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}]

[![fig1][1]][1]


  [1]: https://i.sstatic.net/yzcxE.png