In this problem, two discontinuous functions are not needed, only one is sufficient $\alpha/\sigma$. The scale multiplier $10^{-7}$ should be included in the definition of time, Then the task is solved using the code
tm = 5;
\[CapitalOmega]1 =
ImplicitRegion[(x - 2.5)^2 + (y - 2.5)^2 >= 0.09 && (0 <= x <=
5) && (0 <= y <= 5), {x, y}];
\[Sigma] [x_, y_] :=
Piecewise[ {{1, .09 <= ((x - 2.5)^2) + ((y - 2.5)^2 ) <= 0.25}, {
5.95396, ((x - 2.5)^2) + ((y - 2.5)^2 ) > 0.25}}]
u [x_, y_] := 0;
\[Alpha] [x_, y_] :=
Piecewise[ {{2.82965, .09 <= ((x - 2.5)^2) + ((y - 2.5)^2 ) <=
0.25}, {5.02649/5.95396, ((x - 2.5)^2) + ((y - 2.5)^2 ) >
0.25}}]
q [x_, y_] :=
Piecewise[{{0, .09 < ((x - 2.5)^2) + ((y - 2.5)^2 ) < 0.25}, {
1, ((x - 2.5)^2) + ((y - 2.5)^2 ) > 0.25}}]
eq = D[T[t, x, y] , t] - \[Alpha][x, y]*
Laplacian[T[t, x, y], {x, y}] - q[x, y]/\[Sigma][x, y] == 0;
Subscript[\[CapitalGamma], D] = {DirichletCondition[T[t, x, y] == 45,
((x - 2.5)^2 + (y - 2.5)^2 ) == 0.09],
DirichletCondition[T[t, x, y] == 12,
x == 0 || x == 5 || y == 0 || y == 5]};
Temp = NDSolveValue[{eq, Subscript[\[CapitalGamma],
D], T[0, x, y] == 12},
T, {t, 0, tm}, {x, y} \[Element] \[CapitalOmega]1,
Method -> {"FiniteElement", "InterpolationOrder" -> {T -> 2},
"MeshOptions" -> {"MaxCellMeasure" -> 0.001}}];
Table[ContourPlot[Temp[t, x, y], {x, y} \[Element] \[CapitalOmega]1,
Contours -> 20, ColorFunction -> "TemperatureMap",
PlotLegends -> Automatic, PlotRange -> All,
PlotLabel -> Row[{"t=", t}]], {t, .1*tm, tm, .3*tm}]
