Pancreatic β-cells and α-cells are responsible of controlling the level of glucose in the blood. While insulin reduces the level of blood glucose by stimulating its uptake by muscles and adipose tissues and storing it as glycogen in the liver, glucagon supplies another source of glucose (produced through liver gluconeogenesis and glucogenolysis) for blood.

Consequently, damage in either β-cells or α-cells can lead to a dysfunction in insulin secretion or glucagon production.

Studies related to mathematical modelling of the relationship between glucose, insulin and β-cells have been the subject of several publications. However, the interaction between glucose, insulin, β-cells, α-cells and glucagon has been ignored.

In this study, we propose a mathematical model dealing with the dynamics of glucose, insulin, β-cells, α-cells and glucagon.

Based on the mathematical models proposed by Boutayeb et al. and Ruby et al. our extended model introduces the dynamics of α-cells on the interaction of glucagon and glucose.

Our study describes the roles of β-cells on secreting insulin and α-cells on producing glucagon, besides contribution of those two hormones on the control of glycaemia.

The dynamics of glucose, insulin, β-cells, α-cells and glucagon are studied using ordinary differential equations.

The proposed model has four equilibrium points: P0 (G=600, I=0, β=0, α=0, J=0), P1 (G=600, I=0, β=0, α=300, J=0), P2 (G=80, I=21, β=900, α=300, J=0.16), P3 (G=73, I=19, β=900, α=0, J=0).

G (mg), I (mU/mL), β (mg), α (mg) and J (mg) represent respectively concentrations of glucose, insulin, β-cell mass, α-cell mass and glucagon.

The first point P0 is an unstable point describing a severe hyperglycaemia with zero levels of β-cell mass and α-cell mass leading to zero levels of insulin and glucagon and consequently raising the level of glucose.

The second one is an unstable point corresponding to a severe hyperglycaemia. Despite the normal value of α-cell mass (300mg), the glucagon cannot be produced in a case of hyperglycaemia.

The third point is a stable point corresponding to a normal state with normal values of glucose, insulin, β-cell mass, α-cell mass and glucagon.

The last equilibrium point P3 is an unstable point describing a hypoglycaemic state justified by zero levels of α-cell mass and glucagon and normal values of insulin and β-cell mass.

A normal pancreatic β-cell mass and α-cell mass maintains the level of glycaemia in the normal range.

The apoptosis of β-cells and α-cells is the origin of insulin and glucagon secretory defects causing severe hyperglycaemic state with high glucose rate in the blood.

The death of β-cells leads to hyperglycaemia even without a deterioration of α-cell mass.

The apoptosis of α-cells causes impairment in the glucagon response to hypoglycaemia, which reduces the level of glucose in the blood.