Mathematical modeling and optimal control of the impact of rumors on the banking crisis

Proposition 2.1

Let S 0 , I 0 , U 0 , and R 0 be given initial conditions such that S 0 ≥ 0 , I 0 ≥ 0 , U 0 ≥ 0 , and R 0 ≥ 0 .

1. A solution of (11) remains positive for all t ∈ [ 0 , t f ) , where t f ∈ ( 0 , ∞ ) .

2. The set

   Ω = ( S , I , U , R ) ∈ R 4 : 0 ≤ S ≤ N 0 , 0 ≤ I ≤ N 0 , 0 ≤ R ≤ N 0 , 0 ≤ U ≤ Γ e

   is positively invariant for system (11). Here, N 0 = S 0 + I 0 + R 0 .

3. The solution of (11) is unique and is defined for all t ≥ 0 .

Proof

Let S 0 , I 0 , U 0 , and R 0 be given initial conditions such that S 0 ≥ 0 , I 0 ≥ 0 , U 0 ≥ 0 and R 0 ≥ 0 .

1. Let t ≥ 0 . From the first equation of (11), we have

   S ( t ) = S 0 exp − ∫ 0 t β 1 I ( s ) f ( U ( s ) ) d s ≥ 0 .

   From the second equation of (11), we have

   I ( t ) = I 0 exp ∫ 0 t ( θ 1 β 1 S ( s ) f ( U ( s ) ) − g ( U ( s ) ) ) d s ≥ 0 .

   By using the third equation of (11), we obtain

   U ( t ) = U 0 exp − e t + ∫ 0 t δ 1 I ( s ) K + U ( s ) d s + Γ ∫ 0 t exp e ( s − t ) + ∫ t s δ 1 I ( u ) K + U ( u ) d u d s ≥ 0 .

   Using assumptions (2) and (4), and the positivity of S ( t ) and I ( t ) , we deduce the positivity of R ( t ) .

2. Let t ≥ 0 . Adding the right-hand side of the first, second, and fourth equation of (11) leads to

   S ˙ ( t ) + I ˙ ( t ) + R ˙ ( t ) = 0 .

   That is, S ( t ) + I ( t ) + R ( t ) = S 0 + I 0 + R 0 .

   From the positivity of S ( t ) , I ( t ) , and R ( t ) , we deduce that

   0 ≤ S ( t ) ≤ S 0 + I 0 + R 0 , 0 ≤ I ( t ) ≤ S 0 + I 0 + R 0 and 0 ≤ R ( t ) ≤ S 0 + I 0 + R 0 .

   By using the third equation of (11) and the positivity of U ( t ) and I ( t ) , we get

   (13) U ˙ ( t ) ≤ Γ − e U ( t ) .

   Thus, by using the standard comparison theorem [25] leads to

   U ( t ) ≤ Γ e + Γ e − U 0 e − e t .

   Hence, if U 0 ≤ Γ e , then lim sup t → ∞ U ( t ) ≤ Γ e . That is, U ( t ) ≤ Γ e for all t ≥ 0 .

3. Consider the function Ψ defined from R + 4 to R 4 by

   Ψ ( S , I , U , R ) = − β 1 S I f ( U ) , θ 1 β 1 S I f ( U ) − g ( U ) I , Γ − e U ( t ) − δ 1 I U K + U , ( 1 − θ 1 ) β 1 S I f ( U ) + g ( U ) I ,

   where R + 4 = { ( X , Y , W , Z ) ∈ R 4 : X ≥ 0 , Y ≥ 0 , W ≥ 0 , Z ≥ 0 } .

   The function Ψ is continuously differentiable on R + 4 and therefore is locally Lipschitz. Using the Cauchy-Lipschitz theorem [23], the Cauchy problem for system (11) admits a unique local solution. From item (ii) of Proposition 2.1, the solution is contained in a compact subset of R 4 . Therefore, it is globally defined.□

Proposition 2.2

The equilibrium point E ∗ = 0 , 0 , Γ e , N 0 , of system (11), is globally asymptotically stable in Ω .

Proof

Let ( S ( t ) , I ( t ) , U ( t ) , and R ( t ) ) be any positive solution of system (11) in Ω . Recall that N 0 = S 0 + I 0 + R 0 . Define the following Lyapunov function:

Clearly, the function L is nonnegative definite in Ω with respect to E ∗ . Calculating the time derivative of the function L along the solution of system (11), we obtain

Since θ 1 ≤ 1 , then d L ( t ) d t ≤ 0 , for all S , I , R > 0 , and d L ( t ) d t = 0 holds if and only if S = 0 , I = 0 and R = N 0 . Thus, L is a Lyapunov function on Ω . So, by LaSalle’s invariance principle [24], it follows that

It follows from the third equation of (11) that

and so, by the comparison theorem [25],

Now, from (15), we have lim sup t → ∞ I ( t ) = 0 . This implies that for a sufficiently small ε > 0 , there exists a constant P 1 > 0 , such that lim sup t → ∞ I ( t ) ≤ ε , for all t > P 1 .

Thus, from the third equation of system (11), it follows that, for t > P 1 ,

Thus, by using the standard comparison theorem [25], we obtain

From (17), we obtain

so that, by letting ε → 0 , in (18), we obtain

It then follows from (16) and (19) that

This infers that

Thus, we have from (15) and (20) that,

Moreover, Ω is an invariant and attracting set of R + 4 . It follows that the largest compact invariant subset in { ( S , I , U , R ) ∈ Ω : d L d t = 0 } is the singleton { E ∗ } . So, by LaSalle’s invariance principle [24], it follows that every solution of system (11), with initial conditions in R + 4 approaches the equilibrium point { E ∗ } as t → ∞ . This completes the proof.□

Lemma 2.1

1. Let t 0 ≥ 0 . For the given initial conditions H t 0 , D t 0 and C t 0 , the system (12) admits a unique maximal solution defined on an open interval ( t 0 − ξ , t 0 + ξ ) , where ξ is a positive real number. This solution is a continuous function of time.

2. All solutions ( H ( t ) , D ( t ) , and C ( t ) ) of (12) with H t 0 ≥ 0 , D t 0 ≥ 0 , and C t 0 ≥ 0 verify H ( t ) ≥ 0 , D ( t ) ≥ 0 , and C ( t ) ≥ 0 for all t ≥ t 0 .

3. Let N ( t ) = H ( t ) + D ( t ) + C ( t ) , where ( H ( t ) , D ( t ) , and C ( t ) ) is a solution of (12) with nonnegative initial conditions. Then, there are constants m and M such that

   (21) 0 < m ≤ lim inf t → ∞ N ( t ) ≤ lim sup t → ∞ N ( t ) ≤ M < ∞ .

4. Given initial conditions H t 0 , D t 0 , and C t 0 , the solution of system (12) is defined on [ t 0 , ∞ ) .

5. There are constant k and T such that, if D ( t ) < ν for t > T , then

   (22) C ( t ) < k ν

   for t sufficiently large.

Proof

1. Let X = ( H , D , C ) . System (12) can be rewritten as X ˙ ( t ) = h ( t , X ( t ) ) , where h is the function defined on [ 0 , ∞ ) × R 3 by

   (23) h ( t , H , D , C ) = Λ ( t ) − β ( t ) H D − a ( t ) H + δ ( t ) D + θ ( t ) C β ( t ) H D − ( μ ( t ) δ ( t ) + γ ( t ) ) D γ ( t ) D − ( μ ( t ) + θ ( t ) ) C .

   The function h is continuously differential on R 3 with respect to the variable ( H , D , C ) and therefore is locally Lipschitz. By using the Cauchy-Liptschitz theorem [23], we obtain the existence and uniqueness of a local solution for problem (12) for given initial conditions.

2. Let t 0 ≥ 0 . We assume that H t 0 ≥ 0 , D t 0 ≥ 0 and C t 0 ≥ 0 . From the second equation of (12), we have

   D ( t ) = D t 0 exp ∫ t 0 t ( β ( s ) H ( s ) − ( δ ( t ) + μ ( t ) + γ ( s ) ) ) d s ,

   that is, D ( t ) assumes only nonnegative values.

   From the third equation of (12), we have

   d d t C ( t ) exp ∫ t 0 t ( μ ( s ) + θ ( s ) ) d s = exp ∫ t 0 t ( μ ( s ) + θ ( s ) ) d s D ( t ) > 0 ,

   which implies that t ↦ C ( t ) exp ∫ t 0 t ( μ ( s ) + θ ( s ) ) d s is an increasing function of t , and therefore, C ( t ) ≥ 0 .

   From the first equation of (12), we have

   H ˙ ( t ) ∣ H = 0 = Λ ( t ) + δ ( t ) D ( t ) + θ ( t ) C ( t ) ≥ 0 ,

   that is, H ( t ) cannot assumes negative values.

3. Adding the three equations of system (12), we have

   H ˙ ( t ) + D ˙ ( t ) + C ˙ ( t ) = Λ ( t ) − a ( t ) H ( t ) − μ ( t ) D ( t ) − μ ( t ) C ( t ) .

   From a ( t ) ≤ μ ( t ) for all t ≥ 0 , we have

   (24) Λ ( t ) − μ ( t ) N ( t ) ≤ N ˙ ( t ) ≤ Λ ( t ) − a ( t ) N ( t ) .

   Let t ↦ N ∗ ( t ) be a solution to the ODE

   (25) N ∗ ˙ ( t ) = Λ ( t ) − a ( t ) N ∗ ( t )

   with initial condition N ∗ ( t 0 ) > 0 , t 0 ≥ 0 . We have

   N ∗ ( t ) = N ∗ ( t 0 ) e − ∫ t 0 t a ( s ) d s + ∫ t 0 t e − ∫ u t a ( s ) d s Λ ( u ) d u .

   Since t ↦ Λ ( t ) is bounded, by ( H 2 ) , we obtain for t 0 sufficiently large and t ≥ t 0 ,

   N ∗ ( t ) ≤ N ∗ ( t 0 ) e − a 1 ( t − t 0 ) + a 2 + ∫ t 0 t e − a 1 ( t − u ) + a 2 Λ ( u ) d u < N ∗ ( t 0 ) e − a 1 ( t − t 0 ) + a 2 + e a 2 a 1 sup t ≥ t 0 Λ ( t ) ( 1 − e − a 1 ( t − t 0 ) ) ,

   and therefore, lim sup t → ∞ N ∗ ( t ) ≤ e a 2 a 1 sup t ≥ t 0 Λ ( t ) ( 1 − e − a 1 ( t − t 0 ) )

   Let t ↦ N ∗ ( t ) be a solution to the ODE

   (26) N ∗ ˙ ( t ) = Λ ( t ) − μ ( t ) N ∗ ( t )

   with initial condition N ∗ ( t 0 ) > 0 , t 0 ≥ 0 . We have

   (27) N ∗ ( t ) = N ∗ ( t 0 ) e − ∫ t 0 t μ ( s ) d s + ∫ t 0 t e − ∫ u t μ ( s ) d s Λ ( u ) d u .

   Since t ↦ μ ( t ) is bounded, by (27) and ( H 2 ) , we obtain for t sufficiently large

   N ∗ ( t ) = N ∗ ( t − ω 2 ) e − ∫ t − ω 2 t μ ( s ) d s + ∫ t − ω 2 t e − ∫ u t μ ( s ) d s Λ ( u ) d u > e − ω 2 sup t ≥ 0 μ ( s ) ∫ t − ω 2 t Λ ( u ) d u ≥ Λ − ω 2 e − ω 2 sup t ≥ 0 μ ( s ) ,

   where Λ − > 0 is a real number such that ∫ t t + ω 2 Λ ( s ) d s > Λ − for t sufficiently large. Therefore, lim inf t → ∞ N ∗ ( t ) ≥ Λ − ω 2 e − ω 2 sup t ≥ 0 μ ( s ) . By using the comparison property together with the solutions of (25) and (26), we deduce that

   ∀ t ≥ t 0 , N ∗ ( t ) ≤ N ( t ) ≤ N ∗ ( t )

   and

   m ≤ lim inf t → ∞ N ∗ ( t ) ≤ lim inf t → ∞ N ( t ) ≤ lim sup t → ∞ N ( t ) ≤ lim sup t → ∞ N ∗ ( t ) ≤ M ,

   where m = Λ − ω 2 e − ω 2 sup t ≥ 0 μ ( s ) and M = e a 2 a 1 sup t ≥ t 0 Λ ( t ) .

4. Let H t 0 , D t 0 , and C t 0 be initial conditions for system (12). From (iii), the solution ( H ( t ) , D ( t ) , C ( t ) ) of (12) is contained in a compact subset of R 3 and, therefore, is globally defined.

5. Assume that D ( t ) < k for all t > T and some k > 0 . From the third equation of (12), we have for t > T ,

   (28) C ˙ ( t ) < γ ( t ) k − ( μ ( t ) + θ ( t ) ) C ( t ) .

   From the assumptions on the functions γ ( t ) , μ ( t ) and θ ( t ) , there exist real numbers γ i , γ s , μ i , μ s , θ i , θ f ∈ ( 0 , 1 ) such that

   (29) γ i ≤ γ ( t ) ≤ γ s , μ i ≤ μ ( t ) ≤ μ s , θ i ≤ θ ( t ) ≤ θ s .

   By using (29), (28) becomes C ˙ ( t ) < γ s k − ( μ i + θ i ) C ( t ) . That is,

   lim sup t → ∞ C ( t ) < γ s k μ i + θ i and C ( t ) < γ s k μ i + θ i

   for t sufficiently large. The proof of Lemma 2.1 is completed.□