Mathematical Study of a Class of Epidemiological Models with Multiple Infectious Stages

S. Bowong; A. Temgoua; Y. Malong; J. Mbang

doi:10.1515/ijnsns-2017-0244

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Mathematical Study of a Class of Epidemiological Models with Multiple Infectious Stages

S. Bowong , A. Temgoua , Y. Malong und J. Mbang

Veröffentlicht/Copyright: 22. November 2019

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Aus der Zeitschrift International Journal of Nonlinear Sciences and Numerical Simulation Band 21 Heft 3-4

Abstract

This paper deals with the mathematical analysis of a general class of epidemiological models with multiple infectious stages for the transmission dynamics of a communicable disease. We provide a theoretical study of the model. We derive the basic reproduction number R0 that determines the extinction and the persistence of the infection. We show that the disease-free equilibrium is globally asymptotically stable whenever R0≤1, while when R0>1, the disease-free equilibrium is unstable and there exists a unique endemic equilibrium point which is globally asymptotically stable. A case study for tuberculosis (TB) is considered to numerically support the analytical results.

Keywords: epidemiological models; global stability; Lyapunov functions; TB mathematical model

JEL Classification: 34A34; 34D23; 34D40; 92D30

Appendix A: Proof of Theorem 3

Herein, we give the proof of Theorem 3 on the GAS of the endemic equilibrium Q∗=(S∗,E∗,Ii∗,,L∗,R∗) where S^*, E, Ii∗ (1 ≤ i ≤ n), L^* and R^* are defined as in eq. (16).

Consider the following Lyapunov function:

(19)U(S,E,Ik,L,R)=(S−S∗lnS)+a1(E−E∗lnE)+a2(L−L∗lnL)+a3(R−R∗lnR)+∑k=1nbk(Ik−Ik∗lnIk),

where a_i (i = 1,2,3) and b_k (k = 1,…,n) are defined as

(20)a1=k(1−r1)A1p+(1−p)k(1−r1),b1=A1A1p+(1−p)k(1−r1),a3=A1γA3[A1p+(1−p)k(1−r1)],a2=a3δ+b1λ+βlS∗A2,bn=a2εn+a3αn+βnS∗Bn,bk=a2εk+a3αk+bk+1ηk+βkS∗Bk,k=2,3,…,n−1.

The time derivative of U(S,E,I_k,L,R) satisfies

(21)U˙=1−S∗SS˙+a11−E∗EE˙+a21−L∗LL˙+a31−R∗RR˙+∑k=1nbk1−Ik∗IkI˙k,=1−S∗SΛ−S∑k=1nβkIk+βlL−μS+a11−E∗E(1−p)S∑k=1nβkIk+βlL−a1A1E+a1A1E∗+a21−L∗L∑k=1nεkIk−a2A2L+a2A2L∗+a31−R∗R∑k=1nαkIk+a3δL−a3δLR∗R−a3A3R+a3A3R∗+b11−I1∗I1pS∑k=1nβkIk+βlL+k(1−r1)E+γR−b1(1−r1)EI1∗I1−b1γRE∗E−b1λLE∗E−b1B1I1+b1B1I1∗+∑k=2nbk1−Ik∗Ikηk−1Ik−1−∑k=2nbkBkIk+∑k=2nbkBkIk∗.

By considering model system (2) at the endemic equilibrium, one has

(22)Λ=S∗∑k=1nβkIk∗+βlL∗+μS∗,a1A1E∗=a1(1−p)S∗∑k=1nβkIk∗+βlL∗,a2A2L∗=a2∑k=1nεkIk∗,a3A3R∗=a3∑k=1nαkIk∗+a3δL∗,b1B1I1∗=b1pS∗∑k=1nβkIk∗+βlL∗+b1k(1−r1)E∗+b1γR∗+b1λL∗,⋮bkBkIk∗=bkηk−1Ik−1∗,k=1,…,n−1.

Introducing eq. (22) into eq. (21) yields

(23)U˙=−μ(S−S∗)2S+S∗∑k=1nβkIk∗+βlL∗1−S∗S+S∑k=1nβkIk+βlL[−1+a1(1−p)+b1p]−a1(1−p)∑k=1nβkIk+βlLSE∗E−b1p∑k=1nβkIk+βlLSI1∗I1+[−a1A1+b1k(1−r1)]E+[−a2A2+a3δ+b1λ+βlS∗]L+[−a3A3+b1γ]R+∑k=1n−1[a2εk+a3αk+bk+1ηk−bkBk+βkS∗]Ik+[a2εn+a3αn−bnBn+βnS∗]In+S∗∑k=1nβkIk∗+βlL∗[a1(1−p)+b1p]+b1k(1−r1)E∗1−EE∗I1∗I1+b1λL∗1−LL∗I1∗I1+a3δL∗1−LL∗R∗R+a3∑k=1nαkIk∗1−R∗RIkIk∗+a2∑k=1nεkIk∗1−L∗LIkIk∗+b1γR∗1−I1∗I1RR∗+∑k=2nbkηk−1Ik−1∗1−Ik∗IkIk−1Ik−1∗.

Now, let (u1,u2,u3,u4,vk)=E∗E,R∗R,L∗L,S∗S,Ik∗Ik, k = 1,…,n. Then, eq. (23) becomes

(24)U˙=−μ(S−S∗)2S+S∗∑k=1nβkIk∗+βlL∗1−u4+S∑k=1nβkIk+βlL[−1+a1(1−p)+b1p]−a1(1−p)∑k=1nβkIk∗vk+βlL∗u3S∗u1u4−b1p∑k=1nβkIk∗vk+βlL∗u3S∗v1u4+[−a1A1+b1k(1−r1)]E+[−a2A2+a3δ+b1λ+βlS∗]L+[−a3A3+b1γ]R+∑k=1n−1[a2εk+a3αk+bk+1ηk−bkBk+βkS∗]Ik+[a2εn+a3αn−bnBn+βnS∗]In+S∗∑k=1nβkIk∗+βlL∗[a1(1−p)+b1p]+b1k(1−r1)E∗1−v1u1+b1λL∗1−v1u3+a3δL∗1−u2u3+a3∑k=1nαkIk∗1−u2vk+a2∑k=1nεkIk∗1−u3vk+b1γR∗1−v1u2+∑k=2nbkηk−1Ik−1∗1−vkvk−1.

The coefficients a_i et b_k are chosen such that the coefficients of S∑k=1nβkIk+βlL, E, L , I₁, I_k (k = 2,…,n - 1), I_n, and R are equal to zero, that is,

(25)−1+a1(1−p)+b1p=0,−a1A1+b1k(1−r1)=0,−a2A2+a3δ+b1λ+βlS∗=0,−a3A3+b1γ=0,a2ε1+a3α1+b2η1−b1B1+β1S∗=0,a2εk+a3αk+bk+1ηk−bkBk+βkS∗=0,k=2,…,n−1,a2εn+a3αn−bnBn+βnS∗=0.

Now, multiplying the fifth equation of (25) by I1∗ and using the expression of b1B1I1∗ defined as in eq. (22) yields

(26)a2ε1I1∗+a3α1I1∗+b2η1I1∗−b1B1I1∗+β1S∗I1∗=a2ε1I1∗+a3α1I1∗+b2η1I1∗−b1pS∗∑k=1nβkIk∗+βlL∗−b1γR∗−b1λL∗+β1S∗I1∗.

On the other hand, from eq. (22), one has

(27)a1A1E∗−a1(1−p)S∗∑k=1nβkIk∗+βlL∗+a3A3R∗−a3∑k=1nαkIk∗−a3δL∗+a2A2L∗−a2∑k=1nεkIk∗+∑k=1n−1bkBkIk∗−∑k=1n−1bkηk−1Ik−1∗+bnBnIn∗−bnηn−1In−1∗=0.

Adding the above equation in the right-hand side of eq. (26) gives

(28)a2ε1I1∗+a3α1I1∗+b2η1I1∗−b1B1I1∗+β1S∗I1∗=[a1A1−b1k(1−r1)]E∗+[−b1γ+a3A3]R∗+[−b1p−a1(1−p)+1]∑k=1nβkIk∗+[−b1p−a1(1−p)+1]βlS∗L∗+∑k=2n−1[βkS∗−a2εk−a3αk−bk+1ηk+bkBk]Ik∗+[−βlS∗−b1λ−a3δ+a2A2]L∗+[−βnS∗+bnBn−a2εn−a3αn]In∗=0.

Note that when the fifth equation of (25) is satisfied, then all equations of (25) are also satisfied. This is why in the sequel, we only consider the following system of equations:

(29)−1+a1(1−p)+b1p=0,−a1A1+b1k(1−r1)=0,−a2A2+a3δ+b1λ+βlS∗=0,−a3A3+b1γ=0,a2εk+a3αk+bk+1ηk−bkBk+βkS∗=0,k=2,…,n−1,a2εn+a3αn−bnBn+βnS∗=0.

Solving the above equations give the expressions of a_i and b_k defined as in eq. (20). Thus, after plugging the expressions of a_i and b_k given as in eq. (20) into eq. (24), one obtains

(30)U˙=−μ(S−S∗)2S+S∗∑k=1nβkIk∗+βlL∗2−u4−a1(1−p)∑k=1nβkIk∗vk+βlL∗u3S∗u1u4−b1p∑k=1nβkIk∗vk+βlL∗u3S∗v1u4+b1k(1−r1)E∗1−v1u1+b1λL∗1−v1u3+a3δL∗1−u2u3+a3∑k=1nαkIk∗1−u2vk+a2∑k=1nεkIk∗1−u3vk+b1γR∗1−v1u2+∑k=2nbkηk−1Ik−1∗1−vkvk−1.

Using the fact that a1(1−p)+b1p=1, eq. (30) becomes

(31)U˙=−μ(S−S∗)2S+a1(1−p)∑k=1nβkS∗Ik∗2−u4−u1vku4+b1p∑k=1nβkIk∗S∗2−u4−v1vku4+a1(1−p)βlL∗S∗2−u4−u1u3u4+b1pβlL∗S∗2−u4−v1u3u4+b1k(1−r1)E∗1−v1u3+b1λL∗1−v1u3+a3δL∗1−u2u3+a3∑k=1nαkIk∗1−u2vk+a2∑k=1nεkIk∗1−u3vk+b1γR∗1−v1u2+∑k=2nbkηk−1Ik−1∗1−vkvk−1.

Now, multiplying the second, third, fourth, and fifth equations of (29) by E^*, L^*, R^*, Ik∗ (k = 2,…,n - 1), and In∗, respectively, gives

(32)−a1A1E∗+b1k(1−r1)E∗=0,−a2A2L∗+a3δL∗+b1λL∗+βlS∗L∗=0,−a3A3R∗+b1γR∗=0,a2εkIk∗+a3αkIk∗+bk+1ηkIk∗−bkBkIk∗+βkS∗Ik∗=0,k=2,…,n−1,a2εnIn∗+a3αnIn∗−bnBnIn∗+βnS∗In∗=0.

Using the expressions of a1A1E∗, a2A2L∗, a3A3R∗, bkBkIk∗ (k = 2,…,n - 1) and bnBnIn∗ given as in eq. (22), then eq. (32) becomes

(33)−a1(1−p)S∗∑k=1nβkIk∗+βlL∗+b1k(1−r1)E∗=0,−a2∑k=1nεkIk∗+a3δL∗+b1λL∗+βlS∗L∗=0,−a3∑k=1nαkIk∗−a3δL∗+b1γR∗=0,a2εkIk∗+a3αkIk∗+bk+1ηkIk∗−bkηk−1Ik−1∗+βkS∗Ik∗=0,k=2,…,n−1,a2εnIn∗+a3αnIn∗−bnηn−1In−1∗+βnS∗In∗=0.

Now, let F₁(u), F₂(u), F₃(u), and G_k(u) (k = 2,…,n) where u=(u1,u2,u3,vk)T be n + 2 functions to be determined later. Then, multiplying the first, second, third, fourth, and fifth equations in (33) by F₁(u), F₂(u), F₃(u), and G_k(u) (k = 2,…,n - 1) and G_n(u), respectively, yields

(34)−a1(1−p)∑k=1nβkS∗Ik∗F1(u)−a1(1−p)S∗βlS∗L∗F1(u)+b1k(1−r1)E∗F1(u)=0,−a2∑k=1nεkIk∗F2(u)+a3δL∗F2(u)+b1λL∗F2(u)+[a1(1−p)+b1p]βlS∗L∗F2(u)=0,−a3∑k=1nαkIk∗F3(u)−a3δL∗F3(u)+b1γR∗F3(u)=0,a2εkIk∗Gk(u)+a3αkIk∗Gk(u)+bk+1ηkIk∗Gk(u)−bkηk−1Ik−1∗Gk(u)+[a1(1−p)+b1p]βkS∗Ik∗Gk(u)=0,k=2,…,n−1,a2εnIn∗Gn(u)+a3αnIn∗Gn(u)−bnηn−1In−1∗Gn(u)+[a1(1−p)+b1p]βnS∗In∗Gn(u)=0.

Adding eq. (34) to the right-hand side of (31) gives

(35)U˙=−μ(S−S∗)2S+a1(1−p)β1S∗I1∗2−u4−u1v1u4−F1+a1(1−p)∑k=2nβkIk∗2−u4−u1vku4−F1+Gk+b1pβ1S∗I1∗2−u4−1u4+b1p∑k=2nβkIk∗S∗2−u4−v1vku4+Gk+a1(1−p)βlL∗S∗2−u4−u1u3u4+F2−F1+b1pβlL∗S∗2−u4−v1u3u4+F2+b1k(1−r1)E∗1−v1u1+F1+b1λL∗1−v1u3+F2+a3δL∗1−u2u3+F2−F3+a3α1I1∗1−u2v1−F3+a3∑k=2nαkIk∗1−u2vk−F3+Gk+a2ε1I1∗1−u3v1−F2+a2∑k=2nεkIk∗1−u3vk+Gk−F2+b1γR∗1−v1u2+F3+b2η1I1∗1−v2v1−G2+∑k=3nbkηk−1Ik−1∗1−vkvk−1+Gk−1−Gk.

Now, we shall choose the functions F₁(u), F₂(u), F₃(u), F₄(u), and G_k(u), k = 2,…,n which make U˙ nonpositive. To do so, the functions F₁(u), F₂(u), F₃(u), and F₄(u) are chosen such that the coefficients of b1k(1−r1)E∗, a2ε1I1∗, b1γR∗, b2η1I1∗ are equal to zero, and the function G_k(u) is chosen such that the coefficient of b2η1I1∗ is equal to zero and the coefficient of αkIk∗ is nonpositive, that is,

(36)F1(u)=−1+v1u1,F2(u)=1−u3v1,F3(u)=−1+v1u2andGk(u)=k−1−v2v1−v3v2−⋯−vkvk−1,k=2,…,n.

After plugging these expressions given as in eq. (36) into eq. (35), one finally obtains

(37)U˙=−μ(S−S∗)2S+a1(1−p)β1S∗I1∗3−u4−u1v1u4−v1u1+a1(1−p)∑k=2nβkIk∗k+2−u4−u1vku4−v1u1−v2v1−v3v2−⋯−vkvk−1+b1pβ1S∗I1∗2−u4−1u4+b1p∑k=2nβkIk∗S∗k+2−u4−v1vku4−v2v1−v3v2−⋯−vkvk−1+a1(1−p)βlL∗S∗4−u4−u1u3u4−u3v1−v1u1+b1pβlL∗S∗3−u4−v1u3u4−u3v1+b1λL∗2−v1u3−u3v1+a3δL∗3−u2u3−u3v1−v1u2+a3α1I1∗2−u2v1−v1u2+a3∑k=2nαkIk∗k+2−u2vk−v1u2−v2v1−v3v2−⋯−vkvk−1+a2∑k=2nεkIk∗k−v2v1−v3v2−⋯vkvk−1−u3vk+u3v1−1.

Since the arithmetic mean exceeds the geometric mean, the following inequalities hold:

(38)3−u4−u1v1u4−v1u1≤0,k+2−u4−u1vku4−v1u1−v2v1−v3v2−⋯−vkvk−1≤0,2−u4−1u4≤0,k+2−u4−v1vku4−v2v1−v3v2−⋯−vkvk−1≤0,4−u4−u1u3u4−u3v1−v1u1≤0,3−u4−v1u3u4−u3v1≤0,2−v1u3−u3v1≤0,3−u2u3−u3v1−v1u2≤0,2−u2v1−v1u2≤0,k+2−u2vk−v1u2−v2v1−v3v2−⋯−vkvk−1≤0.

Now, let

(39)Dk=u3v1+(k−1)−v2v1+v3v2+⋯+vkvk−1+u3vk,k=2,…,n.

The next step is to show that the functions D_k is nonpositive for all v1,u1,u3,vk∈R+ for each value of k. Others expressions are negative by applying Corollary 1. By using Lemma 5 in Appendix B with w = k, Y = v₁, yi=vi+1,i=1,...,k−1, and yk=u3, when vi≤u3,i=1,...,k, one has D_k ≤ 0. Thus, U˙≤0 for all S,E,I_k,L,R ≥ 0, provided that S∗,E∗,Ik∗,L∗,R∗ are positive, where the equality U˙=0 holds only on the straight line S = S^*, E∗/E=Ik∗/Ik=L∗/L=R∗/R, k = 1,…,n. It is easy to see that for model system (2), Q^* is the only equilibrium state on this line. Therefore, by Lyapunov–LaSalle asymptotic stability theorem [38, 39, 40, 41], the positive equilibrium state Q^* is globally asymptotically stable in the positive region Ω⊂R+n+4, except on the S-axis which is the stable manifold for the fixed point Q₀. This achieves the proof.

□

Appendix B: Useful inequalities

In this appendix, we give inequalities which are necessary to demonstrate that the time derivative of the Lyapunov function (19) is nonpositive. A key tool is the arithmetic geometric means inequality, which we state here.

Lemma 4

(Arithmetic geometric means inequality): Letz1,…,zwbe positive real numbers. Then,

(40)z1…zww≤z1+…+zww.

Furthermore, exact equality only occurs ifz1=…=zw. □

An immediate consequence of the arithmetic geometric means inequality follows.

Corollary 1

Lety1,…,ywbe positive real numbers such asy1…yw=1 . Then

(41)w−(y1+y2+⋯+yw)≤0,

Furthermore, exact equality only occurs ify1=…=yw

We also have the following result

Lemma 5

LetY≤y1≤…≤ywbe positive real numbers. Then

(42)ykY+(w−1)−y1Y+y2y1+…+ywyw−1≤0.

□

Proof

We have

ywY+(w−1)−y1Y+y2y1+⋯+ywyw−1=yw−YY−y1−YY+y2−y1y1+⋯+yw−yw−1yw−1,≤yw−YY−y1−YY+y2−y1Y+⋯+yw−yw−1Y,≤0.

□

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Received: 2017-11-07

Accepted: 2019-10-21

Published Online: 2019-11-22

Published in Print: 2020-05-26

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Artikel in diesem Heft

https://doi.org/10.1515/ijnsns-2017-0244

Schlagwörter für diesen Artikel

epidemiological models; global stability; Lyapunov functions; TB mathematical model