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Measles is a reemerging disease that has a devastating impact, especially among children under 5. In this paper, an SEIRS model is developed to investigate a possible outbreak among the population of children under 5 in the Sunyani Municipality. We consider waning immunity or loss of immunity among those who were vaccinated, which leads to secondary attacks among some in the population. Using Routh-Hurwitz criterion, Matrix Theoretic and Goh-Volterra Lyapunov functions, the stability of the model was investigated around the equilibria. We have computed the threshold parameter,
*R*
_{0}, using the Next Generation Matrix method. The disease-free equilibrium is globally stable whenever
*R*
_{0 }≤_{}1 and unstable otherwise. The endemic equilibrium is globally stable when
*R*
* _{0 }*>1.

Measles, a recurrent virus infection has a short term outbreak but its impact is devastating especially among children under five. Severe measles results in pneumonia, which is the disease with high mortality rate in Ghana. Found in [

Mathematical models are strong instruments used extensively to study the spread and control of infectious diseases. One important measure that determines the dynamics of disease models is the threshold parameter known as basic reproductive number R 0 . This parameter measures the number of infectives generated by a single infectious individual introduced into the susceptible population. In measuring the foregoing, researchers in recent literature use the NGM approach. When R 0 ≤ 1 (usually when the disease-free equilibrium exists) the introduction of infectious individual can not generate a large outbreak. On the other hand, when R 0 > 1 , (where the endemic equilibrium exists), the disease will persist in the population. [

In this paper, the matrix theoretic method, a type of Lyapunov function, is used to establish the stability in the disease free system. Later in this study, the Goh-Volterra method is used to establish the stability of the endemic equilibrium. [

We consider a susceptible-exposed-infectious-recovered-susceptible (SEIRS) model with relapse since there is a vaccination parameter that provides a near-permanent immunity against the disease with non-negative initial conditions. E represents the number of latent(exposed) individuals, σ > 0 represents the rate at which the exposed individuals become infectious (i.e.) 1 / σ represents the average latent period. λ > 0 represents the rate at which the susceptible individual become immune and gain near-permanent immunity and γ > 0 also represents the rate at which the infectious individuals go through the successful treatment process of the disease. Near-permanent immunity because there exist a rate ϵ > 0 of the recovered individuals falling back into the susceptible compartment. It is considered that this ϵ R individual have a waning immunity over time so they fall back susceptible to the infectious disease. Already, [

S ′ = α N + ϵ R − ( μ + β I / N + λ ) S E ′ = β S I / N − ( μ E + σ + δ E ) E I ′ = σ E − ( τ + γ ) I R ′ = γ I − ( μ + ϵ ) R + λ S + δ E E (1)

The total population size N t can be determined by N = S ( t ) + E ( t ) + I ( t ) + R (t)

or from the differential equation N ′ = ( α − μ ) N − k E E − k I I , which is derived by adding the equations in (1). Let s = S / N , e = E / N , i = I / N and r = R / N denote the fractions of the classes S, E, I and R in the population, respectively. Then for s ′ = ( N ⋅ S ′ − S ⋅ N ′ ) / N 2 , e ′ = ( N ⋅ E ′ − E ⋅ N ′ ) / N 2 , i ′ = ( N ⋅ I ′ − I ⋅ N ′ ) / N 2 and r ′ = ( N ⋅ R ′ − R ⋅ N ′ ) / N 2 respectively, the following assignments are appropriate;

s ′ = N ⋅ ( α N + ϵ R − ( μ + β I / N + λ ) S ) − S ⋅ ( ( α − μ ) N − k E E − k I I ) N 2 = α + ϵ r − ( α + λ ) s − ( β − k I ) s i + k E s e e ′ = N ⋅ ( β S I / N − ( μ E + σ + δ E ) E ) − E ⋅ ( ( α − μ ) N − k E E − k I I ) N 2 = β s i − ( α + σ + k E + δ E ) e + k E e 2 + k I i e

i ′ = N ⋅ ( σ E − ( τ + γ ) I ) − I ⋅ ( ( α − μ ) N − k E E − k I I ) N 2 = σ e − ( α + k I ) i + k E e i + k I i 2 r ′ = N ⋅ ( γ I − ( μ + ϵ ) R + λ S + δ E E ) − R ⋅ ( ( α − μ ) N − k E E − k I I ) N 2 = γ i − ( α + ϵ ) r + λ s + δ E e + k E e r + k I i r (2)

From the transformation above, the following equations are derived

s ′ = α + ϵ r − ( α + λ ) s − ( β − k I ) s i + k E s e e ′ = β s i − ( α + σ + k E + δ E ) e + k E e 2 + k I i e i ′ = σ e − ( α + k I ) i + k E e i + k I i 2 r ′ = γ i − ( α + ϵ ) r + λ s + δ E e + k E e r + k I i r (3)

subject to the restriction that s + e + i + r = 1 . Let us hereon refer to (3) as the full model. It is also worthy to note that the total population N ( t ) does not appear in Equation (2). We can attribute that to the homogeneity in the system (1). Also, since r appears in the first equation of the Equation (3), we substitute r = 1 − s − e − i into the first equation to get:

s ′ = α + ϵ − ( α + λ + ϵ ) s − ( β − k I ) s i + k E s e − ϵ e − ϵ i e ′ = β s i − ( α + σ + k E + δ E ) e + k E e 2 + k I i e i ′ = σ e − ( α + k I ) i + k E e i + k I i 2 (4)

so that we can focus our attention on the subsystem (4).

s ′ = ( α + ϵ ) − ( α + λ + ϵ ) s − ( β − k I ) s i + k E s e − ϵ e − ϵ i e ′ = β s i − ( α + σ + k E + δ E ) e + k E e 2 + k I i e i ′ = σ e − ( α + k I ) i + k E e i + k I i 2 (5)

The transformation has revealed some dynamics and interactions embedded in our model which is not intended to be measured in this study. Let us rewrite system (5) by making these substitutions; h 1 = α + ϵ ; h 2 = α + λ + ϵ , h 3 = α + σ + k E + δ E ; and h 4 = α + k I to obtain the system of equations:

s ′ = h 1 − h 2 s − ( β − k I ) s i e ′ = β s i − h 3 e i ′ = σ e − h 4 i (6)

For biological considerations we study the system (6) in the region

Ω = { ( s , e , i ) ∈ ℜ + 3 | 0 ≤ s + e + i ≤ 1 }

It can be shown that Ω is positively invariant. In the latter part of this study, we show that the transformed model (3) is similar to the new model (6) by numerical simulation. [

It can be seen that Equation (6) has a disease-free equilibrium is D F E = ( h 1 h 2 , 0 , 0 ) . The Jacobian matrix evaluated at the DFE is given thus,

D F E = ( − h 2 0 h 1 ( k I − β ) h 2 0 − h 3 β h 1 h 2 0 σ − h 4 )

The characteristic polynomial of the DFE system F ( λ ) is given as,

F ( λ ) = λ 3 + ( h 2 + h 3 + h 4 ) λ 2 + ( h 2 h 3 + h 2 h 4 + h 3 h 4 − β σ h 1 h 2 ) λ + h 2 h 3 h 4 − β σ h 1 (7)

From the characteristic polynomial above,

b 1 = h 2 + h 3 + h 4 (8)

b 2 = h 2 h 3 + h 2 h 4 + h 3 h 4 − β σ h 1 h 2 (9)

b 3 = h 2 h 3 h 4 − β σ h 1 (10)

By the Routh-Hurwitz’s criterion, for the characteristic equation to have negative eigenvalues, then b 1 > 0 , b 3 > 0 and b 1 b 2 > b 3 . The Routh-Hurwitz’s criterion is a tool used to establish the negativity or otherwise of the roots of a characteristic equation. Let us first introduce our threshold parameter R 0 .

Following the theory made explicit in [

F = [ ( β − k I ) s i 0 ] and V = [ h 3 e − σ e + h 4 i ] (11)

From (11), we proceed to find F and V as defined in [

F = [ 0 ( β − k I ) s 0 0 ] and V = [ h 3 0 − σ h 4 ] (12)

It can be easily shown that

V − 1 = 1 h 3 h 4 [ h 4 0 σ h 3 ] (13)

Consequently,

R 0 = ρ ( F V − 1 ) = 1 h 3 h 4 [ 0 ( β − k I ) s 0 0 ] × [ h 3 0 − σ h 4 ] = ( β − k I ) σ s D F E h 3 h 4 , but s D F E = h 1 h 2 R 0 = ( β − k I ) h 1 σ h 2 h 3 h 4 (14)

Another construction of the measure R 0 mostly considered by researchers long before the introduction of the new generation matrix (NGM) method was introduced is

R 0 = D F E E E

where E E is the endemic equilibrium.

R 0 = D F E E E = β σ h 1 h 2 h 3 h 4 (15)

We remark that, the NGM’s measure of R 0 (as shown in 14) is equivalent to this measure (15) if and only if k I = 0 . But for the purpose of this study, we choose (14) as our measure for R 0 .

Theorem 1. Denote R 0 = ( β − k I ) h 1 σ h 2 h 3 h 4 . When R 0 ≤ 1 , then the system (6) has only a D F E = ( h 1 h 2 , 0 , 0 ) on the set Ω . When R 0 > 1 then besides the DFE, there exist a unique endemic equilibrium EE where the disease is persistent.

Theorem 2 (Routh-Hurwitz’s Criterion) Consider the characteristic equation

| λ I − A | = λ n + b 1 λ ( n − 1 ) + ⋯ + b ( n − 1 ) λ + b n = 0 (16)

determining the n eigenvalues λ of a real n × n square matrix A, where I is the identity matrix. Then the eigenvalues λ all have negative real parts if Δ 1 > 0 , Δ 2 > 0 , ⋯ , Δ n > 0 , where

Δ k = | b 1 1 0 0 0 0 ⋯ 0 b 3 b 2 b 1 1 0 0 ⋯ 0 b 5 b 4 b 3 b 2 b 1 1 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ b 2 k − 1 b 2 k − 2 b 2 k − 3 b 2 k − 4 b 2 k − 5 b 2 k − 6 ⋯ b k |

The above theorem (2) is reproduced from [

Theorem 3. The disease-free equilibrium DFE is locally stable as R 0 ≤ 1 and unstable as R 0 > 1 .

Proof. It is easy to show from (7) that b 1 > 0 and b 2 > 0 whenever R 0 ≤ 1 . Now for b 1 b 2 > b 3 ,

⇒ b 1 b 2 > b 3 ⇒ ( h 2 + h 3 + h 4 ) ( h 2 h 3 + h 3 h 4 − β σ h 1 h 2 ) > h 2 h 3 h 4 − β σ h 1 ⇒ h 2 2 h 3 + h 2 h 3 h 4 − β σ h 1 + h 2 h 3 2 + h 3 2 h 4 − β σ h 1 h 3 h 2 + h 2 h 3 h 4 + h 3 h 4 2 − β σ h 1 h 4 h 2 > h 2 h 3 h 4 − β σ h 1

And since R 0 ≤ 1 ,

h 2 2 h 3 + h 2 h 3 2 + h 3 2 h 4 + h 3 h 4 2 + h 2 h 3 h 4 > β σ h 1 h 2 ( h 3 + h 4 )

therefore b 1 b 2 > b 3 . The DF system has negative eigenvalues, hence the DF system is locally stable when R 0 ≤ 1 . Again, when R 0 > 1 , then the condition 2 of the Routh-Hurwitz’s criterion is violated; thus b 2 may become less than 0 and therefore the system becomes unstable.

The matrix theoretic method is used to prove the global stability of the DFE.

A Matrix-Theoretic MethodThe matrix-theoretic method is used to prove the sharp threshold statement in theorem (5). It is a systematic method, and it is presented to guide the construction of a Lyapunov function. Taking the same path as [

f ( x , y ) : = ( F − V ) x − F ( x , y ) + V ( x , y ) (17)

Then the equation for the disease compartment can be written as

x ′ = ( F − V ) x − f ( x , y ) (18)

Let ψ T ≤ 0 be the left eigenvector of the non-negative matrix V − 1 F corresponding to the eigenvalue ρ ( V − 1 F ) = ρ ( F V − 1 ) = R 0 . The following result provides a general method to construct a Lyapunov function for [1.1]. [

Theorem 4 Let F, V and f ( x , y ) be defined as in (1.2) and (2.1) in [

Proof. The proof as followed in [

D ′ = D ′ | = ψ V − 1 x ′ = ψ V − 1 ( F − V ) x − ψ V − 1 f ( x , y ) = ( R 0 − 1 ) ψ T x − ψ T V − 1 ( x , y ) (19)

Since ψ T ≥ 0 , V − 1 ≥ 0 , and f ( x , y ) ≥ 0 in the region Ω , the last term is non-positive. If R 0 ≤ 1 , then D 1 ≤ 0 in Ω and thus D is a Lyapunov function for the system [1.1] as defined in [

[

Theorem 5. The DFE is globally stable.

Proof. Taking the left side expansion of (19), the Lyapunov function D as defined in theorem (4) therefore is

D = ψ T V − 1 x = ( σ h 4 2 + σ h 3 h 4 ) e + 1 h 3 i (20)

D ′ = ( R 0 − 1 ) ( σ h 4 e + i ) − ( 2 σ h 3 h 4 + 1 h 4 ) f ( x , y )

it is easy to prove that

f ( x , y ) = 0 (21)

D ′ = ( R 0 − 1 ) ( σ h 4 e + i ) (22)

It is obvious from this equation that when R 0 ≤ 1 then D ′ < 0 and therefore the DFE is globally asymptotically stable and unstable when R 0 > 1 .

Referring to theorem (1), there exists a unique EE. In this section, we establish the stability of the EE using Routh-Hurwitz’s criterion for the proof of local stability and Lyapunov functions for the global stability.

Theorem 6. The EE is locally stable.

Proof. The Jacobian of the EE is given thus

J E E = ( − ( β − k I ) ( β h 1 σ − h 2 h 3 h 4 ) ( β − k I ) h 3 h 4 − h 2 0 − h 3 h 4 ( β − k I ) β σ β ( β h 1 σ − h 2 h 3 h 4 ) ( β − k I ) h 3 h 4 − h 2 − h 3 h 3 h 4 σ 0 σ − h 4 )

Then the characteristic polynomial of the EE would be

F ( λ ) = − β h 1 h 3 h 4 k I σ h 3 h 4 k I − β h 3 h 4 − λ β h 1 h 4 k I σ h 3 h 4 k I − β h 3 h 4 − λ β h 1 h 3 k I σ h 3 h 4 k I − β h 3 h 4 − λ 2 β h 1 k I σ h 3 h 4 k I − β h 3 h 4 + β 2 h 1 h 3 h 4 σ h 3 h 4 k I − β h 3 h 4 + λ β 2 h 1 h 4 σ h 3 h 4 k I − β h 3 h 4 + λ β 2 h 1 h 3 σ h 3 h 4 k I − β h 3 h 4 + λ 2 β 2 h 1 σ h 3 h 4 k I − β h 3 h 4 + h 2 h 3 2 h 4 2 k I h 3 h 4 k I − β h 3 h 4 + λ h 2 h 3 h 4 2 k I h 3 h 4 k I − β h 3 h 4 + λ h 2 h 3 2 h 4 k I h 3 h 4 k I − β h 3 h 4 + λ 2 h 2 h 3 h 4 k I h 3 h 4 k I − β h 3 h 4 − β h 2 h 3 2 h 4 2 h 3 h 4 k I − β h 3 h 4 − λ β h 2 h 3 h 4 2 h 3 h 4 k I − β h 3 h 4 − λ β h 2 h 3 2 h 4 h 3 h 4 k I − β h 3 h 4 − λ 2 β h 2 h 3 h 4 h 3 h 4 k I − β h 3 h 4 − λ h 2 h 4 − λ 2 h 4 − λ h 2 h 3 − λ 2 h 3 − λ 2 h 2 − λ 3 (23)

which is of the form F ( λ ) = λ 3 + b 1 λ 2 + b 2 λ + b 3 where b 1 , b 2 and b 3 are simplified from Equation (23) as

b 1 = h 3 + h 4 + β σ h 1 h 3 h 4

b 2 = β σ h 1 + h 1 h 3 σ

b 3 = β σ h 1 − h 2 h 3 h 4

when F ( λ ) = 0 . It can be seen that b 1 , b 2 and b 3 are all positive. For b 1 b 2 > b 3 ,

b 1 b 2 = ( h 3 + h 4 + β σ h 1 h 3 h 4 ) ( β σ h 1 + h 1 h 3 σ ) > β σ h 1 − h 2 h 3 h 4 = b 3 = β σ h 1 h 3 + h 1 σ + β σ h 1 h 4 + h 1 h 4 h 3 σ + ( β σ h 1 ) 2 h 3 h 4 + β ( σ h 1 ) 2 h 3 2 h 4 > β σ h 1 − h 2 h 3 h 4 = b 3 = ( h 3 + h 4 ) β σ h 1 + h 1 σ + h 1 h 4 h 3 σ + ( β σ h 1 ) 2 h 3 h 4 + β ( σ h 1 ) 2 h 3 2 h 4 > β σ h 1 − h 2 h 3 h 4 = b 3 (24)

Now h 1 < h 3 + h 4 and by extension ( h 3 + h 4 ) β σ h 1 > β σ h 1 . The rest of the terms in the left hand side of this inequality are non-negative and therefore obviously greater than − h 2 h 3 h 4 . Therefore the EE is locally stable.

A general form of Lyapunov functions coined from the first integral form of the Goh-Volterra system which is often used in the literature of mathematical biology is used to prove the global stability of the EE. This function takes the form

L i = ∑ i = 1 n c i ( x i − x i * − x i * ln x i x i * ) (25)

where x is the variables and c i are carefully selected constants. This criterion has been used many times in establishing the stability or otherwise of many disease models and also present in [4, 13, 15].

Theorem 7. The EE is globally stable.

Proof. Let s > s * and i > i * ,

L 1 = s − s * − s * ln s s * (26)

L ′ 1 = − ( s * − s s ) s ′ (27)

= − ( s * − s s ) ( h 1 − h 2 s − ( β − k I ) s i ) , (28)

and the equilibrium relation for h 1 = h 2 s * + ( b − k I ) s * i *

≤ − ( s * − s s ) ( h 2 ( s * − s ) + ( β − k I ) ( s * i * − s i ) ) (29)

L ′ 1 ≤ 0 , whenever s > s * and i > i * in the region ℜ + (30)

In line (26), we define the Goh-Volterra function for the first variable s and differentiate it in the second line (27). In the following line, s ′ is substituted for its relation in (6) and evaluated at the equilibrium relation for h 1 in line (29). Consider that s > s * and i > i * , (which is a necessary condition for (26) to hold). Then the inequality in line (29) is justified. Therefore, L 1 defined above is a Lyapunov function.

Again, let e > e * and L 2 = e − e * − e * ln e e *

L 2 = e − e * − e * ln e e * (31)

L ′ 2 = − ( e * − e e ) e ′ (32)

L ′ 2 = − ( e * − e e ) ( β s i − h 3 e ) (33)

= − ( e * − e ) ( β s i e − h 3 ) (34)

≤ − ( e * − e ) ( β s * i * e * − h 3 ) , (35)

then from the system under study, (6), i * e * = s h 4 and s * = h 3 h 4 b s

≤ − ( e * − e ) ( β h 3 h 4 σ β h 4 σ − h 3 ) = − ( e * − e ) × 0 = 0 (36)

L ′ 2 ≤ 0 (37)

The next Lyapunov function for the e variable is given in line (31), differentiated and evaluated in lines (32) and (33) respectively. The equilibrium relation for β s * i * e * is introduced in lines (35) and (36). Again, the inequality is obviously justified whenever e > e * . The function defined as L 2 = e − e * − e * ln e e * is successful Lyapunov function.

Lastly, let us set i > i * and L 3 = i − i * − i * ln i i * ,

L 3 = i − i * − i * ln i i * (38)

L ′ 3 = − ( i * − i i ) i ′ (39)

L ′ 3 = − ( i * − i i ) ( σ e − h 4 i ) (40)

= − ( i * − i ) ( σ e i − h 4 ) (41)

≤ − ( i * − i ) ( σ e * i * − h 4 ) , e * i * = h 4 s (42)

≤ − ( i * − i ) ( σ h 4 σ − h 4 ) = 0 (43)

L ′ 3 ≤ 0 (44)

The Lyapunov function for the last variable understudy is introduced in line (38) and differentiated in the following line (39). The equilibrium relation for e * / i * is substituted in lines (42) and (43) thereby justifying the inequality.

Therefore L defined as

L = ∑ i = 1 n ( x i − x i * − x i * ln x i x i * ) = L 1 + L 2 + L 3

is a Lyapunov function for the system (6). Arbitrary constants c i can be chosen from ℜ + and any linear combination of L would be a Lyapunov function for the system. Hence the proof.

In this section, we simulate our model to demonstrate the theoretic results found by this study. We only talk about susceptibility and recovery over time since they explain the rest of the compartments. Note that from the schematic diagram (figure [

From

In

Parameters | Value | Sources |
---|---|---|

α | 0.045 | GHS & GIS |

μ | 0.00875 | GHS |

μ E | 0.0111 | GHS |

β | 0.005 | Estimated |

σ | 0.9913 | Estimated |

γ | 0.9912 | Estimated |

δ E | 0.015 | Assumed |

τ | 0.0025 | Assumed |

ϵ | 0.0003 | Assumed |

S 0 ( ∗ ∗ ) = y 1 _ 0 ( ∗ ) | 10000 | Assumed |

E 0 ( ∗ ∗ ) = y 2 _ 0 ( ∗ ) | 3000 | Assumed |

I 0 ( ∗ ∗ ) = y 3 _ 0 ( ∗ ) | 326 | Assumed |

R 0 ( ∗ ∗ ) = y 4 _ 0 ( ∗ ) | 2965 | Assumed |

vaccination cover in the population, the disease will persist in the population for a while and die out. When the time is extended, it would be seen that the disease will completely die out of the system as the exposed individuals approaches zero. It makes sense because, from the schematic diagram (4.1) of the model, the exposed individuals are assumed to go through the infectious stage or attained a natural cure against the disease and move into the recovery stage. Again, we see that the full model and the reduced model are the same.

In

In

In this paper, the global dynamics of measles has been studied in a varying population system such as could be compared to the thriving metropolis of Sunyani in Ghana. A matrix theoretic method and Goh-Volterra types of Lyapunov function were used to establish the stability of the disease-free and endemic equilibria respectively. The basic reproductive number R 0 is defined using the New Generation Matrix method and proved as the threshold parameter. The DFE is globally stable whenever R 0 ≤ 1 and unstable otherwise. The EE is globally stable when R 0 > 1 . Also, the transformation of the original system (1) uncovered some unnecessary measures embedded in the model which was removed to give the approximate system (6). The latter system was used to replace the former and towards the end of this paper, we showed by numerical simulation that approximate system (3) is the same as (6). The numerical simulation also showed the theoretic dynamics discussed in this paper. It showed that more people will gain immunity against the measles and by extension, an outbreak of measles would not impact the community if vaccination cover is 80% annually.

The authors declare no conflicts of interest regarding the publication of this paper.

Otoo, D., Kessie, J.A., Donkoh, E.K., Okyere, E. and Kumi, W. (2019) Global Dynamics of an SEIRS Compartmental Measles Model with Interrupted Vaccination. Applied Mathematics, 10, 588-604. https://doi.org/10.4236/am.2019.107042