetes Mellitus (DM) during pregnancy is a major global public health burden with various adverse outcomes. In this paper, a mathematical model of the population dynamics of DM during pregnancy is developed and analyzed. Four independent variables have been considered, namely the numbers of nonpregnancy nondiabetic women, diabetic nonpregnant women, diabetic pregnant women and diabetic pregnant women with complication. The model is described by a system of ordinary differential equations. The stability of the equilibrium point is analyzed using Routh-Hurwitz criteria. The model is numerically simulated using MATLAB to verify the analytical results. The model has only one nonnegative equilibrium point which is asymptotically stable. The equilibrium solution is further investigated using simple sensitivity analysis. The results of simple sensitivity analysis of the equilibrium solution suggest the key parameters of the model. The equilibrium point of the model indicates the influential parameters that can be controlled to address the issue.

Appendices

FORMULATING ODES BASED ON A FLOW DIAGRAM

The state variables are depicted by the boxes in the flow diagram while the arrows illustrate the movement of people between different states in the population. The flows shown as arrows are calculated using the terms on the right-hand side of the equations: a flow pointing out of a box is taken away from a state variable and is a negative term while a flow pointing into a box is added and is a positive term [6]. Some constant parameters are introduced adjacent to the arrows to represent the proportionality rates of the flows. An illustration of an example of a flow diagram is shown in Fig. A-1.

Formulating differential equations (the governing equations of the model) based on the flow diagram in Fig. 5 (the parameters is shown in Table 5):

$$\frac{{dX}}{{dt}} = + \beta \,\, - \,\,\mu X + \alpha X + \rho Y\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{dY}}{{dt}} = - \mu Y\,\, - ~\,\,\alpha X~\,\, - \,\,\rho Y$$

Fig. 5.

An example of a flow diagram.

Table 5.   Description of parameters in a model

PROOF OF THEOREM 1

Given that the initial conditions $V\left( 0 \right) = {{V}_{0}},~N\left( 0 \right) = {{N}_{0}},~K\left( 0 \right) = {{K}_{0}}$ and $G\left( 0 \right) = {{G}_{0}}$ are nonnegative. It is clear from Eq. (1) that [31]

$$\frac{{dV}}{{dt}} + \left[ {\mu + \xi + \alpha + \sigma ~} \right]V\left( t \right) \geqslant 0,$$

so that,

$$\frac{d}{{dt}}\left[ {V\left( t \right)\exp \left( {\mu + \xi + \alpha + \sigma } \right)t} \right] \geqslant 0.~$$

(A1)

Integrating (A1) with respect to t, gives

$$V\left( t \right) \geqslant V\left( 0 \right)\exp \left[ { - \left( {\mu + \xi + \alpha + \sigma } \right)t} \right] > 0,\,\,\,\,\forall t \geqslant 0.$$

Similarly, it can be shown that N(t) > 0, K(t) > 0 and G(t) > 0 for all time t > 0. This completes the proof.

It is crucial to note that (1)–(4) will be analysed in a feasible region D given by

$$D = \left\{ {\left( {V,N,K,G} \right) \in R_{ + }^{4}~:V + N + K + G = P} \right\},$$

which can be easily verified to be positively invariant with respect to (1)–(4). In what follows, the model is epidemiologically and mathematically well posed in D (see [13]).