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A stochastic SIR epidemic dynamic model with distributed-time-delay, for a two-scale dynamic population is derived. The distributed time delay is the varying naturally acquired immunity period of the removal class of individuals who have recovered from the infection, and have acquired natural immunity to the disease. We investigate the stochastic asymptotic stability of the disease free equilibrium of the epidemic dynamic model, and verify the impact on the eradication of the disease.

The recent advent of high technology in the area of communication, transpor- tation and basic services, multilateral interactions have afforded efficient global mass flow of human beings, animals, goods, equipments and ideas on the earth’s multi-patches surface. As a result of this, the world has become like a neighbor- hood. Furthermore, the national and binational problems have become the multinational problems. This has generated a sense of cooperation and under- standing about the basic needs of human species in the global community. In short, the idea of globalization is spreading in almost all aspects of the human species on the surface of earth. The world today faces the challenge of increas- ingly high rates of globalization of new human infectious diseases and disease strains [

The inclusion of the effects of disease latency or immunity into the epidemic dynamic modeling process leads to more realistic epidemic dynamic models. Furthermore, epidemic dynamic processes in populations exhibiting varying time disease latency or immunity delay periods are represented by differential equation models with distributed time delays. Several studies [

Stochastic models also offer a better representation of the reality. Several stochastic dynamic models describing single and multi-group disease dynamics have been investigated [

In this paper we extend the two-scale network SIR temporary delayed epide- mic dynamic model [

This work is organized as follows. In Section 2, we derive the distributed time acquired immunity delay epidemic dynamic model. In Section 3, we present the model validation results of the epidemic model. In Section 4, we show the stochastic asymptotic stability of the disease free equilibrium.

In this section, we derive the varying immunity delay effect in the SIR disease dynamics of residents of site s i r in region C r of the two-scale population. We recall the general large scale two level stochastic SIR constant temporary delayed epidemic dynamic model studied is given ( [

ρ a u ∫ 0 ∞ I i a r u ( t − s ) f i a r u ( s ) e − δ a u s d s

where e − δ a u s is the probability that an individual who recovered from disease at an earlier time t − s is still alive at time t. Furthermore, f i a r u ( s ) is the integral kernel [

d I i a r u = { [ ∑ k = 1 n r ρ i k r r I i k r r + ∑ q ≠ r M ∑ a = 1 n q ρ i a r q I i a r q − ϱ i r I i i r r − ( γ i r + σ i r + δ i r + d i r ) I i i r r + ∑ u = 1 M ∑ a = 1 n u β i i a r r u S i i r r I a i u r ] d t + [ ∑ u = 1 M ∑ a = 1 n u v i i a r r u S i i r r I a i u r d w i i a r r u ( t ) ] , u = r , a = i [ σ i j r r I i i r r − ϱ j r I i j r r − ( ρ i j r r + δ j r + d j r ) I i j r r + ∑ u = 1 M ∑ a = 1 n u β j i a r r u S i j r r I a j u r ] d t + [ ∑ u = 1 M ∑ a = 1 n u v j i a r r u S i j r r I a j u r d w j i a r r u ( t ) ] , u = r , a = j , j ≠ i , [ γ i l r q I i i r r − ϱ l q I i l r q − ( ρ i l r q + δ l q + d l q ) I i l r q + ∑ u = 1 M ∑ a = 1 n u β l i a q r u S i l r q I a l u q ] d t + [ ∑ u = 1 M ∑ a = 1 n u v l i a q r u S i l r q I a l u q d w l i a q r u ( t ) ] , u = q , a = l , q ≠ r , (2.2)

R i a r u = { [ ∑ k = 1 n r ρ i k r r R i k r r + ∑ q ≠ r M ∑ a = 1 n q ρ i l r q R i l r q + ϱ i r I i i r r − ϱ i r ∫ 0 ∞ I i i r r ( t − s ) f i i r r ( s ) e − δ i r s d s − ( γ i r + σ i r + δ i r ) R i i r r ] d t , u = r , a = i [ σ i j r r R i i r r + ϱ j r I i j r r − ϱ j r ∫ 0 ∞ I i j r r ( t − s ) f i j r r ( s ) e − δ j r s d s − ( ρ i j r r + δ j r ) R i j r r ] d t , u = r , a = j , j ≠ i , [ γ i l r q R i i r r + ϱ l q I i l r q − ϱ l q ∫ 0 ∞ I i l r q ( t − s ) f i l r q ( s ) e − δ l q s d s − ( ρ i l r q + δ l q ) R i l r q ] d t , u = q , a = l , q ≠ r , (2.3)

where all parameters are previously defined. Furthermore, for each r ∈ I ( 1, M ) , and i ∈ I ( 1, n r ) , we have the following initial conditions

( S i a r u ( t ) , I i a r u ( t ) , R i a r u ( t ) ) = ( φ i a 1 r u ( t ) , φ i a 2 r u ( t ) , φ i a 3 r u ( t ) ) , t ∈ [ − ∞ , t 0 ] , φ i a k r u ∈ C ( [ − ∞ , t 0 ] , ℝ + ) , ∀ k = 1,2,3, ∀ r , q ∈ I ( 1, M ) , a ∈ I ( 1, n u ) , i ∈ I ( 1, n r ) , φ i a k r u ( t 0 ) > 0, ∀ k = 1,2,3, (2.4)

where C ( [ − ∞ , t 0 ] , ℝ + ) is the space of continuous functions with the supremum norm

‖ φ ‖ ∞ = S u p − ∞ ≤ t ≤ t 0 | φ ( t ) | . (2.5)

and w is a Wierner process. Furthermore, the random continuous functions φ i a k r u , k = 1 , 2 , 3 are Ϝ 0 -measurable, or independent of w ( t ) for all t ≥ t 0 .

We express the state of system (2.1)-(2.3) in vector form and use it, subse- quently. We denote

x i a r u = ( S i a r u , I i a r u , R i a r u ) T ∈ ℝ 3 x i 0 r u = ( x i 1 r u T , x i 2 r u T , ⋯ , x i , n u r u T ) T ∈ ℝ 3 n u , x 00 r u = ( x 10 r u T , x 20 r u T , ⋯ , x n r 0 r u T ) T ∈ ℝ 3 n r n u , x 00 r 0 = ( x 00 r 1 T , x 00 r 2 T , ⋯ , x 00 r M T ) T ∈ ℝ 3 n r ∑ u = 1 M n u , x 00 00 = ( x 00 10 , x 00 20 , ⋯ , x 00 M 0 ) T ∈ ℝ 3 ( ∑ r = 1 M n r ) ( ∑ u = 1 M n u ) , (2.6)

where r , u ∈ I ( 1, M ) , i ∈ I ( 1, n r ) , a ∈ I i r ( 1, n u ) . We set n = ∑ u = 1 M n u .

Definition 2.1.

1. p-norm in ℝ 3 n 2 : Let z 00 00 ∈ ℝ 3 n 2 be an arbitrary vector defined in (2.6), where z i a r u = ( z i a 1 r u 0 , z i a 2 r u 0 , z i a 3 r u 0 ) T whenever r , u ∈ I ( 1, M ) , i ∈ I ( 1, n r ) , a ∈ I i r ( 1, n u ) . The p-norm on ℝ 3 n 2 is defined as follows

‖ z 00 00 ‖ p = ( ∑ r = 1 M ∑ u = 1 M ∑ i = 1 n r ∑ a = 1 n u ∑ j = 1 3 | z i a j r u 0 | p ) 1 p (2.7)

whenever 1 ≤ p < ∞ , and

z ¯ ≡ ‖ z 00 00 ‖ p = max 1 ≤ r , u ≤ M , 1 ≤ i ≤ n r , 1 ≤ a ≤ n u , 1 ≤ j ≤ 3 | z i a j r u 0 | , (2.8)

whenever p = ∞ . Let

k _ ≡ k 00 min 00 = min 1 ≤ r , u ≤ M , 1 ≤ i ≤ n r , 1 ≤ a ≤ n u | k i a r u | . (2.9)

2. Closed Ball in ℝ 3 n 2 : Let z 00 * 00 ∈ ℝ 3 n 2 be fixed. The closed ball in ℝ 3 n 2 with center at z 00 * 00 and radius r > 0 denoted B ¯ ℝ 3 n 2 ( z 00 * 00 ; r ) is the set

B ¯ ℝ 3 n 2 ( z 00 * 00 ; r ) = { z 00 00 ∈ ℝ 3 n 2 : ‖ z 00 00 − z 00 * 00 ‖ p ≤ r } (2.10)

In addition, from (2.1)-(2.3), define the vector y 00 00 ∈ ℝ n 2 as follows: For i ∈ I ( 1 , n r ) , l ∈ I i r ( 1 , n q ) , r ∈ I ( 1 , M ) and q ∈ I r ( 1, M ) ,

y i a r u = S i a r u + I i a r u + R i a r u ∈ ℝ + = [ 0 , ∞ ) y i 0 r u = ( y i 1 r u , y i 2 r u , ⋯ , y i , n u r u ) T ∈ ℝ + n u , y 00 r u = ( y 10 r u T , y 20 r u T , … , y n r 0 r u T ) T ∈ ℝ + n r n u , y 00 r 0 = ( y 00 r 1 T , y 00 r 2 T , ⋯ , y 00 r M T ) T ∈ ℝ + n r ∑ u = 1 M n u , y 00 00 = ( y 00 10 T , y 00 20 T , ⋯ , y 00 M 0 T ) T ∈ ℝ + ( ∑ r = 1 M n r ) ( ∑ u = 1 M n u ) , (2.11)

and obtain

In the following we state and prove a positive solution process existence theorem for the delayed system (2.1)-(2.3). We utilize the Lyapunov energy function method in our earlier study [

for ( S i a r u , I i a r u ) . We utilize the notations (2.6) and keep in mind that X i a r u = ( S i a r u , I i a r u ) T .

Theorem 3.1. Let r , u ∈ I ( 1, M ) , i ∈ I ( 1, n r ) and a ∈ I ( 1, n u ) . Given any initial conditions (2.4) and (2.5), there exists a unique solution process X i a r u ( t , w ) = ( S i a r u ( t , w ) , I i a r u ( t , w ) ) T satisfying (2.1) and (2.2), for all t ≥ t 0 . More- over, the solution process is positive for all t ≥ t 0 a.s. That is,

S i a r u ( t , w ) > 0 , I i a ( t , w ) r u > 0 , ∀ t ≥ t 0 a.s.

Proof:

It is easy to see that the coefficients of (2.1) and (2.2) satisfy the local Lipschitz condition for the given initial data (2.4). Therefore there exist a unique maximal local solution X i a r u ( t , w ) on t ∈ [ − ∞ , τ e ( w ) ] , where τ e ( w ) is the first hitting time or the explosion time [

{ τ + = sup { t ∈ ( t 0 , τ e ( w ) ) : S i a r u | [ t 0 , t ] > 0 and I i a r u | [ t 0 , t ] > 0 } , τ + ( t ) = min ( t , τ + ) , for t ≥ t 0 . (3.1)

and we show that τ + ( t ) = τ e ( w ) a.s. Suppose on the contrary that P ( τ + ( t ) < τ e ( w ) ) > 0 . Let w ∈ { τ + ( t ) < τ e ( w ) } , and t ∈ [ t 0 , τ + ( t ) ) . Define

{ V ( X 00 00 ) = ∑ r = 1 n M ∑ i = 1 n r ∑ u = 1 M ∑ a = 1 n u V ( X i a r u ) , V ( X i a r u ) = ln ( S i a r u ) + ln ( I i a r u ) , ∀ t ≤ τ + ( t ) . (3.2)

We rewrite (3.2) as follows

V ( X 00 00 ) = ∑ r = 1 M ∑ i = 1 n r [ V ( X i i r r ) + ∑ j ≠ i n r V ( X i j r r ) + ∑ q ≠ r M ∑ l = 1 n q V ( X i l r q ) ] , (3.3)

And (3.3) further implies that

d V ( X 00 00 ) = ∑ r = 1 M ∑ i = 1 n r [ d V ( X i i r r ) + ∑ j ≠ i n r d V ( X i j r r ) + ∑ q ≠ r M ∑ l = 1 n q d V ( X i l r q ) ] , (3.4)

where d V is the Ito-Doob differential operator with respect to the system (2.1)-(2.3). We express the terms on the right-hand-side of (3.4) in the following:

Site Level: From (3.2) the terms on the right-hand-side of (3.4) for the case of u = r , a = i

d V ( X i i r r ) = [ B i r S i i r r + ∑ k ≠ i n r ρ i k r r S i k r r S i i r r + ∑ q ≠ r M ∑ l = 1 n q ρ i a r q S i a r q S i i r r + ρ i r S i i r r ∫ 0 ∞ I i i r r ( t − s ) f i i r r ( s ) e − δ i r s d s − ( γ i r + σ i r + δ i r ) − ∑ u = 1 M ∑ a = 1 n u β i i a r r u I a i u r − 1 2 ∑ u = 1 M ∑ a = 1 n u ( v i i a r r u ) 2 ( I a i u r ) 2 ] d t + [ ∑ k ≠ i n r ρ i k r r I i k r r S i i r r + ∑ q ≠ r M ∑ l = 1 n q ρ i a r q I i a r q S i i r r − ϱ i r − ( γ i r + σ i r + δ i r + d i r ) − ∑ u = 1 M ∑ a = 1 n u β i i a r r u S i i r r I i i r r I a i u r − 1 2 ∑ u = 1 M ∑ a = 1 n u ( v i i a r r u ) 2 ( S i i r r ) 2 ( I i i r r ) 2 ( I a i u r ) 2 ] d t − ∑ u = 1 M ∑ a = 1 n u v i i a r r u I a i u r d w i i a r r u ( t ) + ∑ u = 1 M ∑ a = 1 n u v i i a r r u S i i r r I i i r r I a i u r d w i i a r r u ( t ) (3.5)

Intra-regional Level: From (3.2) the terms on the right-hand-side of (3.4) for the case of u = r , a = j , j ≠ i

d V ( X i j r r ) = [ σ i j r r S i i r r S i j r r + ϱ j r S i j r r ∫ 0 ∞ I i j r r ( t − s ) f i j r r ( s ) e − δ j r s d s − ( ρ i j r r + δ j r ) − ∑ u = 1 M ∑ a = 1 n u β j i a r r u I a j u r − 1 2 ∑ u = 1 M ∑ a = 1 n u ( v j i a r r u ) 2 ( I a j u r ) 2 ] d t + [ σ i j r r I i i r r I i j r r − ϱ j r − ( ρ i j r r + δ j r + d j r ) + ∑ u = 1 M ∑ a = 1 n u β j i a r r u S i j r r I i j r r I a j u r − 1 2 ∑ u = 1 M ∑ a = 1 n u ( v j i a r r u ) 2 ( S i j r r ) 2 ( I i j r r ) 2 ( I a j u r ) 2 ] d t − ∑ u = 1 M ∑ a = 1 n u v j i a r r u I a j u r d w j i a r r u ( t ) + ∑ u = 1 M ∑ a = 1 n u v j i a r r u S i j r r I i j r r I a j u r d w j i a r r u ( t ) (3.6)

Regional Level: From (3.2) the terms on the right-hand-side of (3.4) for the case of u = q , q ≠ r , a = l ,

d V ( X i l r q ) = [ γ i l r q S i i r r S i q r q + ϱ l q S i l r q ∫ 0 ∞ I i l r q ( t − s ) f i l r q ( s ) e − δ l q s d s − ( ρ i l r q + δ l q ) − ∑ u = 1 M ∑ a = 1 n u β l i a q r u I a l u q − 1 2 ∑ u = 1 M ∑ a = 1 n u ( v l i a q r u ) 2 ( I a l u q ) 2 ] d t + [ γ i l r q I i i r r I i l r q − ϱ l q − ( ρ i l r q + δ l q + d l q ) + ∑ u = 1 M ∑ a = 1 n u β l i a q r u S i l r q I i l r q I a l u q − 1 2 ∑ u = 1 M ∑ a = 1 n u ( v l i a q r u ) 2 ( S i l r q ) 2 ( I i l r q ) 2 ( I a l u q ) 2 ] d t − ∑ u = 1 M ∑ a = 1 n u v l i a q r u I a l u q d w l i a q r u ( t ) + ∑ u = 1 M ∑ a = 1 n u v l i a q r u S i l r q I i l r q I a l u q d w l i a q r u ( t ) (3.7)

It follows from (3.5)-(3.7), (3.4), and (3.1) that for t < τ + ( t ) ,

Taking the limit on (3.8) as t → τ + ( t ) , it follows from (3.2) and (3.1) that the left-hand-side V ( X 00 00 ( t ) ) − V ( X 00 00 ( t 0 ) ) ≤ − ∞ (since from (3.2) and (3.1), V ( X i a r u ( τ + ( t ) ) ) = ln S i a r u ( τ + ( t ) ) + ln I i a r u ( τ + ( t ) ) = − ∞ ). This contradicts the finiteness of the right-hand-side of the inequality (3.8). Hence τ + ( t ) = τ e ( w ) a.s. We show subsequently that τ e ( w ) = ∞ .

Let k > 0 be a positive integer such that ‖ φ 00 00 ‖ 1 ≤ k , where the vector of initial values φ 00 00 = ( φ i a r u ) 1 ≤ r , u ≤ M , 1 ≤ i ≤ n r , 1 ≤ a ≤ n u ∈ ℝ 2 n 2 is defined in (2.4). Furthermore, ‖ . ‖ 1 is the p-sum norm (2.7) for the case of p = 1 . We define the stopping time

{ τ k = sup { t ∈ [ t 0 , τ e ) : ‖ X 00 00 ( s ) ‖ 1 ≤ k , s ∈ [ 0 , t ] } τ k ( t ) = min ( t , τ k ) . (3.9)

where from (2.7),

‖ X 00 00 ( s ) ‖ 1 = ∑ r = 1 M ∑ u = 1 M ∑ i = 1 n r ∑ a = 1 n u ( S i a r u ( s ) + I i a r u ( s ) ) . (3.10)

It is easy to see that as k → ∞ , τ k increases. Set lim k → ∞ τ k ( t ) = τ ∞ . Then τ ∞ ≤ τ e a.s. We show in the following that: (1) τ e = τ ∞ a.s. ⇔ P ( τ e ≠ τ ∞ ) = 0 , (2) τ ∞ = ∞ a.s. ⇔ P ( τ ∞ = ∞ ) = 1 .

Suppose on the contrary that P ( τ ∞ < τ e ) > 0 . Let w ∈ { τ ∞ < τ e } and t ≤ τ ∞ . In the same structure form as (3.2) and (3.4), define

{ V 1 ( X 00 00 ) = ∑ r = 1 M ∑ i = 1 n r ∑ u = 1 M ∑ a = 1 n u V ( X i a r u ) , V 1 ( X i a r u ) = e δ a u t ( S i a r u + I i a r u ) , ∀ t ≤ τ k ( t ) . (3.11)

From (3.11), using the expression (3.4), the Ito-Doob differential d V 1 with respect to the system (2.1)-(2.3) is given as follows:

Site Level: From (3.11), the terms of the right-hand-side of (3.4) for the case of u = r , a = i

d V 1 ( X i i r r ) = e δ i r t [ B i r + ∑ k ≠ i n r ρ i k r r S i k r r + ∑ q ≠ r M ∑ l = 1 n q ρ i a r q S i a r q + ϱ i r ∫ 0 ∞ I i i r r ( t − s ) f i i r r ( s ) e − δ i r s d s − ( γ i r + σ i r ) S i i r r ] d t + e δ i r t [ ∑ k ≠ i n r ρ i k r r I i k r r + ∑ q ≠ r M ∑ l = 1 n q ρ i a r q I i a r q − ϱ i r I i i r r − ( γ i r + σ i r + d i r ) I i i r r ] d t (3.12)

Intra-regional Level: From (3.11), the terms of the right-hand-side of (3.4) for the case of u = r , a = j , j ≠ i

d V 1 ( X i j r r ) = e δ i r t [ σ i j r r S i i r r + ϱ j r ∫ 0 ∞ I i j r r ( t − s ) f i j r r ( s ) e − δ j r s d s − ρ i j r r S i j r r ] d t + e δ j r t [ σ i j r r I i i r r + ϱ j r I i j r r − ( ρ i j r r + d j r ) I i j r r ] d t (3.13)

Regional Level: From (3.11), the terms of the right-hand-side of (3.4) for the case of u = q , q ≠ r , a = l

d V 1 ( X i l r q ) = e δ l q t [ γ i l r q S i i r r + ϱ l q ∫ 0 ∞ I i l r q ( t − s ) f i l r q ( s ) e − δ l q s d s − ρ i l r q S i l r q ] d t + e δ l q t [ γ i l r q I i i r r + ϱ l q I i l r q − ( ρ i l r q + d l q ) I i l r q ] d t (3.14)

From (3.12)-(3.14), (3.4), integrating (3.4) over [ t 0 , τ ] leads to the following

From (3.15), we let τ = τ k ( t ) , where τ k ( t ) is defined in (3.9). It is easy to see from (3.15), (3.9), (3.10), and (3.11) that

k = ‖ X 00 00 ( τ k ( t ) ) ‖ 1 ≤ V 1 ( X 00 00 ( τ k ( t ) ) ) (3.16)

Taking the limit on (3.16) as k → ∞ leads to a contradiction because the left-hand-side of the inequality (3.16) is infinite, and the right-hand-side is finite. Hence τ e = τ ∞ a.s. In the following, we show that τ e = τ ∞ = ∞ a.s. We let w ∈ { τ e < ∞ } . Applying some algebraic manipulations and simplifications to (3.15), we have the following

I { τ e < ∞ } V 1 ( X 00 00 ( τ ) ) = I { τ e < ∞ } V 1 ( X 00 00 ( t 0 ) ) + I { τ e < ∞ } ∑ r = 1 M ∑ i = 1 n r B i r δ i r ( e δ i r τ − 1 ) + I { τ e < ∞ } ∑ r = 1 M ∑ i = 1 n r ∑ q = 1 M ∑ l = 1 n q ∫ 0 ∞ f i l r q ( t ) [ ϱ l q ∫ − t t 0 I i l r q ( s ) e δ l q s d s − ϱ l q ∫ τ − t τ I i l r q ( s ) e δ l q s d s ] d t − I { τ e < ∞ } ∑ r = 1 M ∑ i = 1 n r ∫ t 0 τ [ σ i r e δ i r s − ∑ j ≠ i n r σ i j r r e δ j r s ] ( S i i r r + I i i r r ) d s − I { τ e < ∞ } ∑ r = 1 M ∑ i = 1 n r ∫ t 0 τ [ γ i r e δ i r s − ∑ q = 1 M ∑ l = 1 n q γ i l r q e δ l q s ] ( S i i r r + I i i r r ) d s − I { τ e < ∞ } ∑ r = 1 M ∑ i = 1 n r d i r ∫ t 0 τ I i i r r e δ i r s d s − I { τ e < ∞ } ∑ r = 1 M ∑ i = 1 n r ∑ j ≠ i n r d j r ∫ t 0 τ I i j r r e δ j r s d s − I { τ e < ∞ } ∑ r = 1 M ∑ i = 1 n r ∑ q = 1 M ∑ l = 1 n q d l q ∫ t 0 τ I i l r q e δ l q s d s , (3.17)

where I A is the indicator function of the set A.

We recall [

[ σ i r e δ i r s − ∑ j ≠ i n r σ i j r r e δ j r s ] ≥ 0 , ∀ δ i r ≥ δ j r , j ≠ i

and

[ γ i r e δ i r s − ∑ q = 1 M ∑ l = 1 n q γ i l r q e δ l q s ] ≥ 0 , ∀ δ i r ≥ δ l q , q ≠ r , l ∈ I ( 1 , n q )

We now let τ = τ k ( t ) ∧ T in (3.17), ∃ T > 0 , where τ k ( t ) is defined in (3.9). The expected value of (3.17) is estimated as follows

E [ I { τ e < ∞ } V 1 ( X 00 00 ( τ k ( t ) ∧ T ) ) ] ≤ V 1 ( X 00 00 ( t 0 ) ) + ∑ i = 1 n r B i r δ i r e δ i r τ k ( t ) ∧ T + ∑ r = 1 M ∑ i = 1 n r ∑ q = 1 M ∑ l = 1 n q ∫ 0 ∞ f i l r q ( t ) [ ϱ l q ∫ − t t 0 φ i l 2 r q ( s ) e δ l q s d s ] d t (3.18)

Furthermore, from (3.10), (3.11) and the definition of the indicator function I A it follows that

I { τ e < ∞ , τ k ( t ) ≤ T } ‖ X 00 00 ( τ k ( t ) ) ‖ 1 ≤ I { τ e < ∞ } V 1 ( X 00 00 ( τ k ( t ) ∧ T ) ) (3.19)

It follows from (3.18), (3.19) and (3.9) that

P ( { τ e < ∞ , τ k ( t ) ≤ T } ) k = E [ I { τ e < ∞ , τ k ( t ) ≤ T } ‖ X 00 00 ( τ k ( t ) ) ‖ 1 ] ≤ E [ I { τ e < ∞ } V ( X 00 00 ( τ k ( t ) ∧ T ) ) ] ≤ V 1 ( X 00 00 ( t 0 ) ) + ∑ i = 1 n r B i r δ i r e δ i r T + ∑ r = 1 M ∑ i = 1 n r ∑ q = 1 M ∑ l = 1 n q ∫ 0 ∞ f i l r q ( t ) [ ϱ l q ∫ − t t 0 φ i l 2 r q ( s ) e δ l q s d s ] d t (3.20)

It follows immediately from (3.20) that P ( { τ e < ∞ , τ ∞ ≤ T } ) → 0 as k → ∞ . Furthermore, since T < ∞ is arbitrary, we conclude that P ( { τ e < ∞ , τ ∞ < ∞ } ) = 0 . Finally, by the total probability principle,

P ( { τ e < ∞ } ) = P ( { τ e < ∞ , τ ∞ = ∞ } ) + P ( { τ e < ∞ , τ ∞ < ∞ } ) ≤ P ( { τ e ≠ τ ∞ } ) + P ( { τ e < ∞ , τ ∞ < ∞ } ) = 0. (3.21)

Thus from (3.21), τ e = τ ∞ = ∞ a.s. as was required to show.

Remark 3.1. For any r ∈ I ( 1, M ) and i ∈ I ( 1, n r ) , Theorem 3.1 signifies that the number of residents of site s i r of all categories present at home site s i r , or visiting intra and inter-regional sites s j r and s l q respectively, are nonne- gative. This implies that the total number of residents of site s i r present at home site and also visiting sites in regions in their intra and inter-regional accessible domains [

The following result defines an upper bound for the solution process of the system (2.1)-(2.3). We utilize Theorem 3.1 to establish this result.

Theorem 3.2. Suppose the hypotheses of Theorem 3.1 is satisfied. Let μ = min 1 ≤ u ≤ M , 1 ≤ a ≤ n u ( δ a u ) . If

∑ r = 1 M ∑ u = 1 M ∑ i = 1 n r ∑ a = 1 n u y i a r u ( t 0 ) ≤ 1 μ ∑ r = 1 M ∑ i = 1 n r B i r , (3.22)

then

∑ r = 1 M ∑ u = 1 M ∑ i = 1 n r ∑ a = 1 n u y i a r u ( t ) ≤ 1 μ ∑ r = 1 M ∑ i = 1 n r B i r , for t ≥ t 0 a . s . (3.23)

Proof: See ( [

Remark 3.2. From Theorem 3.1 and Theorem 3.2, we conclude that a closed ball B ¯ R 3 n 2 ( 0 → ; r ) in R 3 n 2 under the sum norm ‖ ⋅ ‖ 1 centered at the origin

0 → ∈ R 3 n 2 , with radius r = 1 μ ∑ r = 1 M ∑ i = 1 n r B i r is self-invariant with regard to a two-

scale network dynamics of human epidemic process (2.1)-(2.3) that is under the influence of human mobility process [

is a positive self-invariant set for system (2.1)-(2.3). We shall denote

B ¯ ≡ 1 μ ∑ r = 1 M ∑ i = 1 n r B i r (3.25)

In this section, we study the existence and the asymptotic behavior of the disease free equilibrium state of the system (2.1)-(2.3). The disease free equilibrium is obtained by solving the system of algebraic equations obtained by setting the drift and the diffusion parts of the system of stochastic differential equations to zero. In addition, we utilize the conditions that I = R = 0 in the event when there is no disease in the population. We summarize the results in the following. For any r , u ∈ I ( 1, M ) , i ∈ I ( 1, n r ) and a ∈ I ( 1, n u ) , let

D i r = γ i r + σ i r + δ i r − ∑ a = 1 n r ρ i a r r σ i a r r ρ i a r r + δ a r − ∑ u ≠ r M ∑ a = 1 n u ρ i a r r γ i a r u ρ i a r u + δ a u ≥ δ i r > 0. (4.1)

Furthermore, let ( S i a r u ∗ , I i a r u ∗ , R i a r u ∗ ) be the equilibrium state of the delayed system (2.1)-(2.3). One can see that the disease free equilibrium state is given by E i a r u = ( S i a r u ∗ , 0 , 0 ) , where

S i a r u ∗ = { B i r D i r , for u = r , a = i , B i r D i r σ i j r r ρ i j r r + δ j r , for u = r , a ≠ i , B i r D i r γ i a r u ρ i a r u + δ a u , for u ≠ r . (4.2)

The asymptotic stability property of E i a r u will be established by verifying the conditions of the stochastic version of the Lyapunov second method given in ( [

{ U i a r u = S i a r u − S i a r u * V i a r u = I i a r u W i a r u = R i a r u (4.3)

By employing this transformation, system (2.1)-(2.3) is transformed into the following forms

d V i l r q = { [ ∑ q = 1 M ∑ a = 1 n q ρ i a r q V i a r q − ( ϱ i r + γ i r + σ i r + δ i r + d i r ) W i i r r + ∑ u = 1 M ∑ a = 1 n u β i i a r r u ( S i i r r * + U i i r r ) V a i u r ] d t + [ ∑ u = 1 M ∑ a = 1 n u v i i a r r u ( S i i r r * + U i i r r ) V a i u r d w i i a r r u ( t ) ] , for q = r , l = i [ σ i j r r V i i r r − ( ϱ j r + ρ i j r r + δ j r + d j r ) V i j r r + ∑ u = 1 M ∑ a = 1 n u β j i a r r u ( S i j r r * + U i j r r ) V a j u r ] d t + [ ∑ u = 1 M ∑ a = 1 n u v j i a r r u ( S i j r r * + U i j r r ) V a j u r d w j i a r r u ( t ) ] , for q = r , l = j , j ≠ i , [ γ i l r q V i i r r − ( ϱ l q + ρ i l r q + δ l q + d l q ) V i l r q ∑ u = 1 M ∑ a = 1 n u β l i a q r u ( S i l r q * + U i l r q ) V a l u q ] d t + [ ∑ u = 1 M ∑ a = 1 n u v l i a q r u ( S i l r q * + U i l r q ) V a l u q d w l i a q r u ( t ) ] , for q ≠ r , (4.5)

and

We state and prove the following lemmas that would be useful in the proofs of the stability results.

Lemma 4.1. Let V 1 : ℝ 3 n 2 × ℝ + → ℝ + be a function defined by

{ V 1 ( x ˜ 00 00 ) = ∑ r = 1 M ∑ u = 1 M ∑ i = 1 n r ∑ a = 1 n u V ( x ˜ i a r u ) , V 1 ( x ˜ i a r u ) = ( S i a r u − S i a r u * + I i a r u ) 2 + c i a r u ( I i a r u ) 2 + ( R i a r u ) 2 x ˜ 00 00 = ( U i a r u , V i a r u , W i a r u ) T and c i a r u ≥ 0. (4.7)

Then V 1 ∈ C 2,1 ( ℝ 3 n 2 × ℝ + , ℝ + ) , and it satisfies

b ( ‖ x ˜ 00 00 ‖ ) ≤ V 1 ( x ˜ 00 00 ( t ) ) ≤ a ( ‖ x ˜ 00 00 ‖ ) (4.8)

where

b ( ‖ x ˜ 00 00 ‖ ) = min 1 ≤ r , u ≤ M , 1 ≤ i ≤ n r , 1 ≤ a ≤ n u { c i a r u 2 + c i a r u } ∑ r = 1 M ∑ u = 1 M ∑ i = 1 n r ∑ a = 1 n u [ ( U i a r u ) 2 + ( V i a r u ) 2 + ( W i a r u ) 2 ] a ( ‖ x ˜ 00 00 ‖ ) = max 1 ≤ r , u ≤ M , 1 ≤ i ≤ n r , 1 ≤ a ≤ n u { c i a r u + 2 } ∑ r = 1 M ∑ u = 1 M ∑ i = 1 n r ∑ a = 1 n u [ ( U i a r u ) 2 + ( V i a r u ) 2 + ( W i a r u ) 2 ] . (4.9)

Proof: See ( [

Remark 4.1. Lemma 4.1 shows that the Lyapunov function V defined in (4.7) is positive definite, decrescent and radially unbounded (4.8) function [

We now state the following lemma.

Lemma 4.2. Assume that the hypothesis of Lemma 4.1 is satisfied. Define a Lyapunov functional

V = V 1 + V 2 , (4.10)

where V 1 is defined by (4.7), and

V 2 = 3 ∑ r = 1 M ∑ i = 1 n r ∑ u = 1 M ∑ a = 1 n u [ ( ϱ a u ) 2 μ i a r u ∫ 0 ∞ ( f i a r r ( s ) e − 2 δ a r s ∫ t − s t ( V i a r u ( θ ) ) 2 d θ ) d s ] , (4.11)

Furthermore, let

U i a r u = { [ ∑ u = 1 M ∑ a = 1 n u μ i a r u + ∑ a ≠ i n r ( σ i a r r ) 2 μ i i r r + ∑ a ≠ r M ∑ a = 1 n r ( γ i a r u ) 2 μ i i r r + 2 μ i i r r ] ( γ i r + σ i r + δ i r ) , for u = r , i = a [ ( ρ i a r r ) 2 μ i a r r + μ i i r r + 3 2 μ i a r r ] ( ρ i a r r + δ a r ) , for u = r , a ≠ i [ ( ρ i a r u ) 2 μ i a r u + μ i i r r + 3 2 μ i a r u ] ( ρ i a r u + δ a u ) , for u ≠ r , (4.12)

V i a r u = { 1 2 ∑ u = 1 M ∑ a = 1 n u μ i a r u + 1 2 ∑ v = 1 M ∑ b = 1 n v β i i b r r v ( S i i r r ∗ μ i i r r + μ i i r r ) + 1 2 d i i r r ϱ i r + γ i r + σ i r + δ i r + d i r , for a = i , u = r 1 2 μ i i r r + 1 2 ∑ v = 1 M ∑ b = 1 n v β a i b r r v ( S i a r r ∗ μ i a r r + μ i a r r ) + 1 2 d a i r r ϱ a r + ρ i a r r + δ a r + d a r , for a ≠ i , u = r 1 2 μ i i r r + 1 2 ∑ v = 1 M ∑ b = 1 n v β a i b u r v ( S i i r u ∗ μ i a r u + μ i a r u ) + 1 2 d a i u r ϱ a u + ρ i a r u + δ a u + d a u , for u ≠ r . (4.13)

and

W i a r u = { [ 1 2 ∑ u = 1 M ∑ a = 1 n u μ i a r u + 1 2 ∑ u ≠ r M ∑ a = 1 n r ( γ i a r u ) 2 μ i i r r + 1 2 ∑ a ≠ i n r ( σ i a r r ) 2 μ i i r r + μ i i r r ] ( γ i r + σ i r + δ i r ) , for u = r , a = i , [ 1 2 ( ρ i a r r ) 2 μ i a r r + 1 2 μ i i r r + μ i a r r ] ( ρ i a r r + δ a r ) , for u = r , a ≠ i , [ 1 2 ( ρ i a r u ) 2 μ i a r u + 1 2 μ i i r r + μ i a r u ] ( ρ i a r u + δ a u ) , for u ≠ r (4.14)

for some suitably defined positive numbers μ i a r u and d a i u r , where μ i a r u depends on δ a u , for all r , u ∈ I r ( 1, M ) , i ∈ I ( 1, n ) and a ∈ I i r ( 1, n r ) . Assume that U i a r u ≤ 1 , V i a r u < 1 and W i a r u ≤ 1 . There exist positive numbers ϕ i a r u , ψ i a r u and φ i a r u such that the differential operator LV associated with Ito-Doob type stochastic system (2.1)-(2.3) satisfies the following inequality

L V ( x ˜ 00 00 ) ≤ ∑ r = 1 M ∑ i = 1 n r [ − [ ϕ i i r r ( U i i r r ) 2 + ψ i i r r ( V i i r r ) 2 + φ i i r r ( W i i r r ) 2 ] − ∑ a ≠ i n r [ ϕ i a r r ( U i a r r ) 2 + ψ i a r r ( V i a r r ) 2 + φ i a r r ( W i a r r ) 2 ] − ∑ u ≠ r M ∑ a = 1 n u [ ϕ i a r u ( U i a r r ) 2 + ψ i a r u ( V i a r u ) 2 + φ i a r u ( W i a r u ) 2 ] ] . (4.15)

Moreover,

L V ( x ˜ 00 00 ) ≤ − c V 1 ( x ˜ 00 00 ) (4.16)

where a positive constant c is defined by

c = min 1 ≤ r , u ≤ M , 1 ≤ i ≤ n r , 1 ≤ a ≤ n u { ϕ i a r u , ψ i a r u , φ i a r u } max 1 ≤ r , u ≤ M , 1 ≤ i ≤ n r , 1 ≤ a ≤ n u { C i a r u + 2 } (4.17)

Proof:

The computation of differential operator [

L V 1 ( x ˜ 00 00 ) = ∑ r = 1 M ∑ i = 1 n r [ L V 1 ( x ˜ i i r r ) + ∑ j ≠ i n r L V 1 ( x ˜ i j r r ) + ∑ u ≠ r M ∑ a = 1 n u L V 1 ( x ˜ i a r u ) ] , (4.18)

where,

By using (3.25) and the algebraic inequality

2 a b ≤ a 2 g ( c ) + b 2 g ( c ) (4.22)

where a , b , c ∈ ℝ , and the function g is such that g ( c ) > 0 . The fourteenth term in (4.19)-(4.21) is estimated as follows:

2 ∑ v = 1 M ∑ b = 1 n v c i i r r β i i b r r v ( S i i r r * + U i i r r ) V b i v r V i i r r ≤ ∑ v = 1 M ∑ b = 1 n v c i i r r β i i b r r v ( S i i r r * g i r ( δ i r ) + g i r ( δ i r ) ) ( V i i r r ) 2 + ∑ v = 1 M ∑ b = 1 n v c i i r r β i i b r r v ( S i i r r * g i r ( δ i r ) + B ¯ 2 g i r ( δ i r ) ) ( V b i v r ) 2

2 ∑ a ≠ r n r ∑ v = 1 M ∑ b = 1 n v c i a r r β a i b r r v ( S i a r r * + U i a r r ) V b a v r V i a r r ≤ ∑ a ≠ r n r ∑ v = 1 M ∑ b = 1 n v c i a r r β a i b r r v ( S i a r r * g i r ( δ a r ) + g i r ( δ a r ) ) ( V i a r r ) 2 + ∑ a ≠ r n r ∑ v = 1 M ∑ b = 1 n v c i a r r β a i b r r v ( S i a r r * g i r ( δ a r ) + B ¯ 2 g i r ( δ a r ) ) ( V b i v r ) 2

and

2 ∑ u ≠ r M ∑ a = 1 n u ∑ v = 1 M ∑ b = 1 n v c i a r u β a i b u r v ( S i a r u * + U i a r u ) V b a v u V i a r u ≤ ∑ u ≠ r M ∑ a = 1 n u ∑ v = 1 M ∑ b = 1 n v c i a r u β a i b u r v ( S i a r u * g i r ( δ a u ) + g i r ( δ a u ) ) ( V i a r u ) 2 + ∑ u ≠ r M ∑ a = 1 n u ∑ v = 1 M ∑ b = 1 n v c i a r u β a i b u r v ( S i a r u * g i r ( δ a u ) + B ¯ 2 g i r ( δ a u ) ) ( V b a v u ) 2 (4.23)

Furthermore, by using Cauchy-Swartz and Hölder inequalities and (4.22), the sixth, seventh and eighth terms in (4.19)-(4.21) are estimated as follows:

2 ϱ a u A i a r u ∫ 0 ∞ V i a r u ( t − s ) f i a r u ( s ) e − δ a u s d s ≤ ( ϱ a u ) 2 μ i a r u ∫ 0 ∞ ( V i a r u ( t − s ) ) 2 f i a r u ( s ) e − 2 δ a u s d s + μ i a r u ( A i a r u ) 2 , ∀ r , u ∈ I ( 1 , M ) , i ∈ I ( 1 , n r ) , a ∈ I ( 1 , n u ) , A i a r u ∈ { U i a r u , V i a r u , W i a r u } . (4.24)

From (4.19)-(4.23), (4.18), repeated usage of (3.25) and inequality (4.22) coupled with some algebraic manipulations and simplifications, we have the following inequality

where μ i a r u = g i r ( δ a u ) , g i r is appropriately defined by (4.22). The differential operator LV [

L V ( x ˜ 00 00 ) ≤ L V 1 ( x ˜ 00 00 ) + 3 ∑ r = 1 M ∑ i = 1 n r ∑ u = 1 M ∑ a = 1 n u ( ϱ a u ) 2 μ i a r u ( V i a r u ( t ) ) 2 ∫ 0 ∞ f i a r u ( s ) e − 2 δ a u s d s − 3 ∑ r = 1 M ∑ i = 1 n r ∑ u = 1 M ∑ a = 1 n u ( ϱ a u ) 2 μ i a r u ∫ 0 ∞ ( V i a r u ( t − s ) ) 2 f i a r u ( s ) e − 2 δ a u s d s . (4.26)

We note that ∫ 0 ∞ f i a r u ( s ) e − 2 δ a u s d s ≤ 1 . Furthermore, tt follows from (4.26), (4.25), and some further algebraic manipulations and simplifications that

L V ( x ˜ 00 00 ) ≤ ∑ r = 1 M ∑ i = 1 n r − { [ ϕ i i r r ( U i i r r ) 2 + ψ i i r r ( V i i r r ) 2 + φ i i r r ( W i i r r ) 2 ] + ∑ a ≠ r n r [ ϕ i a r r ( U i a r r ) 2 + ψ i a r r ( V i a r r ) 2 + φ i a r r ( W i a r r ) 2 ] + ∑ u ≠ r M ∑ a = 1 n u [ ϕ i a r u ( U i a r u ) 2 + ψ i a r u ( V i a r u ) 2 + φ i a r u ( W i a r u ) 2 ] } . (4.27)

where, for each r , u ∈ I ( 1, M ) , i ∈ I ( 1, n r ) and a ∈ I ( 1, n u ) , using (4.12), (4.13) and (4.14), we define the constants d a i u r , ϕ i a r u , ψ i a r u and φ i a r u as follows:

d a i u r = ∑ v = 1 M ∑ b = 1 n v c b a v u β a b i u v r ( S b a v u * + B ¯ 2 μ b a v u ) + ∑ v = 1 M ∑ b = 1 n v c b a v u ( v a b i u v r ) 2 ( S b a v u * + B ¯ ) 2 (4.28)

for some positive numbers c i a r u , for all r , u ∈ I r ( 1, M ) , i ∈ I ( 1, n ) and a ∈ I i r ( 1, n r ) .

ϕ i a r u = { 2 ( γ i r + σ i r + δ i r ) ( 1 − U i a r u ) , for u = r , a = i 2 ( ρ i a r r + δ a r ) ( 1 − U i a r u ) , for u = r , a ≠ i 2 ( ρ i a r u + δ a u ) ( 1 − U i a r u ) , for u ≠ r , (4.29)

ψ i a r u = { 2 ( ϱ i r + γ i r + σ i r + δ i r + d i r ) [ c i i r r ( 1 − V i i r r ) + ( 1 − 1 2 E i i r r ) ] , for u = r , a = i 2 ( ϱ a r + ρ i a r r + δ a r + d a r ) [ c i a r r ( 1 − V i a r r ) + ( 1 − 1 2 E i a r r ) ] , for u = r , a ≠ i 2 ( ϱ a u + ρ i a r u + δ a u + d a u ) [ c i a r u ( 1 − V i a r u ) + ( 1 − 1 2 E i a r u ) ] , for u ≠ r (4.30)

and

φ i a r u = { 2 ( γ i r + σ i r + δ i r ) ( 1 − W i a r u ) , for u = r , a = i , 2 ( ρ i a r r + δ a r ) ( 1 − W i a r u ) , for u = r , a ≠ i , 2 ( ρ i a r u + δ a u ) ( 1 − W i a r u ) , for u ≠ r (4.31)

moreover, U i a r u , V i a r u , W i a r u are given in (4.12), (4.13), (4.14) and

E i a r u = { [ 2 ∑ u = 1 M ∑ a = 1 n u μ i a r u + ∑ a ≠ r n r ( 2 + c i a r r ) ( σ i a r r ) 2 μ i i r r + ∑ u = 1 M ∑ a = 1 n u ( 2 + c i a r u ) ( γ i a r u ) 2 μ i i r r + μ i i r r ( ϱ i r + γ i r + σ i r + δ i r + d i r ) + ( ϱ i r + d i r ) 2 μ i i r r + 4 ( γ i r + σ i r + δ i r ) 2 μ i i r r + ( ϱ i r ) 2 μ i i r r + 3 ( ϱ i r ) 2 μ i i r r ( ϱ i r + γ i r + σ i r + δ i r + d i r ) ] , for u = r , a = i , [ ( 2 + c i i r r ) ( ρ i a r r ) 2 μ i a r r + 2 μ i i r r + ( ρ a r + d a r ) 2 μ i a r r + 4 ( ρ i a r r + δ a r ) 2 μ i a r r + μ i a r r + 3 ( ϱ a r ) 2 μ i a r r ( ϱ a r + ρ i a r r + δ a r + d a r ) ] , for u = r , a ≠ i , [ ( 2 + c i i r r ) ( ρ i a r u ) 2 μ i a r u + 2 μ i i r r + ( ρ a u + d a u ) 2 μ i a r u + 4 ( ρ i a r u + δ a u ) 2 μ i a r u + μ i a r u + 3 ( ϱ a u ) 2 μ i a r u ( ϱ a u + ρ i a r u + δ a u + d a u ) ] , for , u ≠ r

Under the assumptions on U i a r u , V i a r u and W i a r u , it is clear that ϕ i a r u , ψ i a r u and φ i a r u are positive for suitable choices of the constants c i a r u > 0 . Thus this proves the inequality (4.15). Now, the validity of (4.16) follows from (4.15) and (4.8), that is,

L V ( x ˜ 00 00 ) ≤ − c V 1 ( x ˜ 00 00 ) ,

where c = min 1 ≤ r , u ≤ M , 1 ≤ i ≤ n r , 1 ≤ a ≤ n u { ϕ i a r u , ψ i a r u , φ i a r u } max 1 ≤ r , u ≤ M , 1 ≤ i ≤ n r , 1 ≤ a ≤ n u { C i a r u + 2 } . This completes the proof. We now formally state the stochastic stability theorems for the disease free equilibria.

Theorem 4.1. Given r , u ∈ I ( 1, M ) , i ∈ I ( 1, n r ) and a ∈ I ( 1, n u ) . Let us assume that the hypotheses of Lemma 4.2 are satisfied. Then the disease free solutions E i a r u , are asymptotically stable in the large. Moreover, the solutions E i a r u are exponentially mean square stable.

Proof:

From the application of comparison result [

Corollary 4.1. Let r ∈ I ( 1, M ) and i ∈ I ( 1, n r ) . Assume that σ i r = γ i r = 0 , for all r ∈ I ( 1, M ) and i ∈ I ( 1, n r ) .

U i a r u = { 1 δ i r 1 [ ∑ u = 1 M ∑ a = 1 n u μ i a r u + 2 μ i i r r ] for u = r , i = a [ ( ρ i a r r ) 2 μ i a r r + μ i i r r + 3 2 μ i a r r ] ( ρ i a r r + δ a r ) , for u = r , a ≠ i [ ( ρ i a r u ) 2 μ i a r u + μ i i r r + 3 2 μ i a r u ] ( ρ i a r u + δ a u ) , for u ≠ r , (4.32)

V i a r u = { 1 2 ∑ u = 1 M ∑ a = 1 n u μ i a r u + 1 2 ∑ v = 1 M ∑ b = 1 n v β i i b r r v ( S i i r r * μ i i r r + μ i i r r ) + 1 2 d i i r r ϱ i r + δ i r + d i r , for a = i , u = r 1 2 μ i i r r + 1 2 ∑ v = 1 M ∑ b = 1 n v β a i b r r v ( S i a r r * μ i a r r + μ i a r r ) + 1 2 d a i r r ϱ a r + ρ i a r r + δ a r + d a r , for a ≠ i , u = r 1 2 μ i i r r + 1 2 ∑ v = 1 M ∑ b = 1 n v β a i b u r v ( S i i r u * μ i a r u + μ i a r u ) + 1 2 d a i u r ϱ a u + ρ i a r u + δ a u + d a u , for u ≠ r . (4.33)

and

W i a r u = { 1 δ i r 1 [ 1 2 ∑ u = 1 M ∑ a = 1 n u μ i a r u + μ i i r r ] , for u = r , a = i , [ 1 2 ( ρ i a r r ) 2 μ i a r r + 1 2 μ i i r r + μ i a r r ] ( ρ i a r r + δ a r ) , for u = r , a ≠ i , [ 1 2 ( ρ i a r u ) 2 μ i a r u + 1 2 μ i i r r + μ i a r u ] ( ρ i a r u + δ a u ) , for u ≠ r (4.34)

The equilibrium state E i i r r is stochastically asymptotically stable provided that U i a r u , W i a r u ≤ 1 and V i a r u < 1 , for all u ∈ I r ( 1, M ) and a ∈ I i r ( 1, n u ) .

Proof: Follows immediately from the hypotheses of Lemma 4.2, (letting σ i r = γ i r = 0 ), the conclusion of Theorem 4.1 and some algebraic manipulations.

Remark 4.2.

1. The presented results about the two-level large scale delayed SIR disease dynamic model depend on the underlying system parameters. In particular, the sufficient conditions are algebraically simple, computationally attractive and explicit in terms of the rate parameters. As a result of this, several scenarios can be discussed and exhibit practical course of action to control the disease. For simplicity, we present an illustration as follows: the conditions of σ i r = γ i r = 0 , ∀ r , i in Corollary 4.1 signify that the arbitrary site s i r is a “sink” in the context of compartmental systems [

structure for many real life scenarios using the presented two level large-scale delay SIR disease dynamic model will appear elsewhere.

2. The stochastic delayed epidemic model (2.1)-(2.3) is a general representa- tion of infection acquired immunity delay in a two-scale network population disease dynamics. The stochastic delayed epidemic model with temporary immunity period ((2.7)-(2.9), [

The presented two-scale network delayed epidemic dynamic model with varying immunity period characterizes the dynamics of an SIR epidemic in a population with various scale levels created by the heterogeneities in the population. Moreover, the disease dynamics is subject to random environmental perturba- tions at the disease transmission stage of the disease. Furthermore, the SIR epidemic confers varying time temporary acquired immunity to recovered individuals immediately after recovery. This work provides a mathematical and probabilistic algorithmic tool to develop different levels nested type disease transmission rates, the variability in the transmission process as well as the distributed time delay in the framework of the network-centric Ito-Doob type dynamic equations. In addition, the concept of distributed delay caused by the acquired immunity period in the dynamics of human epidemics is explored for the first time in the context of complex scale-structured type human meta- populations.

The model validation results are developed and a positively self invariant set for the dynamic model is defined. Moreover, the globalization of the positive solution existence is obtained by applying an energy function method. In addition, using the Lyapunov functional technique, the detailed stochastic asym- ptotic stability results of the disease free equilibria are also exhibited in this paper. Moreover, the system parameter values dependent threshold values controlling the stochastic asymptotic stability of the disease free equilibrium are also defined. Furthermore, a deduction to the stochastic asymptotic stability results for a simple real life scenario is illustrated. We note, further detail study of the stochastic SIR human epidemic dynamic model with varying immunity period for two scale network mobile population exhibiting several real life human mobility patterns will appear elsewhere.

We note that the disease dynamics is subject to random environmental perturbations from other related sub-processes such as the mobility, recovery, birth and death processes. The variability due to the disease transmission incorporated in the epidemic dynamic model will be extended to the variability in the mobility, recovery and birth and death processes. A further detailed study of the oscillation of the epidemic process about the ideal endemic equilibrium of the dynamic epidemic model will also appear else where. In addition, a detailed study of the hereditary features of the infectious agent such as the time-lag to infectiousness of exposed individuals in the population is currently underway and it will also appear elsewhere.

This research was supported by the Mathematical Science Division, US Army Research Office, Grant No. W911NF-12-1-0090.

Wanduku, D. and Ladde, G.S. (2017) The Global Analysis of a Stochastic Two-Scale Network Epidemic Dynamic Model with Varying Immunity Period. Journal of Applied Mathematics and Physics, 5, 1150-1173. https://doi.org/10.4236/jamp.2017.55101