Need help with a Coupled Delayed SIR model Differential equation

Question

I am working on a Delayed Coupled SIR model. The model's equations are as follows

  SS1 = NDSolve[
      {x1'[t] == - x1[t] ((0.3/(80*10^6)) y1[t] + (0.5/(80*10^6)) y2[t]),
       x2'[t] == - x2[t] ((0.3/(50*10^6)) y1[t] + (0.5/(50*10^6)) y2[t]),
       y1'[t] == x1[t] ((0.3/(80*10^6)) y1[t] + 0.5/(80*10^6) y2[t]) - 
                 x1[t - 14] ((0.3/(80*10^6)) y1[t - 14] + (0.5/(80*10^6)) y2[t - 14]),
       y2'[t] == x2[t] ((0.5/(50*10^6)) y1[t] + (0.5/(50*10^6)) y2[t]) - 
                 x2[t - 10] ((0.5/(50*10^6)) y1[t - 10] + (0.5/(50*10^6)) y2[t - 10]),
       z1'[t] == x1[t - 14] ((0.3/(80*10^6)) y1[t - 14] + (0.5/(80*10^6)) y2[t - 14]),
       z2'[t] == x2[t - 10] ((0.5/(50*10^6)) y1[t - 10] + (0.5/(50*10^6)) y2[t - 10]),
       x1[0] == 80*10^6, y1[0] == 150, y1[t /; t <= 0] == E^t, z1[0] == 0,
       x2[0] == 50*10^6, y2[t /; t <= 0] == E^t, y2[0] == 100,  
       z2[0] == 0  }, {x1, y1, z1, x2, y2, z2}, {t, 0, 200}] 

As you can see, I have tried to couple two SIR models with different coefficients (and delays). I am having issues with introducing a delay (or a disjoint) in the coupling. I wished to give the mixed coefficient values based on time ie

$\beta_{12}$ = {$a_1$ : t<$t_0$, 0 :($t_0$<t<$t_1$), $a_2$ :($t_1$<t)

The mixed and pure coefficients are constant in the above equation, and I couldn't go about how to incorporate this in my model.

Answer 1

This is not an answer but an example to show how to introduce delay equations of a SIR model.

Manipulate[
 sol = First@NDSolve[
 {
 SS'[t] == \[Lambda] - \[Beta] SS[t] II[t] - (\[Mu] + \[Nu]) SS[t],                                          
 SS[t /; t <= 0] == .990,
 II'[t] == Exp[-\[Mu]*\[Tau]] \[Beta] SS[t - \[Tau]] II[t - \[Tau]] - (\[Mu] + \[Alpha] + \[Gamma]) II[t],   
 II[t /; t <= 0] == .010,
 RR'[t] == \[Gamma] II[t] + \[Nu] SS[t] - \[Mu]*RR[t],                                                    
 RR[t /; t <= 0] == .0
 }, {SS, II, RR}, {t, 0, 365}];

Show[Plot[Evaluate[{#[t], #[t - \[Tau]]} /. sol], {t, 21, time}, 
  PlotLabel -> "SIR model with delay", PlotRange -> All] & /@ {SS, II, RR}],

Delimiter,
{{\[Tau], 7, "incubation period (in days) (: "}, 0, 21, 1},
{{\[Beta], .25, "transmission rate: "}, 0, 1, .05},
{{\[Gamma], .07, "recovery rate in days: "}, 0, 1, .005},
{{\[Alpha], .001, "death rate from virus: "}, 0, 1, .001},
{{\[Nu], .001, "vaccination rate: "}, 0, 1, .001},

Delimiter,
{{\[Lambda], 0, "birth rate: "}, 0, 1, .001},
{{\[Mu], 0, "natural death rate: "}, 0, 1, .001},
{{time, 100, "duration of the pandemic (in days): "}, 1, 365, 7}
]