Table 1 Stochastic dynamics of the model

Infection of a

S _s →S _s−1

$\frac{{\mathrm{\bigwedge }}_s{S}_s}{N}\mathrm{\Delta }t+o\left(\mathrm{\Delta }t\right)$

susceptible in S _s

Infection of a

S _p →S _p −1

$\frac{{\mathrm{\bigwedge }}_s{S}_p}{N}\mathrm{\Delta }t+o\left(\mathrm{\Delta }t\right)$

susceptible in S _p

Increase in exposed

E _s →E _s + 1

$\frac{{\mathrm{\bigwedge }}_s{S}_s}{N}\mathrm{\Delta }t+o\left(\mathrm{\Delta }t\right)$

class E _s

Increase in exposed

E _p →E _p + 1

$\frac{{\mathrm{\bigwedge }}_s{S}_p}{N}\mathrm{\Delta }t+o\left(\mathrm{\Delta }t\right)$

class E _p

Decrease in exposed

E _s →E _s −1

γ _s E _s Δt + o(Δt)

class E _s

Decrease in exposed

E _p →E _p −1

γ _s E _p Δt + o(Δt)

class E _p

Increase in asymptomatic

A _s →A _s + 1

(1−p _s )γ _s E _s Δt + o(Δt)

infection A _s

Increase in asymptomatic

A _p →A _p + 1

(1−p _p )γ _p E _p Δt + o(Δt)

infection A _p

Recovery from asymptomatic

A _s →A _s −1

α _s A _s Δt + o(Δt)

infection A _s

Recovery from asymptomatic

A _p →A _p −1

α _p A _p Δt + o(Δt)

infection A _p

Increase in symptomatic

I _s →I _s + 1

p _s γ _s E _s Δt + o(Δt)

infection I _s

Increase in symptomatic

I _p →I _p + 1

p _p γ _p E _p Δt + o(Δt)

infection I _p

Recovery from symptomatic

I _s →I _s−1

α _s I _s Δt + o(Δt)

infection I _s

Recovery from symptomatic

infection I _p