Parameter | Value | Parameter | Value | Parameter | Value |
ρ | 10 | b3 | 0.8 | γ | 0.05 |
α | 0.01 | d1 | 0.2 | τ | Varied |
η1 | Varied | d2 | 1 | ζ | 0.1 |
η2 | Varied | d3 | 5 | σ | Varied |
b1 | 0.6 | μ | 0.1 | π | 0.1 |
b2 | 0.7 | ε | 1.5 |
Clusters of COVID-19 in high-risk settings, such as schools, have been deemed a critical driving force of the major epidemic waves at the societal level. In Japan, the vaccination coverage among students remained low up to early 2022, especially for 5–11-year-olds. The vaccination of the student population only started in February 2022. Given this background and considering that vaccine effectiveness against school transmission has not been intensively studied, this paper proposes a mathematical model that links the occurrence of clustering to the case count among populations aged 0–19, 20–59, and 60+ years of age. We first estimated the protected (immune) fraction of each age group either by infection or vaccination and then linked the case count in each age group to the number of clusters via a time series regression model that accounts for the time-varying hazard of clustering per infector. From January 3 to May 30, 2022, there were 4,722 reported clusters in school settings. Our model suggests that the immunity offered by vaccination averted 226 (95% credible interval: 219–232) school clusters. Counterfactual scenarios assuming elevated vaccination coverage with faster roll-out reveal that additional school clusters could have been averted. Our study indicates that even relatively low vaccination coverage among students could substantially lower the risk of clustering through vaccine-induced immunity. Our results also suggest that antigenically updated vaccines that are more effective against the variant responsible for the ongoing epidemic may greatly help decrease not only the incidence but also the unnecessary loss of learning opportunities among school-age students.
Citation: Yuta Okada, Hiroshi Nishiura. Vaccine-induced reduction of COVID-19 clusters in school settings in Japan during the epidemic wave caused by B.1.1.529 (Omicron) BA.2, 2022[J]. Mathematical Biosciences and Engineering, 2024, 21(9): 7087-7101. doi: 10.3934/mbe.2024312
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Clusters of COVID-19 in high-risk settings, such as schools, have been deemed a critical driving force of the major epidemic waves at the societal level. In Japan, the vaccination coverage among students remained low up to early 2022, especially for 5–11-year-olds. The vaccination of the student population only started in February 2022. Given this background and considering that vaccine effectiveness against school transmission has not been intensively studied, this paper proposes a mathematical model that links the occurrence of clustering to the case count among populations aged 0–19, 20–59, and 60+ years of age. We first estimated the protected (immune) fraction of each age group either by infection or vaccination and then linked the case count in each age group to the number of clusters via a time series regression model that accounts for the time-varying hazard of clustering per infector. From January 3 to May 30, 2022, there were 4,722 reported clusters in school settings. Our model suggests that the immunity offered by vaccination averted 226 (95% credible interval: 219–232) school clusters. Counterfactual scenarios assuming elevated vaccination coverage with faster roll-out reveal that additional school clusters could have been averted. Our study indicates that even relatively low vaccination coverage among students could substantially lower the risk of clustering through vaccine-induced immunity. Our results also suggest that antigenically updated vaccines that are more effective against the variant responsible for the ongoing epidemic may greatly help decrease not only the incidence but also the unnecessary loss of learning opportunities among school-age students.
In the last decades, many researchers have formulated various mathematical models to characterize the human immune system reaction on invading viruses [1,2,3,4,5,6]. The two mean immune system reactions are the cell-mediated immunity and the humoral immunity. The cell-mediated immunity is based on Cytotoxic T Lymphocytes (CTLs) which kill the infected cells, while the humoral immunity is based on antibodies which are produced by B cells and neutralize the free viruses from the plasma. Some existing models describe the virus dynamics under the effect of cell-mediated immune response (see e.g., [7,8,9,10], see also [11] and the references therein) or humoral immune response [12,13,14,15,16,17]. Wodarz [18] has formulated a virus dynamics model with five compartments; susceptible cells (S), infected cells (I), virus particles (V), B cells (A) and CTL cells (B) as:
{˙S(t)=ρ−αS(t)−ηS(t)V(t),˙I(t)=ηS(t)V(t)−bI(t)−μC(t)I(t),˙V(t)=dI(t)−γA(t)V(t)−εV(t),˙A(t)=τA(t)V(t)−ζA(t),˙C(t)=σC(t)In(t)−πC(t). | (1.1) |
The model has been extended in [19,20,21,22,23], but with virus-to-cell transmission. Cell-to-cell infection plays an important role in increasing the number of infected cells. Mathematical models of virus dynamics with both virus-to-cell and cell-to-cell transmissions have been studied in several works (see e.g., [24,25,26,27,28,29,30,31,32,33,34]). In very recent works [35], both CTL cells and B cells have been incorporated into the viral infection models with both cell-to-cell and virus-to-cell transmissions. However, in [35], only one class of infected cells (actively infected cells) is considered. It has been reported in [36] and [37] that the time from the contact of viruses and susceptible cells to the death of the cells can be modeled by dividing the process into n short stages I1→I2→....→In. In [38], virus dynamics models with multi-staged infected cells, humoral immunity and with only virus-to-cell infection have been studied.
The aim of the present paper is to formulate a virus dynamics model by incorporating (ⅰ) multi-staged infected cells, (ⅱ) both cell-mediated and humoral immune responses (ⅱ) both cell-to-cell and virus-to-cell infections as:
{˙S(t)=ρ−αS(t)−η1S(t)V(t)−η2S(t)In(t),˙I1(t)=η1S(t)V(t)+η2S(t)In(t)−b1I1(t),˙Ik(t)=dk−1Ik−1(t)−bkIk(t), k=2,...,n−1,˙In(t)=dn−1In−1(t)−bnIn(t)−μC(t)In(t),˙V(t)=dnIn(t)−γA(t)V(t)−εV(t),˙A(t)=τA(t)V(t)−ζA(t),˙C(t)=σC(t)In(t)−πC(t), | (1.2) |
where, Ik, k=1,2,...,n represents the concentration of the i-th stage of infected cells. The model assumes that the susceptible cells are infected by virus particles at rate η1S(t)V(t) and by infected cells at rate η2S(t)In(t).
Let Ωj>0, j=1,2,...,n+3 and define
Θ={(S,I1,...,In,V,A,C)∈Rn+4≥0:0≤S,I1≤Ω1,0≤Ik≤Ωk,0≤C≤Ωn+1, 0≤V≤Ωn+2,0≤A≤Ωn+3, k=2,...,n}. |
Proposition 1. The compact set Θ is positively invariant for system (1.2).
Proof. We have
˙S∣S=0=ρ>0, ˙I1∣I1=0=η1SV+η2SIn≥0 ∀ S,V,In≥0,˙Ik∣Ik=0=dk−1Ik−1≥0, ∀ Ik−1≥0, k=2,...,n,˙V∣V=0=dnIn≥0, ∀In≥0, ˙A∣A=0=0, ˙C∣C=0=0. |
This insures that, S(t)>0, Ik(t)≥0, k=1,...,n, V(t)≥0, A(t)≥0, and C(t)≥0 for all t≥0.
To show the boundedness of S(t) and I1(t) we let Ψ1(t)=S(t)+I1(t), then
˙Ψ1=ρ−αS−b1I1≤ρ−ϕ1(S+I1)=ρ−ϕ1Ψ1, |
where ϕ1=min{α,b1}. It follows that,
Ψ1(t)≤e−ϕ1t(Ψ1(0)−ρϕ1)+ρϕ1. |
Hence, 0≤Ψ1(t)≤Ω1 if Ψ1(0)≤Ω1 for t≥0, where Ω1=ρϕ1. Since S(t)>0 and I1(t)≥0, then 0≤S(t),I1(t)≤Ω1 if S(0)+I1(0)≤Ω1. From the fourth equation of system (1.2) in case of k=2, we have
˙I2=d1I1−b2I2≤d1Ω1−b2I2. |
It follows that, 0≤I2(t)≤Ω2 if I2(0)≤Ω2, where Ω2=d1Ω1b2. Similarly, we can show0≤Ik(t)≤Ωk if Ik(0)≤Ωk, where Ωk=dk−1Ωk−1bk, k=3,...,n−1. Further, we let Ψ2(t)=In(t)+μσC(t), then
˙Ψ2=dn−1In−1−bnIn−μπσC≤dn−1Ωn−1−ϕ2(In+μσC)=dn−1Ωn−1−ϕ2Ψ2, |
where ϕ2=min{bn,π}. It follows that, 0≤Ψ2(t)≤Ωn if Ψ2(0)≤Ωn, where Ωn=dn−1Ωn−1ϕ2. Since In(t)≥0 and C(t)≥0, then 0≤In(t)≤Ωn and 0≤C(t)≤Ωn+1 if In(0)+μσC(0)≤Ωn, where Ωn+1=σμΩn. Finally, let Ψ3(t)=V(t)+γτA(t), then
˙Ψ3=dnIn−εV−γζτA≤dnΩn−ϕ3(V+γτA)=dnΩn−ϕ3Ψ3, |
where ϕ3=min{ε,ζ}. It follows that, 0≤Ψ3(t)≤Ωn+2 if Ψ3(0)≤Ωn+2, where Ωn+2=dnΩnϕ3. It follows that, 0≤V(t)≤Ωn+2 and 0≤A(t)≤Ωn+3 if V(0)+γτA(0)≤Ωn+2, where Ωn+3=τγΩn+2.
In this section, we derive five threshold parameters which guarantee the existence of the equilibria of the model.
Lemma 1. System (1.2) has five threshold parameters ℜ0>0, ℜA1>0, ℜC1>0, ℜC2>0 and ℜA2>0 with ℜC1<ℜ0 such that
(ⅰ) if ℜ0≤1, then there exists only one steady state Ɖ0,
(ⅱ) if ℜA1≤1 and ℜC1≤1<ℜ0, then there exist only two equilibria Ɖ0 and ˉƉ,
(ⅲ) if ℜA1>1 and ℜC2≤1, then there exist only three equilibria Ɖ0, ˉƉ and ˆƉ,
(ⅳ) if ℜC1>1 and ℜA2≤1, then there exist only three equilibria Ɖ0, ˉƉ and ∨Ɖ, and
(ⅴ) if ℜA2>1 and ℜC2>1, then there exist five equilibria Ɖ0, ˉƉ, ˆƉ, ∨Ɖ and ˜Ɖ.
Proof. Let (S,I1,...,In,V,A,C) be any equilibrium of system (1.2) satisfying the following equations:
ρ−αS−η1SV−η2SIn=0, | (3.1) |
η1SV+η2SIn−b1I1=0, | (3.2) |
dk−1Ik−1−bkIk=0, k=2,...,n−1, | (3.3) |
dn−1In−1−bnIn−μCIn=0, | (3.4) |
dnIn−γAV−εV=0, | (3.5) |
(τV−ζ)A=0, | (3.6) |
(σIn−π)C=0. | (3.7) |
We find that system (1.2) admits five equilibria.
(ⅰ) Infection-free equilibrium Ɖ0=(S0,n+3⏞0,...,0,0), where S0=ρ/α.
(ⅱ) Chronic-infection equilibrium with inactive immune response ˉƉ=(ˉS,ˉI1,...,ˉIn,ˉV,0,0), where
ˉS=(n∏i=1bidi)εdnη1dn+η2ε,ˉIk=εαdndk(η1dn+η2ε)(k∏i=1dibi)(n∏i=1bidi)((η1dn+η2ε)S0εdn(n∏i=1dibi)−1), k=1,2,...,n,ˉV=αdnη1dn+η2ε((η1dn+η2ε)S0εdn(n∏i=1dibi)−1). |
Therefore, ˉƉ exists when
(η1dn+η2ε)S0εdn(n∏i=1dibi)>1. |
At the equilibrium ˉƉ the disease persists while the immune response is inhibited. The basic infection reproductive ratio for system (1.2) is defined as:
ℜ0=(η1dn+η2ε)S0εdn(n∏i=1dibi). |
The parameter ℜ0 determines whether the disease will progress or not. In terms of ℜ0, we can write
ˉS=S0ℜ0,ˉIk=εαdndk(η1dn+η2ε)(k∏i=1dibi)(n∏i=1bidi)(ℜ0−1), k=1,2,...,n,ˉV=αdnη1dn+η2ε(ℜ0−1). |
(ⅲ) Chronic-infection equilibrium with only active humoral immune response ˆƉ=(ˆS,ˆI1,...,ˆIn,ˆV,ˆA,0), where
ˆS=τρατ+η1ζ+η2τˆIn, ˆIk=(k∏i=1dibi)ρ(η1ζ+η2τˆIn)dk(ατ+η1ζ+η2τˆIn), k=1,2,...,n−1,ˆV=ζτ, ˆA=εγ(dnτεζˆIn−1), |
where
ˆIn=−ϖ2+√ϖ22−4ϖ1ϖ32ϖ1 | (3.8) |
is the positive solution of
ϖ1ˆI2n+ϖ2ˆIn+ϖ3=0, |
with
ϖ1=(n∏i=1bidi)dnη2τ, ϖ2=(n∏i=1bidi)dn(η1ζ+ατ)−ρη2τ, ϖ3=−η1ρζ. | (3.9) |
We note that ˆƉ exists when dnτεζˆIn>1. Let us define the active humoral immunity reproductive ratio
ℜA1=dnτεζˆIn=dnˆInεˆV, | (3.10) |
which determines when the humoral immune response is activated. Thus, ˆA=εγ(ℜA1−1).
(ⅳ) Chronic-infection equilibrium with only active cell-mediated immune response ∨Ɖ=(ˇS,ˇI1,...,ˇIn,ˇV,0,ˇC), where
ˇS=εσρπ(η1dn+η2ε)+αεσ, ˇIn=πσ, ˇV=dnπεσ=dnεˇIn,ˇIk=(k∏i=1dibi)ρπ(η1dn+η2ε)dk[π(η1dn+η2ε)+αεσ], k=1,2,...,n−1,ˇC=bnμ[σρ(η1dn+η2ε)dn[π(η1dn+η2ε)+αεσ](n∏i=1dibi)−1]. |
We note that ∨Ɖ exists when σρ(η1dn+η2ε)dn[π(η1dn+η2ε)+αεσ](n∏i=1dibi)>1. The active cell-mediated immunity reproductive ratio is stated as:
ℜC1=σρ(η1dn+η2ε)dn[π(η1dn+η2ε)+αεσ](n∏i=1dibi)=ℜ01+π(η1dn+η2ε)αεσ. |
The parameter ℜC1 determines when the cell-mediated immune response is activated. Thus, ˇC=bnμ(ℜC1−1) and ℜC1<ℜ0.
(ⅴ) Chronic-infection equilibrium with both active humoral and cell-mediated immune responses ˜Ɖ=(˜S,˜I1,...,˜In,˜V,˜A,˜C), where
˜S=ρτσατσ+η1ζσ+η2τπ, ˜In=πσ=ˇIn, ˜V=ζτ=ˆV,˜Ik=(k∏i=1dibi)ρ(η1ζσ+η2τπ)dk[ατσ+η1ζσ+η2τπ], k=1,2,...,n−1,˜C=bnμ[σρ(η1ζσ+η2τπ)dnπ[ατσ+η1ζσ+η2τπ](n∏i=1dibi)−1],˜A=εγ(dnπτεσζ−1). |
It is obvious that ˜Ɖ exists when σρ(η1ζσ+η2τπ)dnπ[ατσ+η1ζσ+η2τπ](n∏i=1dibi)>1 and dnπτεσζ>1. Now we define
ℜC2=σρ(η1ζσ+η2τπ)dnπ[ατσ+η1ζσ+η2τπ](n∏i=1dibi) and ℜA2=dnπτεσζ=τζˇV, |
where ℜC2 refers to the competed cell-mediated immunity reproductive ratio and appears as the average number of T cells activated due to infectious cells in the scene that the humoral immune response has been constructed, while, ℜA2 refers to the competed humoral immunity reproductive ratio and appears as the average number of B cells activated due to mature viruses in the scene that the cell-mediated immune response has been constructed. Clearly, ˜Ɖ exists when ℜC2>1 and ℜA2>1 and we can write ˜C=bnμ(ℜC2−1) and ˜A=εγ(ℜA2−1).
The five threshold parameters are given as follows:
ℜ0=(η1dn+η2ε)S0εdn(n∏i=1dibi), ℜA1=dnτεζˆIn=dnˆInεˆV, ℜC1=σρ(η1dn+η2ε)dn[π(η1dn+η2ε)+αεσ](n∏i=1dibi)ℜC2=σρ(η1ζσ+η2τπ)dnπ[ατσ+η1ζσ+η2τπ](n∏i=1dibi) and ℜA2=dnπτεσζ=τζˇV. |
We define the active humoral immunity reproductive ratio ℜAhumoral which comes from the limiting (linearized) A-dynamics near A=0 as:
ℜAhumoral=ˉVˆV. |
Lemma 2. (ⅰ) if ℜA1<1, then ℜAhumoral<1,
(ⅱ) if ℜA1>1, then ℜAhumoral>1,
(ⅲ) if ℜA1=1, then ℜAhumoral=1,
Proof. (ⅰ) Let ℜA1<1, then from Eq. 3.10 we have ˆIn<εˆVdn. Then, using Eq. 3.8 we get
−ϖ2+√ϖ22−4ϖ1ϖ32ϖ1<εˆVdn, |
which leads to
(2ϖ1εˆVdn+ϖ2)2−(ϖ22−4ϖ1ϖ3)>0. |
Using Eq. 3.9 we derive
4η2τζεˆV(η1dn+η2ε)(n∏i=1bidi)2[1−ρ(η1dn+η2ε)−εαdn(n∏i=1bidi)εˆV(η1dn+η2ε)(n∏i=1dibi)]>0⟹4η2τζεˆV(η1dn+η2ε)(n∏i=1bidi)2[1−ρ(η1dn+η2ε)(n∏i=1dibi)−εαdnεˆV(η1dn+η2ε)]>0⟹4η2τζεˆV(η1dn+η2ε)(n∏i=1bidi)2[1−ˉVˆV]>0⟹4η2τζεˆV(η1dn+η2ε)(n∏i=1bidi)2[1−ℜAhumoral]>0. |
Thus, ℜAhumoral<1. Using the same argument one can easily confirm part (ⅱ) and (ⅲ).
The global stability of the each equilibria will be investigated by constructing Lyapunov functions using the method presented [39,40,41,42,43,44,45]. Let us define the function ϝ:(0,∞)→[0,∞) as ϝ(υ)=υ−1−lnυ. Denote (S,I1,...,In,V,A,C)=(S(t),I1(t),...,In(t),V(t),A(t),C(t)). The following equalities will be used:
0∏i=1bidi=1, 0∏i=1di=1, | (4.1) |
b1I∗1+n−1∑k=2(k−1∏i=1bidi)bkI∗k+(n−1∏i=1bidi)bnI∗n=n∑k=1(k−1∏i=1bidi)bkI∗k=n∑k=1(k∏i=1bidi)dkI∗k,n−1∑k=2(k−1∏i=1bidi)dk−1Ik−1+(n−1∏i=1bidi)dn−1In−1=n∑k=2(k−1∏i=1bidi)dk−1Ik−1,n−1∑k=2(k−1∏i=1bidi)bkIk+(n−1∏i=1bidi)bnIn=n∑k=2(k−1∏i=1bidi)bkIk,n−1∑k=2(k−1∏i=1bidi)dk−1Ik−1I∗kIk+(n−1∏i=1bidi)dn−1In−1I∗nIn=n∑k=2(k−1∏i=1bidi)dk−1Ik−1I∗kIk, | (4.2) |
n−1∑k=2(k−1∏i=1bidi)(dk−1Ik−1−bkIk)=b1I1−(n−1∏i=1bidi)dn−1In−1, | (4.3) |
where I∗∈{ˉI,ˆI,ˇI,˜I}.
Theorem 1. If ℜ0≤1, then the infection-free equilibrium Ɖ0 is globally asymptotically stable.
Theorem 2. Suppose that ℜA1≤1 and ℜC1≤1<ℜ0, then the chronic-infection equilibrium with inactive immune response ˉƉ is globally asymptotically stable.
Theorem 3. If ℜA1>1 and ℜC2≤1, then the chronic-infection equilibrium with only active humoral immune response ˆƉ is globally asymptotically stable.
Theorem 4. Suppose that ℜC1>1 and ℜA2≤1, then the chronic-infection equilibrium with only active cell-mediated immune response ∨Ɖ is globally asymptotically stable.
Theorem 5. If ℜA2>1 and ℜC2>1, then the chronic-infection equilibrium with both active humoral and cell-mediated immune responses ˜Ɖ is globally asymptotically stable.
The proofs of Theorems 1–5 are given in a Supplementary.
In this section, we perform some numerical simulations in case of three stages of infected cells i.e. n=3.
{˙S=ρ−αS−η1SV−η2SI3,˙I1=η1SV+η2SI3−b1I1,˙I2=d1I1−b2I2,˙I3=d2I2−b3I3−μCI3,˙V=d3I3−γAV−εV,˙A=τAV−ζA,˙C=σCI3−πC. | (5.1) |
The threshold parameters ℜ0, ℜA1, ℜC1, ℜC2, and ℜA2 for system (5.1) are given by:
ℜ0=d1d2(η1d3+η2ε)S0b1b2b3ε, ℜA1=d3τεζˆI3, ℜC1=d1d2σρ(η1d3+η2ε)b1b2b3[π(η1d3+η2ε)+αεσ],ℜC2=d1d2σρ(η1ζσ+η2τπ)b1b2b3π[ατσ+η1ζσ+η2τπ], and ℜA2=d3πτεσζ, |
where
ˆI3=d1d2η2ρτ−b1b2b3(ζη1+ατ)+√−4b1b2b3d1d2Cη1τ+(η2ρτd1d2−b1b2b3(ζη1+ατ))22b1b2b3η1τ. |
Table 1 contains the values of the parameters of model (5.1).
Parameter | Value | Parameter | Value | Parameter | Value |
ρ | 10 | b3 | 0.8 | γ | 0.05 |
α | 0.01 | d1 | 0.2 | τ | Varied |
η1 | Varied | d2 | 1 | ζ | 0.1 |
η2 | Varied | d3 | 5 | σ | Varied |
b1 | 0.6 | μ | 0.1 | π | 0.1 |
b2 | 0.7 | ε | 1.5 |
The results of Theorems 1–5 will be investigated by choosing the values of η1, η2, τ and σ under three different initial conditions for model (5.1) as follows:
Initial–1: (S(0),I1(0),I2(0),I3(0),V(0),A(0),C(0))=(800,3,1,1,2,3,10), (Solid lines in the figures)
Initial–2: (S(0),I1(0),I2(0),I3(0),V(0),A(0),C(0))=(700,0.5,2,2,3,4,5), (Dashed lines in the figures)
Initial–3: (S(0),I1(0),I2(0),I3(0),V(0),A(0),C(0))=(300,0.1,0.5,0.5,1.5,2,2.5). (Dotted lines in the figures)
Stability of Ɖ0: η1=η2=0.0001, τ=0.001 and σ=0.01. For this set of parameters, we have ℜ0=0.26<1, ℜA1=0.10<1, ℜC1=0.18<1 and ℜC2=0.31<1. Figure 1 illustrates that the solution trajectories starting from different initial conditions reach the equilibrium Ɖ0=(1000,0,0,0,0,0,0). This ensures that Ɖ0 is globally asymptotically stable according to the result of Theorem 1. In this situation the viruses will be died out.
Stability of ˉƉ: η1=η2=0.001, τ=0.001 and σ=0.01. With such choice we get, ℜA1=0.18<1 and ℜC1=0.48<1<ℜ0=2.58 and ˉƉ exists with ˉƉ=(387.68,10.21,2.92,3.65,12.15,0,0). Thus, Lemma 1 is verified. Figure 2 shows that the solution trajectories starting from different initial conditions tend to ˉƉ and this support Theorem 2. This case represents the persistence of the viruses but with inhibited humoral and cell-mediated immune responses.
Stability of ˆƉ: η1=η2=0.001, τ=0.07 and σ=0.05. Then, we calculate ℜ0=2.58>1, ℜA1=2.94>1 and ℜC2=0.76<1. The numerical results show that ˆƉ=(787.99,3.53,1.01,1.26,1.43,58.34,0) which confirm Lemma 1. The global stability result given in Theorem 3 is illustrated by Figure 3. This situation represents the case when the infection is chronic and the humoral immune response is active, while the cell-mediated immune response is inhibited.
Stability of ∨Ɖ: η1=η2=0.001, τ=0.05 and σ=0.2. Then, we calculate ℜ0=2.58>1, ℜC1=2.12>1 and ℜA2=0.83<1. The results presented in Lemma 1 and Theorem 4 show that the equilibrium ∨Ɖ exists and it is globally asymptotically stable. Figure 4 supports the results of Theorem 4, where the solution trajectories of the system starting from different initial conditions reach the equilibrium point ∨Ɖ =(821.91,2.97,0.85,0.50,1.67,0,8.96). This situation represents the case when the infection is chronic and the cell-mediated immune response is active, while the humoral immune response is inhibited.
Stability of ˜Ɖ: η1=η2=0.001, τ=0.07 and σ=0.2. Then, we calculate ℜ0=2.58>1 and ℜA2=1.17>1, ℜC2=1.92>1. The numerical results show that ˜Ɖ=(838.32,2.69,0.77,0.50,1.43,5.00,7.40) which ensure Lemma 1. Moreover, the global stability result given in Theorem 5 is demonstrated in Figure 5. It can be seen that the solution trajectories of the system starting from different initial conditions converge to the equilibrium ˜Ɖ. This situation represents the case when the infection is chronic and both immune responses are active.
We consider system (5.1) under the effect of two types of treatment as:
{˙S=ρ−αS−(1−ϵ1)η1SV−(1−ϵ2)η2SI3,˙I1=(1−ϵ1)η1SV+(1−ϵ2)η2SI3−b1I1,˙I2=d1I1−b2I2,˙I3=d2I2−b3I3−μCI3,˙V=d3I3−γAV−εV,˙A=τAV−ζA,˙C=σCI3−πC, | (5.2) |
where, the parameter ϵ1∈[0,1] is the efficacy of antiretroviral therapy in blocking infection by virus-to-cell mechanism, and ϵ2∈[0,1] is the efficacy of therapy in blocking infection by cell-to-cell mechanism [47].
The basic reproduction number of system (5.2) is given by
ℜ0,(5.2)(ϵ1,ϵ2)=(1−ϵ1)ℜ01+(1−ϵ2)ℜ02, |
where
ℜ01=d1d2d3η1S0b1b2b3ε, ℜ02=d1d2η2S0b1b2b3. |
When the cell-to-cell transmission is neglected, system (5.2) leads to the following system:
{˙S=ρ−αS−(1−ϵ1)η1SV,˙I1=(1−ϵ1)η1SV−b1I1,˙I2=d1I1−b2I2,˙I3=d2I2−b3I3−μCI3,˙V=d3I3−γAV−εV,˙A=τAV−ζA,˙C=σCI3−πC. | (5.3) |
The basic reproduction number of system (5.3) is given by
ℜ0,(5.3)(ϵ1)=(1−ϵ1)ℜ01. |
Without loss of generality we let ϵ1=ϵ2=ϵ. Now we calculate the minimum drug efficacy ϵ which stabilize the infection-free equilibrium for systems (5.2) and (5.3). For system (5.2) one can determine the minimum drug efficacy ϵmin(5.2) such that ℜ0,(5.2)(ϵ)≤1 for all ϵmin(5.2)≤ϵ≤1 as:
ϵmin(5.2)=max{1−1ℜ01+ℜ02,0}. | (5.4) |
For system (5.3) the minimum drug efficacy ϵmin(5.3) such that ℜ0,(5.3)(ϵ)≤1, ϵmin(5.3)≤ϵ≤1 is given by:
ϵmin(5.3)=max{1−1ℜ01,0}. | (5.5) |
Comparing Eqs. (5.5) and (5.4) we get that ϵmin(5.3)≤ϵmin(5.2). Therefore, if we apply drugs with ϵ such that ϵmin(5.3)≤ϵ<ϵmin(5.2), this guarantee that ℜ0,(5.3)(ϵ)≤1 and then Ɖ0 of system (5.3) is globally asymptotically stable, however, ℜ0,(5.2)>1 and then Ɖ0 of system (5.2) is unstable. Therefore, more accurate drug efficacy ϵ is determined when using the model with both virus-to-cell and cell-to-cell transmissions. This shows the importance of considering the effect of the cell-to-cell transmission in the virus dynamics.
Now we perform numerical simulation for both systems (5.2) and (5.3). Using the values given in Table 1 and choosing η1=0.001, η2=0.005, τ=0.07 and σ=0.2. Then we get
ϵmin(5.3)=0.496, ϵmin(5.2)=0.7984. |
Now we select ϵ=0.5 and choose the initial condition as follows:
Initial–4: (S(0),I1(0),I2(0),I3(0),V(0),A(0),C(0))=(900,3,1,0.5,2,3,5).
From Figure 6 we can see that the trajectory of model (5.3) tends to Ɖ0, while the trajectory of model (5.2) tends to ˜Ɖ. It means that if one design treatment using model (5.3) where the cell-to-cell transmission is neglected, then this treatment will not suffice to clear the viruses from the body.
On the other hand, we choose ϵ=0.8 and consider the following initial condition:
Initial–5: (S(0),I1(0),I2(0),I3(0),V(0),A(0),C(0))=(920,0.5,0.5,0.5,2,3,3).
From Figure 7 we can see that the trajectories of both systems (5.2) and (5.3) tend to Ɖ0. Therefore, this treatment will suffice to clear the viruses from the body.
In this paper, we formulated and analyzed a virus dynamics model with both CTL and humoral immune responses. We incorporated both virus-to-cell and cell-to-cell transmissions. We assumed that the infected cells pass through n stages to produce mature viruses. We showed that the solutions of the system are nonnegative and bounded, which ensures the well-posedness of the proposed model. Further, we obtained five threshold parameters, ℜ0 (the basic infection reproductive ratio), ℜA1 (the active humoral immunity reproductive ratio), ℜC1 (the active cell-mediated immunity reproductive ratio), ℜC2 (the competed cell-mediated immunity reproductive ratio), and ℜA2 (the competed humoral immunity reproductive ratio). The global asymptotic stability of the five equilibria Ɖ0, ˉƉ, ˆƉ, ∨Ɖ, ˜Ɖ was investigated by constructing Lyapunov functions and applying LaSalle's invariance principle. To support our theoretical results, we conducted some numerical simulations. We note that the incorporation of cell-to-cell transmission mechanism into the viral infection model increases the basic reproduction number ℜ0, since ℜ0=ℜ01+ℜ02>ℜ01. Therefore, neglecting the cell-to-cell transmission will lead to under-evaluated basic reproduction number. Model with two types of treatment was presented. We showed that more accurate drug efficacy which is required to clear the virus from the body is calculated by using our proposed model.
There are some factors that can extend our model (1.2):
a. The infected cells may begin to present the viral antigen earlier than when they reach the terminal stage n (i.e. at stage m where m≤n). Therefore, infected cells Im, Im+1,...,In are subject to be targeted by the CTL immune response.
b. Model (1.2) is formulated by assuming that the virus is purely lytic, that is, only the bursting cells are capable of releasing the free virions. However, many viruses are somewhat mixed, in the sense that they are partially lytic and partially budding, where the release of free virions can be from the infected cells Im, Im+1,...,In.
c. The cell-to-cell infection mechanism can also be expanded to the contact between susceptible cells with infected cells Im, Im+1,...,In.
d. The loss of virions upon the infection could also be added to the model. In fact, there is some speculation that the virions may be indiscriminately entering not only the susceptible cells, but also the cells that are already infected [26,53].
Then, taking into account the above factors will leads to the following model:
{˙S(t)=ρ−αS(t)−η1S(t)V(t)−n∑k=mηkS(t)Ik(t),˙I1(t)=η1S(t)V(t)+η2S(t)In(t)−b1I1(t),˙I2(t)=d1I1(t)−b2I2(t)⋮˙Im−1(t)=dm−2Im−2(t)−bm−1Im−1(t),˙Ik(t)=dk−1Ik−1(t)−bkIk(t)−μkC(t)Ik(t), k=m,m+1,...,n,˙V(t)=n∑k=mδkIk(t)−γA(t)V(t)−εV(t)−ˉη1S(t)V(t)−V(t)n∑k=1ϰkIk(t),˙A(t)=τA(t)V(t)−ζA(t),˙C(t)=n∑k=mσkC(t)Ik(t)−πC(t), | (6.1) |
where, n∑k=mηkSIk represent the incidence rates due to the contact of the infected cells Im,Im+1,...,In with susceptible cells. The term ˉη1SV is the loss of virus upon entry of a susceptible cell. The term Vn∑k=1ϰkIk represents the absorption of free virions into already infected cells I1,I2,...,In. The production rate of the viruses and the activation rate of the CTL cells are modeled by n∑k=mδkIk and n∑k=mσkCIk, respectively. The k-stage infected cells Ik, are attacked by CTL cells at rate μkCIk, k=m,m+1,...,n. Analysis of system (6.1) is not straightforward, therefore we leave it for future works.
It is commonly observed that in viral infection processes, time delay is inevitable. Herz et al. [59] formulated an HIV infection model with intracellular delay and they obtained the analytic expression of the viral load decline under treatment and used it to analyze the viral load decline data in patients. Several viral infection models presented in the literature incorporated discrete delays (see, e.g., [36] and [44]) or distributed delays (see, e.g., [7,23] and [48,49,50]). In these papers, the global stability of equilibria was proven by utilizing global Lyapunov functional that was motivated by the work in [51] and [52]. Model (6.1) can be extended to incorporate distributed time delays. Moreover, considering age structure of the infected class or diffusion in the virus dynamics model will lead to PDE model [54,55,56,57,58]. These extensions require more investigations, therefore we leave it for future works.
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (DG–14–247–1441). The authors, therefore, gratefully acknowledge the DSR technical and financial support.
There is no conflicts of interest.
Proof of Theorem 1. Constructing a Lyapunov function:
Φ0(S,I1,...,In,V,A,C)=S0ϝ(SS0)+n∑k=1(k−1∏i=1bidi)Ik+η1S0εV+γη1S0τεA+μσ(n−1∏i=1bidi)C. | (6.2) |
It is seen that, Φ0(S,I1,...,In,V,A,C)>0 for all S,I1,...,In,V,A,C>0, and Φ0 has a global minimum at Ɖ0. We calculate dΦ0dt along the solutions of model (1.2) as:
dΦ0dt=(1−S0S)˙S+˙I1+n−1∑k=2(k−1∏i=1bidi)˙Ik+(n−1∏i=1bidi)˙In+η1S0ε˙V+γη1S0τε˙A+μσ(n−1∏i=1bidi)˙C. | (6.3) |
Using (4.3), we have
n∑k=1(k−1∏i=1bidi)˙Ik=η1SV+η2SIn−b1I1+n−1∑k=2(k−1∏i=1bidi)(dk−1Ik−1−bkIk)+(n−1∏i=1bidi)(dn−1In−1−bnIn−μCIn)=η1SV+η2SIn−(n∏i=1bidi)dnIn−μ(n−1∏i=1bidi)CIn. |
Then,
dΦ0dt=(1−S0S)(ρ−αS)+η1S0εdnIn+η2S0In−(n∏i=1bidi)dnIn−γζη1S0τεA−μπσ(n−1∏i=1bidi)C. |
Using S0=ρ/α, we obtain
dΦ0dt=−α(S−S0)2S+(n∏i=1bidi)dn(ℜ0−1)In−γζη1S0τεA−μπσ(n−1∏i=1bidi)C. | (6.4) |
Therefore, dΦ0dt≤0 for all S,In,A,C>0 with equality holding when S(t)=S0 and In(t)=A(t)=C(t)=0 for all t. Let Υ0={(S(t),I1(t),...,In(t),V(t),A(t),C(t)):dΦ0dt=0} and Υ′0 is the largest invariant subset of Υ0. We note that, the solutions of system (1.2) are confined to Υ0 [46]. The set Υ0 is invariant and contains elements which satisfy In(t)=0. Then, ˙In(t)=0 and from Eq. 3.4 we have
0=˙In(t)=dn−1In−1(t). |
It follows that, In−1(t)=0 for all t. Since we have In−1(t)=0, then ˙In−1(t)=0 and from Eq. 3.3, we have ˙In−1(t)=dn−2In−2=0 which yields In−2(t)=0. Consequently, we obtain Ik(t)=0, where k=1,...,n. Moreover, since S(t)=S0 we have ˙S(t)=0 and Eq. 3.1 implies that
0=˙S(t)=ρ−αS0−η1S0V. |
which insures that V(t)=0. Noting that ℜ0≤1, then Ɖ0 is globally asymptotically stable using LaSalle's invariance principle.
Proof of Theorem 2. Let us define a function Φ1(S,I1,...,In,V,A,C) as:
Φ1=ˉSϝ(SˉS)+n∑k=1(k−1∏i=1bidi)ˉIkϝ(IkˉIk)+η1ˉSεˉVϝ(VˉV)+γη1ˉSτεA+μσ(n−1∏i=1bidi)C. |
Calculating dΦ1dt as:
dΦ1dt=(1−ˉSS)(ρ−αS−η1SV−η2SIn)+(1−ˉI1I1)(η1SV+η2SIn−b1I1)+n−1∑k=2(k−1∏i=1bidi)(1−ˉIkIk)(dk−1Ik−1−bkIk)+(n−1∏i=1bidi)(1−ˉInIn)(dn−1In−1−bnIn−μCIn)+η1ˉSε(1−ˉVV)(dnIn−γAV−εV)+γη1ˉSτε(τVA−ζA)+μσ(n−1∏i=1bidi)(σInC−πC). | (6.5) |
Collecting terms of Eq. 6.5 and using Eqs. 4.2 and 4.3, we derive
dΦ1dt=(1−ˉSS)(ρ−αS)+η2ˉSIn−(n∏i=1bidi)dnIn−η1SVˉI1I1−η2SInˉI1I1−n∑k=2(k−1∏i=1bidi)dk−1Ik−1ˉIkIk+n∑k=1(k∏i=1bidi)dkˉIk+(n−1∏i=1bidi)μCˉIn+η1ˉSεdnIn−η1ˉSεdnInˉVV+η1ˉSˉV+η1ˉSεγAˉV−γη1ˉSτεζA−μπσ(n−1∏i=1bidi)C. | (6.6) |
Using the equilibrium conditions for ˉƉ:
ρ=αˉS+η1ˉSˉV+η2ˉSˉIn, η1ˉSˉV+η2ˉSˉIn=(k∏i=1bidi)dkˉIk=(n∏i=1bidi)εˉV, k=1,...,n. |
We obtain
dΦ1dt=(1−ˉSS)(αˉS−αS)+(η1ˉSˉV+η2ˉSˉIn)(1−ˉSS)+η2ˉSIn−(n∏i=1bidi)dnIn+η1ˉSεdnIn−η1ˉSˉVSVˉI1ˉSˉVI1−η2ˉSˉInSInˉI1ˉSˉInI1−(η1ˉSˉV+η2ˉSˉIn)n∑k=2Ik−1ˉIkˉIk−1Ik+n(η1ˉSˉV+η2ˉSˉIn)+η1ˉSˉV−η1ˉSεdnInˉVV+μ(n−1∏i=1bidi)(ˉIn−πσ)C+η1ˉSγε(ˉV−ζτ)A. | (6.7) |
Since we have
ˉS=(n∏i=1bidi)εdnη1dn+η2ε, |
then
η2ˉSIn−(n∏i=1bidi)dnIn+η1ˉSεdnIn=0. |
Also we have when k=n,
dnε=ˉVˉIn⟹η1ˉSεdnInˉVV=η1ˉSˉVInˉVˉInV. |
Therefor Eq. 6.7 becomes
dΦ1dt=−α(S−ˉS)2S+η1ˉSˉV[(n+2)−ˉSS−SVˉI1ˉSˉVI1−n∑k=2Ik−1ˉIkˉIk−1Ik−InˉVˉInV]+η2ˉSˉIn[(n+1)−ˉSS−SInˉI1ˉSˉInI1−n∑k=2Ik−1ˉIkˉIk−1Ik]+μ(n−1∏i=1bidi)(ˉIn−ˇIn)C+η1ˉSγε(ˉV−ˆV)A=−α(S−ˉS)2S+η1ˉSˉV[(n+2)−ˉSS−SVˉI1ˉSˉVI1−n∑k=2Ik−1ˉIkˉIk−1Ik−InˉVˉInV]+η2ˉSˉIn[(n+1)−ˉSS−SInˉI1ˉSˉInI1−n∑k=2Ik−1ˉIkˉIk−1Ik]+εασ+π(η1dn+η2ε)σ(η1dn+η2ε)(ℜC1−1)C+η1ˉSγε(ˉV−ˆV)A. | (6.8) |
Since the arithmetical mean is greater than or equal to the geometrical mean, then
ˉSS+SVˉI1ˉSˉVI1+n∑k=2Ik−1ˉIkˉIk−1Ik+InˉVˉInV≥n+2 and ˉSS+SInˉI1ˉSˉInI1+n∑k=2Ik−1ˉIkˉIk−1Ik≥n+1. |
From Lemma 2 we have ˉV<ˆV and since ℜC1≤1<ℜ0 then dΦ1dt≤0 for all S,Ik,V,A,C>0 with equality holding when S(t)=ˉS, Ik(t)=ˉIk, k=1,2,...,n, V(t)=ˉV, and A(t)=C(t)=0 for all t. It can be easily verified that Υ′1={ˉƉ} is the largest invariant subset of Υ1={(S(t),I1(t),...,In(t),V(t),A(t),C(t)):dΦ1dt=0}[46]. Then, ˉƉ is globally asymptotically stable using LaSalle's invariance principle.
Proof of Theorem 3. The candidate Lyapunov function is
Φ2(S,I1,...,In,V,A,C)=ˆSϝ(SˆS)+n∑k=1(k−1∏i=1bidi)ˆIkϝ(IkˆIk)+η1ˆSˆVdnˆInˆVϝ(VˆV)+γη1ˆSˆVτdnˆInˆAϝ(AˆA)+μσ(n−1∏i=1bidi)C. | (6.9) |
We calculate dΦ2dt as:
dΦ2dt=(1−ˆSS)(ρ−αS−η1SV−η2SIn)+(1−ˆI1I1)(η1SV+η2SIn−b1I1)+n−1∑k=2(k−1∏i=1bidi)(1−ˆIkIk)(dk−1Ik−1−bkIk)+(n−1∏i=1bidi)(1−ˆInIn)(dn−1In−1−bnIn−μCIn)+η1ˆSˆVdnˆIn(1−ˆVV)(dnIn−γAV−εV)+γη1ˆSˆVτdnˆIn(1−ˆAA)(τVA−ζA)+μσ(n−1∏i=1bidi)(σInC−πC). | (6.10) |
Collecting terms of Eq. 6.10 and using Eqs. 4.2 and 4.3, we derive
dΦ2dt=(1−ˆSS)(ρ−αS)+η1ˆSV+η2ˆSIn−(n∏i=1bidi)dnIn−η1SVˆI1I1−η2SInˆI1I1−n∑k=2(k−1∏i=1bidi)dk−1Ik−1ˆIkIk+n∑k=1(k∏i=1bidi)dkˆIk+(n−1∏i=1bidi)μCˆIn+η1ˆSˆVInˆIn−η1ˆSˆVdnˆInεV−η1ˆSˆVInˆVˆInV+η1ˆSˆVdnˆInεˆV+η1ˆSˆVdnˆInγˆVA−γη1ˆSˆVτdnˆInζA−γη1ˆSˆVdnˆInVˆA+γη1ˆSˆVτdnˆInζˆA−μπσ(n−1∏i=1bidi)C. | (6.11) |
Using the equilibrium conditions for ˆƉ:
ρ=αˆS+η1ˆSˆV+η2ˆSˆIn, η1ˆSˆV+η2ˆSˆIn=(k∏i=1bidi)dkˆIk=(n∏i=1bidi)[εˆV+γˆVˆA], k=1,...,n. | (6.12) |
We obtain
dΦ2dt=(1−ˆSS)(αˆS−αS)+(η1ˆSˆV+η2ˆSˆIn)(1−ˆSS)+η1ˆSˆVVˆV+η2ˆSˆInInˆIn−(n∏i=1bidi)dnIn−η1ˆSˆVSVˆI1ˆSˆVI1−η2ˆSˆInSInˆI1ˆSˆInI1−(η1ˆSˆV+η2ˆSˆIn)n∑k=2Ik−1ˆIkˆIk−1Ik+n(η1ˆSˆV+η2ˆSˆIn)+η1ˆSˆVInˆIn−η1ˆSˆVdnˆIn[εˆV+γˆVˆA]VˆV−η1ˆSˆVInˆVˆInV+η1ˆSˆVdnˆIn[εˆV+γˆVˆA]+μ(n−1∏i=1bidi)(ˆIn−πσ)C=−α(S−ˆS)2S+(η1ˆSˆV+η2ˆSˆIn)(1−ˆSS)+(η1ˆSˆV+η2ˆSˆIn)InˆIn−(n∏i=1bidi)dnIn−η1ˆSˆVSVˆI1ˆSˆVI1−η2ˆSˆInSInˆI1ˆSˆInI1−(η1ˆSˆV+η2ˆSˆIn)n∑k=2Ik−1ˆIkˆIk−1Ik+n(η1ˆSˆV+η2ˆSˆIn)−η1ˆSˆVInˆVˆInV+η1ˆSˆV+μ(n−1∏i=1bidi)(ˆIn−πσ)C. | (6.13) |
Using Eq. 6.12 in case of k=n we get
(η1ˆSˆV+η2ˆSˆIn)InˆIn−(n∏i=1bidi)dnIn=(η1ˆSˆV+η2ˆSˆIn)InˆIn−(n∏i=1bidi)dnˆInInˆIn=0. |
Thus, Eq. 6.13 will become
dΦ2dt=−α(S−ˆS)2S+(η1ˆSˆV+η2ˆSˆIn)(1−ˆSS)−η1ˆSˆVSVˆI1ˆSˆVI1−η2ˆSˆInSInˆI1ˆSˆInI1−(η1ˆSˆV+η2ˆSˆIn)n∑k=2Ik−1ˆIkˆIk−1Ik+n(η1ˆSˆV+η2ˆSˆIn)+η1ˆSˆV−η1ˆSˆVInˆVˆInV+μ(n−1∏i=1bidi)(ˆIn−˜In)C. | (6.14) |
Eq. 6.14 can be written as
dΦ2dt=−α(S−ˆS)2S+η1ˆSˆV[(n+2)−ˆSS−SVˆI1ˆSˆVI1−n∑k=2Ik−1ˆIkˆIk−1Ik−InˆVˆInV]+η2ˆSˆIn[(n+1)−ˆSS−SInˆI1ˆSˆInI1−n∑k=2Ik−1ˆIkˆIk−1Ik]+μ(n−1∏i=1bidi)(ˆIn−˜In)C. | (6.15) |
Thus, if ℜC2≤1, then ˜Ɖ dose not exist since ˜C=bnμ(ℜC2−1)≤0. This guarantee that ˙C(t)=σ(In(t)−πσ)C(t)=σ(In(t)−˜In)C(t)≤0 for all C>0, which implies that ˆIn<˜In. Hence dΦ2dt≤0 for all S,Ik,V,A,C>0 with equality holding when S(t)=ˆS, Ik(t)=ˆIk, k=1,2,...,n, V(t)=ˆV, and C(t)=0 for all t. We note that, the solutions of system (1.2) are tend to Υ′2 the largest invariant subset of Υ2={(S(t),I1(t),...,In(t),V(t),A(t),C(t)):dΦ2dt=0} [46]. For each element of Υ′2 we have In(t)=ˆIn, V(t)=ˆV, then ˙V(t)=0 and from Eq. 3.5 we have
0=˙V(t)=dnˆIn−γA(t)ˆV−εˆV=0, |
which gives A(t)=ˆA. Therefore, Υ′2={ˆƉ}. Applying LaSalle's invariance principle we get ˆƉ is globally asymptotically stable.
Proof of Theorem 4. Define a function Φ3(S,I1,...,In,V,A,C) as:
Φ3=ˇSϝ(SˇS)+n∑k=1(k−1∏i=1bidi)ˇIkϝ(IkˇIk)+η1ˇSεˇVϝ(VˇV)+γη1ˇSτεA+μσ(n−1∏i=1bidi)ˇCϝ(CˇC). |
We calculate dΦ3dt as:
dΦ3dt=(1−ˇSS)(ρ−αS−η1SV−η2SIn)+(1−ˇI1I1)(η1SV+η2SIn−b1I1)+n−1∑k=2(k−1∏i=1bidi)(1−ˇIkIk)(dk−1Ik−1−bkIk)+(n−1∏i=1bidi)(1−ˇInIn)(dn−1In−1−bnIn−μCIn)+η1ˇSε(1−ˇVV)(dnIn−γAV−εV)+γη1ˇSτε(τVA−ζA)+μσ(n−1∏i=1bidi)(1−ˇCC)(σInC−πC). | (6.16) |
Collecting terms of Eq. 6.16 and using Eqs. 4.2 and 4.3, we derive
dΦ3dt=(1−ˇSS)(ρ−αS)+η2ˇSIn−(n∏i=1bidi)dnIn−η1SVˇI1I1−η2SInˇI1I1−n∑k=2(k−1∏i=1bidi)dk−1Ik−1ˇIkIk+n∑k=1(k∏i=1bidi)dkˇIk+(n−1∏i=1bidi)μCˇIn+η1ˇSεdnIn−η1ˇSεdnInˇVV+η1ˇSˇV+η1ˇSεγAˇV−γη1ˇSτεζA−μπσ(n−1∏i=1bidi)C−μ(n−1∏i=1bidi)ˇCIn+μπσ(n−1∏i=1bidi)ˇC. | (6.17) |
Using the equilibrium conditions for ∨Ɖ:
ρ=αˇS+η1ˇSˇV+η2ˇSˇIn, ˇIn=πσ, ˇV=dnπεσ=dnεˇIn,η1ˇSˇV+η2ˇSˇIn=(k−1∏i=1bidi)dk−1ˇIk−1=(n∏i=1bidi)dnˇIn+μ(n−1∏i=1bidi)ˇCˇIn, k=1,...,n, |
we obtain
dΦ3dt=(1−ˇSS)(αˇS−αS)+(η1ˇSˇV+η2ˇSˇIn)(1−ˇSS)+η2ˇSIn−(n∏i=1bidi)dnIn+η1ˇSεdnIn−μ(n−1∏i=1bidi)ˇCIn−η1ˇSˇVSVˇI1ˇSˇVI1−η2ˇSˇInSInˇI1ˇSˇInI1−(η1ˇSˇV+η2ˇSˇIn)n∑k=2Ik−1ˇIkˇIk−1Ik+n(η1ˇSˇV+η2ˇSˇIn)−η1ˇSˇVInˇVˇInV+η1ˇSˇV+η1ˇSγζετ(τˇVζ−1)A. | (6.18) |
Since we have in case of k=n:
η2ˇSIn−(n∏i=1bidi)dnIn+η1ˇSεdnIn−μ(n−1∏i=1bidi)ˇCIn=[η2ˇSˇIn−(n∏i=1bidi)dnˇIn+η1ˇSεdnˇIn−μ(n−1∏i=1bidi)ˇCˇIn]InˇIn=[η1ˇSˇV+η2ˇSˇIn−(n∏i=1bidi)dnˇIn−μ(n−1∏i=1bidi)ˇCˇIn]InˇIn=0. |
Then,
dΦ3dt=−α(S−ˇS)2S+η1ˇSˇV[(n+2)−ˇSS−SVˇI1ˇSˇVI1−n∑k=2Ik−1ˇIkˇIk−1Ik−InˇVˇInV]+η2ˇSˇIn[(n+1)−ˇSS−SInˇI1ˇSˇInI1−n∑k=2Ik−1ˇIkˇIk−1Ik]+η1ˇSγζετ(ℜA2−1)A. | (6.19) |
Hence, if ℜA2=τˇVζ≤1, then dΦ3dt≤0 for all S,Ik,V,A,C>0 with equality holding when S(t)=ˇS, Ik(t)=ˇIk, k=1,2,...,n, V(t)=ˇV and A(t)=0 for all t. It can be easily verified that the largest invariant subset of Υ3={(S(t),I1(t),...,In(t),V(t),A(t),C(t)):dΦ3dt=0} is Υ′3={∨Ɖ} [46]. Applying LaSalle's invariance principle we get that ∨Ɖ is globally asymptotically stable.
Proof of Theorem 5. Define Φ4(S,I1,...,In,V,A,C) as:
Φ4=˜Sϝ(S˜S)+n∑k=1(k−1∏i=1bidi)˜Ikϝ(Ik˜Ik)+η1˜S˜Vdn˜In˜Vϝ(V˜V)+γη1˜S˜Vτdn˜In˜Aϝ(A˜A)+μσ(n−1∏i=1bidi)˜Cϝ(C˜C). |
Calculating dΦ4dt as:
dΦ4dt=(1−˜SS)(ρ−αS−η1SV−η2SIn)+(1−˜I1I1)(η1SV+η2SIn−b1I1)+n−1∑k=2(k−1∏i=1bidi)(1−˜IkIk)(dk−1Ik−1−bkIk)+(n−1∏i=1bidi)(1−˜InIn)(dn−1In−1−bnIn−μCIn)+η1˜S˜Vdn˜In(1−˜VV)(dnIn−γAV−εV)+γη1˜S˜Vτdn˜In(1−˜AA)(τVA−ζA)+μσ(n−1∏i=1bidi)(1−˜CC)(σInC−πC). | (6.20) |
Collecting terms of Eq. 6.20 and using Eqs. 4.2 and 4.3, we obtain
dΦ4dt=(1−˜SS)(ρ−αS)+η1˜SV+η2˜SIn−(n∏i=1bidi)dnIn−η1SV˜I1I1−η2SIn˜I1I1−n∑k=2(k−1∏i=1bidi)dk−1Ik−1˜IkIk+n∑k=1(k∏i=1bidi)dk˜Ik+(n−1∏i=1bidi)μC˜In+η1˜S˜VIn˜In−η1˜S˜Vdn˜InεV−η1˜S˜VIn˜V˜InV+η1˜S˜Vdn˜Inε˜V+η1˜S˜Vdn˜Inγ˜VA−γη1˜S˜Vτdn˜InζA−γη1˜S˜Vdn˜InVˆA+γη1˜S˜Vτdn˜InζˆA−μπσ(n−1∏i=1bidi)C−μ(n−1∏i=1bidi)˜CIn+μπσ(n−1∏i=1bidi)˜C. | (6.21) |
Using the equilibrium conditions for ˜Ɖ:
ρ=α˜S+η1˜S˜V+η2˜S˜In, ˜In=πσ=ˇIn, ˜V=ζτ=ˆV, dn˜In=ε˜V+γ˜V˜A,η1˜S˜V+η2˜S˜In=(k−1∏i=1bidi)dk−1˜Ik−1=(n∏i=1bidi)dn˜In+μ(n−1∏i=1bidi)˜C˜In, k=1,...,n. |
We obtain
dΦ4dt=(1−˜SS)(α˜S−αS)+(η1˜S˜V+η2˜S˜In)(1−˜SS)+η1˜SV+η2˜SIn−(n∏i=1bidi)dnIn−η1˜S˜VSV˜I1˜S˜VI1−η2˜S˜InSIn˜I1˜S˜InI1−(η1˜S˜V+η2˜S˜In)n∑k=2Ik−1˜Ik˜Ik−1Ik+n(η1˜S˜V+η2˜S˜In)+η1˜S˜VIn˜In−η1˜S˜Vdn˜In[ε˜V+γ˜V˜A]V˜V−η1˜S˜VIn˜V˜InV+η1˜S˜Vdn˜In[ε˜V+γ˜V˜A]−μ(n−1∏i=1bidi)˜CIn. | (6.22) |
Since we have
η1˜SV−η1˜S˜Vdn˜In[ε˜V+γ˜V˜A]V˜V=0, |
and
η2˜SIn−(n∏i=1bidi)dnIn−μ(n−1∏i=1bidi)˜CIn+η1˜S˜VIn˜In=[η1˜S˜V+η2˜S˜In−(n∏i=1bidi)dn˜In−μ(n−1∏i=1bidi)˜C˜In]In˜In=0. |
Then, Eq. 6.22 will be reduced to the form
dΦ4dt=−α(S−˜S)2S+(η1˜S˜V+η2˜S˜In)(1−˜SS)−η1˜S˜VSV˜I1˜S˜VI1−η2˜S˜InSIn˜I1˜S˜InI1−(η1˜S˜V+η2˜S˜In)n∑k=2Ik−1˜Ik˜Ik−1Ik+n(η1˜S˜V+η2˜S˜In)−η1˜S˜VIn˜V˜InV+η1˜S˜V=−α(S−˜S)2S+η1˜S˜V[(n+2)−˜SS−SV˜I1˜S˜VI1−n∑k=2Ik−1˜Ik˜Ik−1Ik−In˜V˜InV]+η2˜S˜In[(n+1)−˜SS−SIn˜I1˜S˜InI1−n∑k=2Ik−1˜Ik˜Ik−1Ik]. | (6.23) |
Hence, dΦ4dt≤0 for all S,Ik,V,A,C>0 with equality holding when S(t)=˜S, Ik(t)=˜Ik, k=1,2,...,n, and V(t)=˜V for all t. It can be easily verified that the largest invariant subset of Υ4={(S(t),I1(t),...,In(t),V(t),A(t),C(t)):dΦ3dt=0} is Υ′4={˜Ɖ} [46]. LaSalle's invariance principle implies that ˜Ɖ is globally asymptotically stable.
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1. | N.H. AlShamrani, A.M. Elaiw, H. Batarfi, A.D. Hobiny, H. Dutta, Global stability analysis of a general nonlinear scabies dynamics model, 2020, 138, 09600779, 110133, 10.1016/j.chaos.2020.110133 |
Parameter | Value | Parameter | Value | Parameter | Value |
ρ | 10 | b3 | 0.8 | γ | 0.05 |
α | 0.01 | d1 | 0.2 | τ | Varied |
η1 | Varied | d2 | 1 | ζ | 0.1 |
η2 | Varied | d3 | 5 | σ | Varied |
b1 | 0.6 | μ | 0.1 | π | 0.1 |
b2 | 0.7 | ε | 1.5 |