
The emergence of cooperative quasi-species consortia (QS-C) thinking from the more accepted quasispecies equations of Manfred Eigen, provides a conceptual foundation from which concerted action of RNA agents can now be understood. As group membership becomes a basic criteria for the emergence of living systems, we also start to understand why the history and context of social RNA networks become crucial for survival and function. History and context of social RNA networks also lead to the emergence of a natural genetic code. Indeed, this QS-C thinking can also provide us with a transition point between the chemical world of RNA replicators and the living world of RNA agents that actively differentiate self from non-self and generate group identity with membership roles. Importantly the social force of a consortia to solve complex, multilevel problems also depend on using opposing and minority functions. The consortial action of social networks of RNA stem-loops subsequently lead to the evolution of cellular organisms representing a tree of life.
Citation: Luis P. Villarreal, Guenther Witzany. Social Networking of Quasi-Species Consortia drive Virolution via Persistence[J]. AIMS Microbiology, 2021, 7(2): 138-162. doi: 10.3934/microbiol.2021010
The emergence of cooperative quasi-species consortia (QS-C) thinking from the more accepted quasispecies equations of Manfred Eigen, provides a conceptual foundation from which concerted action of RNA agents can now be understood. As group membership becomes a basic criteria for the emergence of living systems, we also start to understand why the history and context of social RNA networks become crucial for survival and function. History and context of social RNA networks also lead to the emergence of a natural genetic code. Indeed, this QS-C thinking can also provide us with a transition point between the chemical world of RNA replicators and the living world of RNA agents that actively differentiate self from non-self and generate group identity with membership roles. Importantly the social force of a consortia to solve complex, multilevel problems also depend on using opposing and minority functions. The consortial action of social networks of RNA stem-loops subsequently lead to the evolution of cellular organisms representing a tree of life.
Smallpox, cholera, AIDS, COVID-19 and other infectious disease epidemics have wreaked immense havoc on the economy and way of life of the populace. Many mathematical models have been developed by researchers to explore the dynamical behavior of infectious diseases and thus control their transmission and gain a deeper understanding of these diseases [1,2,3,4,5,6]; among which, higher-order networks are widely used the spreading dynamics [7,8,9]. Compartmental models, which were originally established by Kermack and McKendrick [10], constitute a class of representative infectious disease models that includes the SIR model [11], SIS model [12], SIRS model [13], SEIR model [14], SI1⋯IkR [15] model and other variations [16,17,18,19].
Most epidemic models only concentrate on the transmission of a unique infectious disease; however, the host population may actually be at risk of more than two infectious diseases with complicated interaction, and they could occur as parallel, competitive or stimulative. A large percentage of people at risk for HIV infection is also at risk for HBV infection due to shared mechanisms of transmission [20]. Casalegno et al. [21] discovered that during the first half of fall 2009, in France, rhinovirus interference slowed the influenza pandemic and affected the transmission of the H1N1 virus. During the COVID-19 pandemic, it was discovered in [22] that the SARS-COV-2 Delta (B.1.617.2) variant had replaced the Alpha (B.1.1.7) variation on a significant scale, which is related to the Delta version's earlier invasion and superior transmissibility. In this paper, we only focus on two epidemics spreading parallelly, and we assume that an epidemic caused by one virus prevents the occurrence of the other. For related works, we recommend the references [24,25] and the references therein.
The rate at which new infections emerge, known as the disease incidence, is a crucial variable in mathematical models of infectious disease dynamics. The incidence rate has different forms which are commonly used as follows. It is assumed that the exposure rate is proportionate to the whole population and that the mass-action (bilinear) incidence is βSI [26]. The standard incidence is βSI/N, and it requires the assumption that the number of people exposed to a sick person per unit time is constant [27]. If the exposure rate is saturation of the susceptible S or infective I, the incidence will be the saturation incidence βSI/(1+aS) or βSI/(1+aI) [23]. Other incidence forms, such as the nonlinear incidence rate βSIp/(1+αSq) and Beddington-DeAngelis incidence βSI/(1+aS+bI), have been discussed in [29].
A particular Crowley-Martin functional response function was proposed in 1975 [30], and it is widely used in prey-predator models [31], eco-epidemic models [32] and epidemic models [33,34,35]. In infectious disease models, the Crowley-Martin incidence is represented by βSI/(1+aS)(1+bI), which takes into account the interaction between susceptible and infected populations, where a measures the preventive effect of susceptible individuals and b measures the treatment effect with respect to infected individuals.
For these reasons, this paper presents a deterministic epidemic model with the double epidemic hypothesis and Crowley-Martin nonlinear incidence term. We divided the population into three compartments: the susceptible population S, the infected population I1 infected with virus D1 and the infected population I2 infected with virus D2. In addition, susceptible individuals enter at a rate of constant N, βi is the rate of transmission from a susceptible person to an infected person, the natural and causal mortality rates of the population are m and δi respectively, and αi is the rate of infected people transitioning to the susceptible class. The flowchart of disease transmission and progression is as shown in Figure 1; we formulate the following dynamical model:
dSdt=N−β1SI1(1+a1S)(1+b1I1)−β2SI2(1+a2S)(1+b2I2)+α1I1+α2I2−mS,dI1dt=β1SI1(1+a1S)(1+b1I1)−(m+α1+δ1)I1,dI2dt=β2SI2(1+a2S)(1+b2I2)−(m+α2+δ2)I2. | (1.1) |
In fact, disease transmission is quite sensitive to disturbances caused by external environmental factors, such as temperature, light, rainstorms and human intervention. These stochastic factors could have a significant impact on almost all parameters of the model in multiple ways [36,37,38,39,40]. The transmission rate β oscillates around an average value as a result of the environment's ongoing oscillations brought on by the impact of white noise β+σ˙B(t), where B(t) represents the standard Brownian motions and σ>0 is the intensity of environmental fluctuations. Then, we obtain a stochastic epidemic model as follows:
{dS=(N−β1SI1(1+a1S)(1+b1I1)−β2SI2(1+a2S)(1+b2I2)+α1I1+α2I2−mS)dt−σ1SI11+a1S+b1I1+a1b1SI1dB1(t)−σ2SI21+a2S+b2I2+a2b2SI2dB2(t),dI1=(β1SI1(1+a1S)(1+b1I1)−(m+α1+δ1)I1)dt+σ1SI1(1+a1S)(1+b1I1)dB1(t),dI2=(β2SI2(1+a2S)(1+b2I2)−(m+α2+δ2)I2)dt+σ2SI2(1+a2S)(1+b2I2)dB2(t). | (1.2) |
The following describes how this manuscript is structured. In Section 2, we discuss the dynamics of deterministic systems, especially for the asymptotic stability of equilibria. In Section 3, we establish the extinction and persistence conditions of the corresponding stochastic system. In Section 4, through a number of numerical simulations, we explore the effects of the perturbation strength σi and parameters ai and bi on the dynamics of the system. The paper ends with a short discussion and conclusion.
Prior to investigating the stochastic system, it is also essential to ascertain the dynamical behaviors of the deterministic system.
For the deterministic system (1.1) or the stochastic system (1.2), we obtain
ddt(S+I1+I2)=N−m(S+I1+I2)−δ1I1−δ2I2≤N−m(S+I1+I2). |
This implies that
lim supt→∞(S+I1+I2)≤Nm. |
We denote
Γ={(S(t),I1(t),I2(t))∈R3+:S(t)+I1(t)+I2(t)≤Nm,t≥0}; |
then, regarding the solutions of system (1.1), Γ is a positively invariant set.
Utilizing the next generation matrix method [41,42], we can obtain the basic reproduction number:
Ri=βiN(m+aiN)(m+αi+δi),i=1,2. |
The equilibrium equation is listed as follows:
N−β1SI1(1+a1S)(1+b1I1)−β2SI2(1+a2S)(1+b2I2)+α1I1+α2I2−mS=0,β1SI1(1+a1S)(1+b1I1)−(m+α1+δ1)I1=0,β2SI2(1+a2S)(1+b2I2)−(m+α2+δ2)I2=0. | (2.1) |
System (1.1) has four possible equilibria:
(i) disease-free equilibrium E0=(Nm,0,0);
(ii) boundary equilibrium E1=(¯S1,¯I1,0), where 0<ˉS1<Nm,¯I1>0;
(iii) boundary equilibrium E2=(¯S2,0,¯I2), where 0<ˉS2<Nm,¯I2>0;
(iv) endemic equilibrium E3=(S∗,I∗1,I∗2), where S∗,I∗1,I∗2>0.
By (2.1), when I2=0 and I1≠0, we obtain the boundary equilibrium
E1=(N−(m+δ1)¯I1m,¯I1,0), |
and ¯I1 is the positive root of
f(¯I1)=X¯I12+Y¯I1+Z=0, |
where
X=a1b1(m+δ1)>0; |
when R1>1, we have
Y=(m+a1N)m+δ1N(−b1Nm+δ1+a1Nm+a1N−R1)<0,Z=(m+a1N)(R1−1)>0. |
If Δ=Y2−4XZ=0, there exists a unique E1 equilibrium with ¯I1=−Y2X; if Δ=Y2−4XZ>0, then system (1.1) has two E1 equilibria with ¯I1=−Y±√Y2−4XZ2X.
Similarly, by (2.1), when I1=0 and I2≠0, we obtain the boundary equilibrium
E2=(N−(m+δ2)¯I2m,0,¯I2); |
if R2>1 then there may exist one or two-type equilibria E2 as well.
By equilibrium equation (2.1), when Ri>1,i=1,2, the endemic equilibrium E3=(S∗,I∗1,I∗2) exists and satisfies
E3=(N−(m+δ1)I∗1−(m+δ2)I∗2m,I∗1,I∗2), |
where the relationship between I∗1 and I∗2 satisfies
I∗2=(m+α1+δ1)(1+b1I∗1)[m+a1(N−(m+δ1)I∗1)]−β1(N−(m+δ1I∗1))(m+δ2)(a1(m+α1+δ1)(1+b1I∗1−β1)). |
Theorem 1. 1). If R1<1 and R2<1, then the disease-free equilibrium E0 is locally asymptotically stable.
2). If R1>1 and equilibrium E1 exists, then the boundary equilibrium E1 is locally asymptotically stable.
3). If R2>1 and equilibrium E2 exists, then the boundary equilibrium E2 is locally asymptotically stable.
4). If R1>1, R2>1 and equilibrium E3 exists, then the endemic equilibrium E3 is locally asymptotically stable.
The proof is given in Appendix A.
Lemma 1 ([[43]]). For any initial value (S(0),I1(0),I2(0))∈R3+, there exists a unique solution (S(t),I1(t),I2(t)) to system (1.2) on t≥0, and the solution will remain in R3+ with probability 1, i.e., (S(t),I1(t),I2(t))∈R3+ for all t≥0 a.s.
Proof. The proof of Lemma 1 is similar to that in Theorem 2.1 of [44]; we therefore omit it here.
Lemma 2 ([[29]]). Γ is an almost positive invariant set of system (1.2), that is, if (S(0),I1(0),I2(0))∈Γ, then P(S(t),I1(t),I2(t)∈Γ)=1 for all t≥0.
Define the stochastic basic reproduction numbers
Rsi=βiN(m+aiN)(m+αi+δi)−σ2iN22(m+aiN)2(m+αi+δi)=Ri−σ2iN22(m+aiN)2(m+αi+δi),i=1,2. |
We focus on disease extinction and persistence in this subsection since stochastic systems have distinct extinction and persistence conditions compared to deterministic systems. First, the following lemma is presented to demonstrate the extinction and persistence of diseases.
Lemma 3 ([[42,45]]). Let (S(t),I1(t),I2(t)) be a solution of system (1.2) with initial value (S(0),I1(0),I2(0))∈R3+. Then
limt→+∞1t∫t0σiS(τ)(1+aiS(τ))(1+biIi(τ))dBi(τ)=0,limt→+∞1t∫t0σiS(τ)dBi(τ)=0,i=1,2. |
Theorem 2. Suppose that one of the following two assumptions is satisfied:
(H1) σi>ˆσi:=max{√βi(Nai+m)N,βi√2(m+αi+δi)},i=1,2;
(H2) σi≤√βi(Nai+m)N and Rsi<1,i=1,2.
Then the solution (S(t),I1(t),I2(t)) of system (1.2) with any initial value (S(0),I1(0),I2(0))∈Γ satisfies that
limt→+∞S(t)=Nm,limt→+∞I1(t)=limt→+∞I2(t)=0. |
Proof. By using Itˆo's formula, we have
dlnIi(t)=(βiS(1+aiS)(1+biIi)−σ2iS22((1+aiS)(1+biIi))2−(m+αi+δi))dt+σiS(1+aiS)(1+biIi)dBi(t),i=1,2. | (3.1) |
Case 1: Under assumption (H1), integrating both sides of (3.1), we have
lnIi(t)=∫t0(βiS(τ)(1+aiS(τ))(1+biIi(τ))−σ2iS2(τ)2((1+aiS(τ))(1+biIi(τ)))2)dτ−(m+αi+δi)t+Qi(t)+lnIi(0)=−σ2i2∫t0(S(τ)(1+aiS(τ))(1+biIi(τ))−βiσ2i)2dτ−(m+αi+δi)t+β2i2σ2it+Qi(t)+lnIi(0)≤−(m+αi+δi)t+β2i2σ2it+lnIi(0)+Qi(t), | (3.2) |
where
Qi(t)=∫t0σiS(τ)(1+aiS(τ))(1+biIi(τ))dBi(τ); | (3.3) |
dividing both sides of (3.2) by t, we have
lnIi(t)t≤−(m+αi+δi−β2i2σ2i)+Qi(t)t+lnIi(0)t; | (3.4) |
by Lemma 3, we have
limt→+∞Qi(t)t=0; |
since σi>βi√2(m+αi+δi) for i=1,2, taking the limit superior on both sides of (3.4) leads to
lim supt→+∞lnIi(t)t≤−(m+αi+δi−β2i2σ2i)<0. |
Thus, limt→+∞Ii(t)=0 a.s.
Case 2: Under assumption (H2), similar to the calculation in Case 1, we have
lnIi(t)t=1t{∫t0(βiS(τ)(1+aiS(τ))(1+biIi(τ))−σ2iS2(τ)2((1+aiS(τ))(1+biIi(τ)))2−(m+αi+δi))dτ+Qi(t)+lnIi(0)}=1t{∫t0Ψ(S(τ)(1+aiS(τ))(1+biIi(τ)))dτ+Qi(t)+lnIi(0)}, |
where the function Υ(x) is defined as
Υ:x↦−12σ2ix2+βix−(m+αi+δi). |
Take note that Υ(x) increases monotonically for x∈[0,βiσ2i] and x<NaiN+m; thus, when σi≤√βi(Nai+m)N, we have
lnIi(t)t≤βiNm+aiN−σ2iN22(m+aiN)2−(m+αi+δi)+1t(Qi(t)+lnIi(0))=(m+αi+δi)(Rsi−1)+1t(Qi(t)+lnIi(0)). | (3.5) |
Taking the limit superior of both sides of (3.5) leads to
lim supt→+∞lnIi(t)t≤(m+αi+δi)(Rsi−1)<0; |
which implies that limt→+∞Ii(t)=0,i=1,2.
We suppose that 0<Ii(t)<εi(i=1,2) for all t≥0; by the first equation of system (1.2), we have
dS(t)dt≥N−(m+β1ε1+β2ε2+σ1ε1|˙B1(t)|+σ2ε2|˙B2(t)|)S(t). | (3.6) |
Because ε1→0 and ε2→0, if we divide (3.6) by the limit inferior on both sides, we have
lim inft→+∞S(t)≥Nm. | (3.7) |
Combining this with lim supt→+∞S(t)≤Nm, it is easy to see that
limt→+∞S(t)=Nma.s. |
We will present the persistence result of system (1.2) in the following theorem, whose proof is given in Appendix B.
Theorem 3. If we assume that the solution to system (1.2) is (S(t),I1(t),I2(t)) and that (S(0),I1(0),I2(0))∈Γ is the initial value, then we get the following:
(i) The disease I2 becomes extinct and the disease I1 becomes permanent in the mean if Rs1>1, Rs2<1 and the disturbance intensity satisfies that σ2≤√β2(a2+mN). Additionally, I1 satisfies
lim inft→+∞⟨I1(t)⟩≥(m+a1N)(m+α1+δ1)β1(m+δ1)+b1(m+a1N)(m+α1+δ1)(Rs1−1). |
(ii) The disease I1 becomes extinct and the disease I2 becomes permanent in the mean if Rs1<1, Rs2>1 and the disturbance intensity satisfies that σ1≤√β1(a1+mN). Additionally, I2 satisfies
lim inft→+∞⟨I2(t)⟩≥(m+a2N)(m+α2+δ2)β2(m+δ2)+b2(m+a2N)(m+α2+δ2)(Rs2−1). |
(iii) If Rsi>1, then the two infectious diseases Ii are permanent in mean; moreover, Ii satisfies
lim inft→+∞⟨2∑i=1Ii(t)⟩≥1Δmax2∑i=1ai(m+αi+δi)(Rsi−1), |
where
Δmax=2∑i=1(β1+β2m(m+δi)+bi(m+αi+δi)). |
In this section of this paper, we will continue with our investigation of the deterministic system and the stochastic system by using the numerical method. Before looking at how changes in the environment influence the spread of diseases and the effect of the parameters ai and bi on the dynamics of the disease, we first compare the extinction conditions for the same parameter values for the stochastic system and the deterministic system.
To simulate the behavior of the stochastic system (1.2), we made use of Milstein's method [45,46]. Following [23,47], with the exception of σi, the other parameter values of system (1.1) and system (1.2) were derived as given in Table 1. Then, we chose initial values as (S(0),I1(0),I2(0))=(15,10,5).
{Sk+1=Sk+(N−β1SkI1k(1+a1Sk)(1+b1I1k)−β2SkI2k(1+a2Sk)(1+b2I2k)+α1I1k+α2I2k−mSk)Δt−(σ1SkI1k(1+a1Sk)(1+b1I1k)+σ2SkI2k(1+a2Sk)(1+b2I2k))√Δtξk−(σ212(SkI1k(1+a1Sk)(1+b1I1k))+σ222(SkI2k(1+a2Sk)(1+b2I2k)))(ξ2k−1)Δt,I1k+1=σ1SkI1k(1+a1Sk)(1+b1I1k)√Δtξk+σ212(SkI1k(1+a1Sk)(1+b1I1k))(ξ2k−1)Δt+I1k+(β1SkI1k(1+a1Sk)(1+b1I1k)−(m+α1+δ1)I1k)Δt,I2k+1=σ2SkI2k(1+a2Sk)(1+b2I2k)√Δtξk+σ222(SkI2k(1+a2Sk)(1+b2I2k))(ξ2k−1)Δt+I2k+(β2SkI2k(1+a2Sk)(1+b2I2k)−(m+α2+δ2)I2k)Δt. |
Parameter | Description | Value | Sources |
N | Influx of susceptible population | 1 | [23] |
m | Rate of natural death | 0.1 | [23,47] |
β1 | Transmission coefficient for virus D1 | 1.2 | [23,47] |
β2 | Transmission coefficient for virus D2 | 1.5 | [23,47] |
a1 | Preventive effect of virus D1 | 1 | [23] |
a2 | Preventive effect of virus D2 | 1.5 | [23] |
b1 | Treatment effect of virus D1 | 2 | [23] |
b2 | Treatment effect of virus D2 | 1 | [23] |
δ1 | Rate of virus D1-related death | 0.2 | [23,47] |
δ2 | Rate of virus D2-related death | 0.4 | [23,47] |
α1 | Recovery rate for virus D1 | 0.9 | [23,47] |
α2 | Recovery rate for virus D2 | 0.9 | [23,47] |
In order to investigate the dynamical differences that exist between systems (1.1) and (1.2), we give five examples for numerical simulations.
Example 1. When I2 is facing extinction in a deterministic system, the stochastic perturbation could change I1 from prevalence to extinction. When α1 is changed to 0.7, σ1=σ2=1, R1=1.091>1, Rs1=1.091−0.379=0.712<1, R2=0.9375<1 and Rs2=0.9375−0.3125=0.625<1. According to our previous analysis results, disease I1 is prevalent and disease I2 is subject to extinction in the deterministic system (1.1); however, disease I1 and I2 are both extinct in the stochastic system (1.2) (see Figure 2(a)).
Example 2. When I1 is subject to extinction in a deterministic system, the stochastic perturbation could change I2 from a prevalence to extinction condition. When α2 is changed to 0.7, δ2 is 0.2, a2=1.3, σ1=σ2=1, R1=0.909<1,Rs1=0.909−0.416=0.496<1,R2=1.071>1 and Rs2=1.071−0.255=0.816<1. According to our previous analysis results, disease I1 goes to extinction and disease I2 is persistent in the deterministic system (1.1); also, disease I1 and I2 both go to extinction in the stochastic system (1.2) (see Figure 2(b)).
Example 3. When I1 and I2 are extinct in a deterministic system, the stochastic perturbation could change I1 from a prevalence to extinction condition. When δ2=0.1,σ1=1 and σ2=0.1, it follows that R1=1.091>1,Rs1=1.091−0.413=0.678<1,R2=1.041>1 and Rs2=1.041−0.001=1.04>1. According to our previous analysis results, both disease I1 and disease I2 are persistent in the deterministic system (1.1); disease I1 goes to extinction and I2 is persistent in the stochastic system (1.2) (see Figure 2(c)).
Example 4. When I1 and I2 are extinct in a deterministic system, the stochastic perturbation could change I2 from a prevalence to extinction condition. When δ2=0.1,σ1=0.1 and σ2=1, R1=1.091>1,Rs1=1.091−0.0041=1.0869>1,R2=1.041>1 and Rs2=1.041−0.195=0.846<1. According to our previous analysis results, both disease I1 and disease I2 are persistent in the deterministic system; also, disease I1 is persistent and I2 goes to extinction in the stochastic system (1.2) (see Figure 2(d)).
Example 5. When I1 and I2 are extinct in a deterministic system, the stochastic perturbation could change I1 and I2 from a prevalence to extinction condition. When δ2=0.1,σ1=1 and σ2=1, R1=1.091>1, Rs1=1.091−0.413=0.678<1, R2=1.041>1 and Rs2=1.041−0.195=0.846<1. According to our previous analysis results, both disease I1 and disease I2 are persistent in the deterministic system (1.1). In the stochastic system (1.2), both disease I1 and disease I2 go extinct (see Figure 2(e)).
By (H1) in Theorem 3.1, we can see that when the strengths of the perturbations are large, Rsi loses its meaning and the diseases go to extinction. We chose different perturbation strengths for when σi=0,0.3,0.9 to observe the trend of the disease. When σi is larger, the infectious disease Ii goes to extinction (see Figure 3). These simulations support our results for (H1) in Theorem 3.1 well.
It should be noted that ai and bi of the Crowley-Martin incidence are key parameters. In this subsection, we discuss the effects of parameters ai and bi on the population and trend of infections by presenting some numerical simulations.
First, we study the influence of preventive effects ai on the population of infective individuals in the deterministic system (1.1). For the case that the parameters in Table 1 are fixed in Table 1, we chose five different sets of values for ai. For the deterministic system (1.1), it can be shown that the bigger the value of ai, the quicker the extinction of disease Ii (see Figure 4). Second, we wanted to investigate the influence of the parameter ai on the population of infective individuals in the stochastic system (1.2). We choose the perturbation intensity as σi=0.3,i=1,2. At last, we observed the effect of ai in the stochastic system (1.2) (see Figure 5). Similarly, we wanted to study the influence of treatment effects bi on the population of infected individuals in deterministic and stochastic systems (see Figures 6 and 7).
From the above numerical simulations, we conclude that a larger ai leads to a lower infected prevalence Ii(t), and it may result in the extinction in deterministic and stochastic systems. This is because the parameter ai affects Ri and Rsi. Additionally, we found that the infected population Ii(t) also decreases when bi increases, but bi cannot lead to extinction.
In this paper, we have proposed and studied a class of stochastic double disease models with Crowley-Martin incidence. We discussed the existence conditions and stability of the equilibrium points. E0 is locally asymptotically stable when the basic reproduction number Ri<1; E1 is locally asymptotically stable when R1>1 and R2<1; E2 is locally asymptotically stable when R2>1 and R1<1; and E3 is locally asymptotically stable when the basic reproduction number Ri>1. Subsequently, we have given the stochastic basic reproduction number R∗i of the stochastic system and proven the stochastic extinction and persistence of the system. Finally, numerical simulations show that appropriate stochastic perturbations σi can control the spread of the disease, but larger stochastic perturbations can cause the disease to go extinct; the protection effect ai can cause the disease to go extinct; the treatment effect bi can reduce the number of infected individuals, but it cannot cause the disease to go extinct. Therefore, when treatment is given to infected individuals, protective measures for susceptible individuals are more necessary to completely eliminate the virus.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Suli Liu was supported by the National Natural Science Foundation of China (Grant number 12301627), the Science and Technology Research Projects of the Education Office of Jilin Province, China (JJKH20211033KJ) and the Technology Development Program of Jilin Province, China (20210508024RQ).
The authors declare that there is no conflict of interest.
Proof. The system (1.1) has the following Jacobian matrix
J=(a11a12a13a21a220a310a33), | (A1) |
where
a11=−β1I1(1+a1S)2(1+b1I1)−β2I2(1+a2S)2(1+b2I2)−m,a12=−β1S(1+a1S)(1+b1I1)2+α1,a13=−β2S(1+a2S)(1+b2I2)2+α2,a21=β1I1(1+a1S)2(1+b1I1),a22=β1S(1+a1S)(1+b1I1)2−(m+α1+δ1),a31=β2I2(1+a2S)2(1+b2I2),a33=β2S(1+a2S)2(1+b2I2)−(m+α2+δ2). |
Case 1. The evaluation of the Jacobian matrix at E0 is represented by
J(E0)=(−m−β1Nm+a1N+α1−β2Nm+a2N+α20β1Nm+α1N−(m+α1+δ1)000β2Nm+α2N−(m+α2+δ2)), |
which has the following eigenvalues:
λ1=−m,λ2=β1Nm+α1N−(m+α1+δ1)=(m+α1+δ1)(R1−1),λ3=β2Nm+α2N−(m+α2+δ2)=(m+α2+δ2)(R2−1). |
If R1<1 and R2<1, then λ1, λ2 and λ3<0. The disease-free equilibrium E0 is locally asymptotically stable.
Case 2. The evaluation of the Jacobian matrix at E1 is represented by
J(E1)=(a11a12a13a21a22000a33), |
where
a11=−β1ˉI1(1+a1¯S1)2(1+b1¯I1)−m,a12=−β1¯S1(1+a1¯S1)(1+b1¯I1)2+α1,a13=−β2¯S11+a2¯S1+α2,a21=β1¯I1(1+a1¯S1)2(1+b1¯I1),a22=β1¯S1(1+a1¯S1)(1+b1¯I1)2−(m+α1+δ1),a33=β2¯S11+a2¯S1−(m+α2+δ2). |
Using the equilibrium equation (2.1)
β1¯S1(1+a1¯S1)(1+b1¯I1)=m+α1+δ1; |
we obtain
a22=m+α1+δ11+b1¯I1−(m+α1+δ1)<0; |
one of the three eigenvalues of matrices J(E1)'s is represented by
λ1=β2¯S11+a2¯S1−(m+α2+δ2)<0, |
where R2<1 and ¯S1<Nm is used. What follows is the characteristic equation:
λ2+A1λ+A2=0, |
where
A1=−(a11+a22)=β1¯I1(1+a1¯S1)2(1+b1¯I1)−β1¯S1(1+a1¯S1)(1+b1¯I1)2+(m+α1+δ1)+m=β1¯I1(1+a1¯S1)2(1+b1¯I1)+β1b1¯S1¯I1(1+a1¯S1)(1+b1¯I1)2+m>0, |
A2=a11a22−a21a12=(−β1¯I1(1+a1¯S1)2(1+b1¯I1)−m)(β1¯S1(1+a1¯S1)(1+b1¯I1)2−(m+α1+δ1))=(m+δ1)β1¯I1(1+b1¯I1)−β1m¯S1(1+a1¯S1)(1+a1¯S1)2(1+b1¯I1)2+m(m+α1+δ1)=¯I1(m+δ1)(1+b1¯I1)+β1mb1¯S1¯I1(1+a1S)(1+a1¯S1)2(1+b1¯I1)2>0. |
By the Routh-Hurwitz condition, if R1>1 and R2<1, the boundary equilibrium E1 is locally asymptotically stable. The proof process for Cases 2 and 3 are similar, so they are omitted.
Case 4. The evaluation of the Jacobian matrix at E3 is represented by
J(E3)=(−a21−a31−m−a22−(m+δ1)−a33−(m+δ2)β1I∗1(1+a1S∗)2(1+b1I∗1)−β1b1S∗I∗1(1+a1S∗)(1+b1I∗1)20β2I∗2(1+a2S∗)2(1+b2I∗2)0−β2b2S∗I∗2(1+a2S∗)(1+b2I∗2)2); |
what follows is the characteristic equation:
λ3+A1λ2+A2λ+A3=0, |
where
A1=β1I∗1(1+b1I∗1+b1S∗+a1b1S∗2)(1+a1S∗)2(1+b1I∗1)2+β2I∗2(1+b2I∗2+b2S∗+a2b2S∗2)(1+a2S∗)2(1+b2I∗2)2:=C1+C2+m>0, |
A2=(−a21−a31−m)a22+(−a21−a31−m)a33+a22a33+a33a31+(m+δ2)a31+a22a21+(m+δ1)a21=a22a33−a22a31−a21a33+a21(m+δ1)−ma22+a31(m+δ2)−ma33=β1β2S∗I∗1I∗2(b1b2S∗(1+a1S∗)(1+a2S∗)+b1(1+a1S∗)(1+b2)2+b2(1+a2S∗(1+b1)2))((1+a1S∗)(1+b1I∗1)(1+a2S∗)(1+b2I∗2))2+β1I∗1((1+b1I∗1)(m+δ1)+mb1S∗(1+a1S∗))(1+a1S∗)2(1+b1I∗1)2+β2I∗2((1+b2I∗2)(m+δ2)+mb2S∗(1+a2S∗))(1+a2S∗)2(1+b2I∗2)2:=C3+C4+C5>0, |
A3=−(m+δ1)a21a33−(m+δ2)a22a31+ma22a33=β1β2b2S∗I∗1I∗2(m+δ1)(1+a1S∗)2(1+b1I∗1)(1+a2S∗)(1+b2I∗2)2+β1β2b1S∗I∗1I∗2(m+δ2)(1+a1S∗)(1+b1I∗1)2(1+a2S∗)2(1+b2I∗2)+β1β2mb1b2I∗1I∗2S∗2(1+a1S∗)(1+b1I∗1)2(1+a2S∗)(1+b2I∗2)2>0. |
Then
A1A2−A3=(C1+C2+m)(C3+C4+C5)−A3=(C1+C22)C5+C1C3+C2C4+m(C3+C4+C5)+C1C4+C2C3−A3:=p+q, |
where p>0 when R1>1 and R2>1, and we have
q=C1C4+C2C3−A3=β1β2I∗1I∗2(1+b2I∗2)((1+b1I∗1)(m+δ1)+mb1S(1+a1S))+b1S(1+a1S)(mb2S(1+a2S))((1+a1S∗)(1+b1I∗1)(1+a2S∗))2+β1β2I∗1I∗2(1+b1I1)((1+b2I2)(m+δ2)+mb2S(1+a2S))((1+a1S∗)(1+b1I∗1)(1+a2S∗))2>0. |
When both R1 and R2 are bigger than 1, the Routh-Hurwitz criteria show that the endemic equilibrium E3 is locally asymptotically stable.
Proof. Part (i). It is easy to see from the theorem that limt→+∞I2(t)=0. Since Rs1>1, there exists a ε small enough such that 0<I2(t)<ε for all t large enough; we have
β(N−(m+δ1)ε)(Na1+m)(m+α1+δ1)−σ21N22(Na1+m)(m+α1+δ1)>1. |
On the sides of the system (1.2), dividing by t>0 and integrating from 0 to t gives
Θ(t)≜S(t)−S(0)t+I1(t)−I1(0)t+I2(t)−I2(0)t=N−m⟨S(t)⟩−(m+δ1)⟨I1(t)⟩−(m+δ2)⟨I2(t)⟩≥N−m⟨S(t)⟩−(m+δ1)⟨I1(t)⟩−(m+δ2)ε; |
when ⟨f(t)⟩=1t∫t0f(θ)dθ is defined for an integrable function f on [0,+∞), one may get
⟨S(t)⟩≥N−(m+δ2)εm−m+δ1m⟨I1(t)⟩−Θ(t)m. |
By using Itˆo′s formula, it follows that
d((1+a1Nm)lnI1+b1(1+a1Nm)I1)=((1+a1Nm)β1S(1+a1S)(1+b1I1)−(1+a1Nm)(m+α1+δ1)−(1+a1Nm)σ21S22(1+a1S)2(1+b1I1)2)dt+b1(1+a1Nm)(β1SI1(1+a1S)(1+b1I1)−(m+α1+δ1)I1)dt+(1+a1Nm)σ1S(1+a1S)(1+b1I1)dB1(t)+b1(1+a1Nm)σ1SI1(1+a1S)(1+b1I1)dB1(t)≥(β1S(1+b1I1)−(1+a1Nm)(m+α1+δ1)−σ212(1+a1Nm)(Nm+Na1)2)dt+b1(1+a1Nm)(β1SI1(1+a1Nm)(1+b1I1)−(m+α1+δ1)I1)dt+σ1S(t)dB1(t)≥(β1S−(1+a1Nm)(m+α1+δ1)−b1(1+a1Nm)(m+α1+δ1)I1−σ22(Nm)21+a1Nm)dt | (A2) |
+σ1SdB1(t). | (A3) |
On the sides of (A3), dividing by t>0 and integrating from 0 to t gives
(1+a1Nm)lnI1(t)−lnI1(0)t+b1(1+a1Nm)I1(t)−I1(0)t≥β1⟨S(t)⟩−m+a1Nm(m+α1+δ1)−b1m+a1Nm(m+α1+δ1)⟨I1(t)⟩−σ21(Nm)22(m+a1N)m+Q2t≥β1(N−(m+δ2)εm−m+δ2m⟨I1(t)⟩−Θ(t)m)−m+a1Nm(m+α1+δ1)+Q2(t)t−b1(m+a1N)m(m+α1+δ1)⟨I1(t)⟩−σ21(Nm)22(m+a1N)m=m+a1Nm(m+α1+δ1)(β1(N−(N+δ2)ε)(m+a1N)(m+α1+δ1)−σ21(Nm)22(m+a1Nm)2(m+α1+δ1)−1)−(β1(m+δ1)m+b1(m+a1N)m(m+α1+δ1))⟨I1(t)⟩−β1Θ(t)m+Q2(t)t, | (A4) |
where Q2(t)=∫t0σ1S(τ)dB1(τ). It is possible to rewrite the inequality (A4) as
⟨I1(t)⟩≥1Δ{m+a1Nm(m+α1+δ1)(β1(N−(N+δ2)ε)(m+a1N)(m+α1+δ1)−σ21(Nm)22(m+a1Nm)2(m+α1+δ1)−1)−β1Θ(t)m+Q2(t)t−1tm+a1Nm[(lnI1(t)−lnI1(0))+b1(I1(t)−I1(0))]}≥{1Δ{m+a1Nm(m+α1+δ1)(β1(N−(N+δ2)ε)(m+a1N)(m+α1+δ1)−σ21(Nm)22(m+a1Nm)2(m+α1+δ1)−1)−β1Θ(t)m+Q2(t)t+1tm+a1Nm[lnI1(0)−b1(I1(t)−I1(0))]},0<I1(t)<1,1Δ{a1(m+α1+δ1)(β1(N−(N+δ2)ε)(m+a1N)(m+α1+δ1)−σ21(Nm)22(m+a1Nm)2(m+α1+δ1)−1)−β1Θ(t)m+Q2(t)t−1tm+a1Nm[(lnI1(t)−lnI1(0))+b1(I1(t)−I1(0))]},1<I1(t), | (A5) |
where Δ=β1(m+δ1)m+b1(m+a1N)(m+α1+δ1).
By Lemma 3, we get that limt→+∞Q2(t)t=0. We can observe that I1(t)≤Nm; thus, we have that limt→+∞I1(t)t=0 and limt→+∞lnI1(t)t=0 as I1(t)≥1 and limt→+∞Θ(t)=0.
When the limit inferior of both sides of (A5) are taken into account, we have
lim inft→+∞⟨I1(t)⟩≥(1+a1Nm)(m+α1+δ1)Δ(β1(N−(N+δ2)ε)m(1+a1Nm)(m+α1+δ1)−σ21(Nm)22(1+a1Nm)2(m+α1+δ1)−1)≥0. |
Allowing for ε→0, we have
lim inft→+∞⟨I1(t)⟩≥(m+a1N)(m+α1+δ1)β1(m+δ1)+b1(m+a1N)(m+α1+δ1)(Rs1−1). |
Due to the fact that the methods of proving parts (ii) and (i) are similar, this step will not be repeated. Part (iii). Take note that
⟨S(t)⟩=Nm−m+δ1m⟨I1(t)⟩−m+δ2m⟨I2(t)⟩−Θ(t)m. | (A6) |
Define
V(t)=ln[I(1+a1Nm)1(t)I(1+a2Nm)2(t)]+[b1(1+a1Nm)I1(t)+b2(1+a2Nm)I2(t)]. | (A7) |
Consequently, V(t) is bounded. We have
D+V(t)=2∑i=1((1+aiNm)βiS(1+aiS)(1+biIi)−(1+aiNm)(m+αi+δi)−(1+aiNm)σ2iS22((1+aiS)(1+biIi))2)dt+2∑i=1(1+aiNm)σiS(1+aiS)(1+biIi)dBi(t)+2∑ibi(1+aiNm)(βiSIi(1+aiS)(1+biIi)−(m+αi+δi)Ii)dt+2∑ibi(1+aiNm)σiSIi(1+aiS)(1+biIi)dBi(t)≥2∑i=1((1+aiNm)βiS1+aiNm+biIi+aibiNmIi−(1+aiNm)(m+αi+δi)−(1+aiNm)σ2iS22((1+aiS)(1+biIi))2)dt+2∑i=1(1+aiNm)σiS(1+aiNm)(1+biIi)dBi(t)+2∑ibi(1+aiNm)(βiSIi(1+aiNm)(1+biIi)−(m+αi+δi)Ii)dt+2∑ibi(1+aiNm)σiSIi1+aiNm+biIi+aibiNmIidBi(t)≥((β1+β2)S−2∑i(m+αi+δi)(1+biIi)(1+aiNm)+2∑i=1σ2i(Nm)22(m+aiN)m)dt+2∑iσiSdBi(t). | (A8) |
On the sides of (A8), dividing by t>0 and integrating from 0 to t gives
V(t)−V(0)t≥2∑i=1(βi⟨S(t)⟩−(m+αi+δi)(1+aiNm)(1+bi⟨Ii(t)⟩)−σ2i(Nm)22(m+aiN)m+Qit)=(β1+β2)Nm−2∑i=1(m+αi+δi)−2∑i=1σ2i(Nm)22(1+aiNm)−β1+β2mΘ(t)+2∑i=1Qit−2∑i=1(βim(m+δi)+bi(m+αi+δi)(1+aiNm))⟨Ii(t)⟩≥2∑i=1(1+aiNm)(m+αi+δi)(βiN(m+aiN)(m+αi+δi)−σ2iN22(1+aiNm)(m+αi+δi)−1)−Δmax[⟨I1(t)⟩+⟨I2(t)⟩]−2∑i=1(βimΘ(t)−Qit), | (A9) |
where
Qi(t)=∫t0σiS(τ)dBi(τ),Δmax=2∑i=1[β1+β2m(m+δi)+bi(m+αi+δi)]. |
It is possible to rewrite the inequality (A9) as
⟨2∑i=1Ii(t)⟩≥1Δmax(2∑i=1(1+aiNm)(m+αi+δi)(βiN(m+aiN)(m+αi+δi)−σ2iN22(1+aiNm)(m+αi+δi)−1)+V(0)−V(t)t−2∑i=1βimΘ(t)+2∑i=1Qit). | (A10) |
By Lemma 3, we have that limt→+∞Qi(t)t=0 for i=1,2. And we can see that limt→+∞Θ(t)=0 and limt→+∞V(t)t=0.
Taking the limit inferior of both sides of (A10) yields
lim inft→+∞⟨2∑i=1Ii(t)⟩≥1Δmax2∑i=1(1+aiNm)(m+αi+δi)(Rsi−1)>0. |
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1. | Wen-Xuan Li, Chen Xu, Hui-Lai Li, Wen-Da Xu, A SIQRS epidemic model incorporating psychological factors for COVID-19, 2025, 29, 0354-9836, 1077, 10.2298/TSCI2502077L |
Parameter | Description | Value | Sources |
N | Influx of susceptible population | 1 | [23] |
m | Rate of natural death | 0.1 | [23,47] |
β1 | Transmission coefficient for virus D1 | 1.2 | [23,47] |
β2 | Transmission coefficient for virus D2 | 1.5 | [23,47] |
a1 | Preventive effect of virus D1 | 1 | [23] |
a2 | Preventive effect of virus D2 | 1.5 | [23] |
b1 | Treatment effect of virus D1 | 2 | [23] |
b2 | Treatment effect of virus D2 | 1 | [23] |
δ1 | Rate of virus D1-related death | 0.2 | [23,47] |
δ2 | Rate of virus D2-related death | 0.4 | [23,47] |
α1 | Recovery rate for virus D1 | 0.9 | [23,47] |
α2 | Recovery rate for virus D2 | 0.9 | [23,47] |
Parameter | Description | Value | Sources |
N | Influx of susceptible population | 1 | [23] |
m | Rate of natural death | 0.1 | [23,47] |
β1 | Transmission coefficient for virus D1 | 1.2 | [23,47] |
β2 | Transmission coefficient for virus D2 | 1.5 | [23,47] |
a1 | Preventive effect of virus D1 | 1 | [23] |
a2 | Preventive effect of virus D2 | 1.5 | [23] |
b1 | Treatment effect of virus D1 | 2 | [23] |
b2 | Treatment effect of virus D2 | 1 | [23] |
δ1 | Rate of virus D1-related death | 0.2 | [23,47] |
δ2 | Rate of virus D2-related death | 0.4 | [23,47] |
α1 | Recovery rate for virus D1 | 0.9 | [23,47] |
α2 | Recovery rate for virus D2 | 0.9 | [23,47] |