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Research article

Dynamics of a stochastic epidemic model with quarantine and non-monotone incidence

  • Received: 09 December 2022 Revised: 03 March 2023 Accepted: 08 March 2023 Published: 03 April 2023
  • MSC : 34D30, 60H10, 92D25

  • In this paper, a stochastic SIQR epidemic model with non-monotone incidence is investigated. First of all, we consider the disease-free equilibrium of the deterministic model is globally asymptotically stable by using the Lyapunov method. Secondly, the existence and uniqueness of positive solution to the stochastic model is obtained. Then, the sufficient condition for extinction of the stochastic model is established. Furthermore, a unique stationary distribution to stochastic model will exist by constructing proper Lyapunov function. Finally, numerical examples are carried out to illustrate the theoretical results, with the help of numerical simulations, we can see that the higher intensities of the white noise or the bigger of the quarantine rate can accelerate the extinction of the disease. This theoretically explains the significance of quarantine strength (or isolation measures) when an epidemic erupts.

    Citation: Tingting Wang, Shulin Sun. Dynamics of a stochastic epidemic model with quarantine and non-monotone incidence[J]. AIMS Mathematics, 2023, 8(6): 13241-13256. doi: 10.3934/math.2023669

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  • In this paper, a stochastic SIQR epidemic model with non-monotone incidence is investigated. First of all, we consider the disease-free equilibrium of the deterministic model is globally asymptotically stable by using the Lyapunov method. Secondly, the existence and uniqueness of positive solution to the stochastic model is obtained. Then, the sufficient condition for extinction of the stochastic model is established. Furthermore, a unique stationary distribution to stochastic model will exist by constructing proper Lyapunov function. Finally, numerical examples are carried out to illustrate the theoretical results, with the help of numerical simulations, we can see that the higher intensities of the white noise or the bigger of the quarantine rate can accelerate the extinction of the disease. This theoretically explains the significance of quarantine strength (or isolation measures) when an epidemic erupts.



    Let A denote the class of functions f which are analytic in the open unit disk Δ={zC:|z|<1}, normalized by the conditions f(0)=f(0)1=0. So each fA has series representation of the form

    f(z)=z+n=2anzn. (1.1)

    For two analytic functions f and g, f is said to be subordinated to g (written as fg) if there exists an analytic function ω with ω(0)=0 and |ω(z)|<1 for zΔ such that f(z)=(gω)(z).

    A function fA is said to be in the class S if f is univalent in Δ. A function fS is in class C of normalized convex functions if f(Δ) is a convex domain. For 0α1, Mocanu [23] introduced the class Mα of functions fA such that f(z)f(z)z0 for all zΔ and

    ((1α)zf(z)f(z)+α(zf(z))f(z))>0(zΔ). (1.2)

    Geometrically, fMα maps the circle centred at origin onto α-convex arcs which leads to the condition (1.2). The class Mα was studied extensively by several researchers, see [1,10,11,12,24,25,26,27] and the references cited therein.

    A function fS is uniformly starlike if f maps every circular arc Γ contained in Δ with center at ζ Δ onto a starlike arc with respect to f(ζ). A function fC is uniformly convex if f maps every circular arc Γ contained in Δ with center ζ Δ onto a convex arc. We denote the classes of uniformly starlike and uniformly convex functions by UST and  UCV, respectively. For recent study on these function classes, one can refer to [7,9,13,19,20,31].

    In 1999, Kanas and Wisniowska [15] introduced the class k-UCV (k0) of k-uniformly convex functions. A function fA is said to be in the class k-UCV if it satisfies the condition

    (1+zf(z)f(z))>k|zf(z)f(z)|(zΔ). (1.3)

    In recent years, many researchers investigated interesting properties of this class and its generalizations. For more details, see [2,3,4,14,15,16,17,18,30,32,35] and references cited therein.

    In 2015, Sokół and Nunokawa [33] introduced the class MN, a function fMN if it satisfies the condition

    (1+zf(z)f(z))>|zf(z)f(z)1|(zΔ).

    In [28], it is proved that if (f)>0 in Δ, then f is univalent in Δ. In 1972, MacGregor [21] studied the class B of functions with bounded turning, a function fB if it satisfies the condition (f)>0 for zΔ. A natural generalization of the class B is B(δ1) (0δ1<1), a function fB(δ1) if it satisfies the condition

    (f(z))>δ1(zΔ;0δ1<1), (1.4)

    for details associated with the class B(δ1) (see [5,6,34]).

    Motivated essentially by the above work, we now introduce the following class k-Q(α) of analytic functions.

    Definition 1. Let k0 and 0α1. A function fA is said to be in the class k-Q(α) if it satisfies the condition

    ((zf(z))f(z))>k|(1α)f(z)+α(zf(z))f(z)1|(zΔ). (1.5)

    It is worth mentioning that, for special values of parameters, one can obtain a number of well-known function classes, some of them are listed below:

    1. k-Q(1)=k-UCV;

    2. 0-Q(α)=C.

    In what follows, we give an example for the class k-Q(α).

    Example 1. The function f(z)=z1Az(A0) is in the class k-Q(α) with

    k1b2bb(1+α)[b(1+α)+2]+4(b=|A|). (1.6)

    The main purpose of this paper is to establish several interesting relationships between k-Q(α) and the class B(δ) of functions with bounded turning.

    To prove our main results, we need the following lemmas.

    Lemma 1. ([8]) Let h be analytic in Δ with h(0)=1, β>0 and 0γ1<1. If

    h(z)+βzh(z)h(z)1+(12γ1)z1z,

    then

    h(z)1+(12δ)z1z,

    where

    δ=(2γ1β)+(2γ1β)2+8β4. (2.1)

    Lemma 2. Let h be analytic in Δ and of the form

    h(z)=1+n=mbnzn(bm0)

    with h(z)0 in Δ. If there exists a point z0(|z0|<1) such that |argh(z)|<πρ2(|z|<|z0|) and |argh(z0)|=πρ2 for some ρ>0, then z0h(z0)h(z0)=iρ, where

    :{n2(c+1c)(argh(z0)=πρ2),n2(c+1c)(argh(z0)=πρ2),

    and (h(z0))1/ρ=±ic(c>0).

    This result is a generalization of the Nunokawa's lemma [29].

    Lemma 3. ([37]) Let ε be a positive measure on [0,1]. Let ϝ be a complex-valued function defined on Δ×[0,1] such that ϝ(.,t) is analytic in Δ for each t[0,1] and ϝ(z,.) is ε-integrable on [0,1] for all zΔ. In addition, suppose that (ϝ(z,t))>0, ϝ(r,t) is real and (1/ϝ(z,t))1/ϝ(r,t) for |z|r<1 and t[0,1]. If ϝ(z)=10ϝ(z,t)dε(t), then (1/ϝ(z))1/ϝ(r).

    Lemma 4. ([22]) If 1D<C1, λ1>0 and (γ2)λ1(1C)/(1D), then the differential equation

    s(z)+zs(z)λ1s(z)+γ2=1+Cz1+Dz(zΔ)

    has a univalent solution in Δ given by

    s(z)={zλ1+γ2(1+Dz)λ1(CD)/Dλ1z0tλ1+γ21(1+Dt)λ1(CD)/Ddtγ2λ1(D0),zλ1+γ2eλ1Czλ1z0tλ1+γ21eλ1Ctdtγ2λ1(D=0).

    If r(z)=1+c1z+c2z2+ satisfies the condition

    r(z)+zr(z)λ1r(z)+γ21+Cz1+Dz(zΔ),

    then

    r(z)s(z)1+Cz1+Dz,

    and s(z) is the best dominant.

    Lemma 5. ([36,Chapter 14]) Let w, x and\ y0,1,2, be complex numbers. Then, for (y)>(x)>0, one has

    1. 2G1(w,x,y;z)=Γ(y)Γ(yx)Γ(x)10sx1(1s)yx1(1sz)wds;

    2. 2G1(w,x,y;z)= 2G1(x,w,y;z);

    3. 2G1(w,x,y;z)=(1z)w2G1(w,yx,y;zz1).

    Firstly, we derive the following result.

    Theorem 1. Let 0α<1 and k11α. If fk-Q(α), then fB(δ), where

    δ=(2μλ)+(2μλ)2+8λ4(λ=1+αkk(1α);μ=kαk1k(1α)). (3.1)

    Proof. Let f=, where is analytic in Δ with (0)=1. From inequality (1.5) which takes the form

    (1+z(z)(z))>k|(1α)(z)+α(1+z(z)(z))1|=k|1α(z)+α(z)αz(z)(z)|,

    we find that

    ((z)+1+αkk(1α)z(z)(z))>kαk1k(1α),

    which can be rewritten as

    ((z)+λz(z)(z))>μ(λ=1+αkk(1α);μ=kαk1k(1α)).

    The above relationship can be written as the following Briot-Bouquet differential subordination

    (z)+λz(z)(z)1+(12μ)z1z.

    Thus, by Lemma 1, we obtain

    1+(12δ)z1z, (3.2)

    where δ is given by (3.1). The relationship (3.2) implies that fB(δ). We thus complete the proof of Theorem 3.1.

    Theorem 2. Let 0<α1, 0<β<1, c>0, k1, nm+1(m N ), ||n2(c+1c) and

    |αβ±(1α)cβsinβπ2|1. (3.3)

    If

    f(z)=z+n=m+1anzn(am+10)

    and fk-Q(α), then fB(β0), where

    β0=min{β:β(0,1)}

    such that (3.3) holds.

    Proof. By the assumption, we have

    f(z)=(z)=1+n=mcnzn(cm0). (3.4)

    In view of (1.5) and (3.4), we get

    (1+z(z)(z))>k|(1α)(z)+α(1+z(z)(z))1|.

    If there exists a point z0Δ such that

    |arg(z)|<βπ2(|z|<|z0|;0<β<1)

    and

    |arg(z0)|=βπ2(0<β<1),

    then from Lemma 2, we know that

    z0(z0)(z0)=iβ,

    where

    ((z0))1/β=±ic(c>0)

    and

    :{n2(c+1c)(arg(z0)=βπ2),n2(c+1c)(arg(z0)=βπ2).

    For the case

    arg(z0)=βπ2,

    we get

    (1+z0(z0)(z0))=(1+iβ)=1. (3.5)

    Moreover, we find from (3.3) that

    k|(1α)(z0)+α(1+z0(z0)(z0))1|=k|(1α)((z0)1)+αz0(z0)(z0)|=k|(1α)[(±ic)β1]+iαβ|=k(1α)2(cβcosβπ21)2+[αβ±(1α)cβsinβπ2]21. (3.6)

    By virtue of (3.5) and (3.6), we have

    (1+z(z0)(z0))k|(1α)(z0)+α(1+z0(z0)(z0))1|,

    which is a contradiction to the definition of k-Q(α). Since β0=min{β:β(0,1)} such that (3.3) holds, we can deduce that fB(β0).

    By using the similar method as given above, we can prove the case

    arg(z0)=βπ2

    is true. The proof of Theorem 2 is thus completed.

    Theorem 3. If 0<β<1 and 0ν<1. If fk-Q(α), then

    (f)>[2G1(2β(1ν),1;1β+1;12)]1,

    or equivalently, k-Q(α)B(ν0), where

    ν0=[2G1(2β(1μ),1;1β+1;12)]1.

    Proof. For

    w=2β(1ν), x=1β, y=1β+1,

    we define

    ϝ(z)=(1+Dz)w10tx1(1+Dtz)wdt=Γ(x)Γ(y) 2G1(1,w,y;zz1). (3.7)

    To prove k-Q(α)B(ν0), it suffices to prove that

    inf|z|<1{(q(z))}=q(1),

    which need to show that

    (1/ϝ(z))1/ϝ(1).

    By Lemma 3 and (3.7), it follows that

    ϝ(z)=10ϝ(z,t)dε(t),

    where

    ϝ(z,t)=1z1(1t)z(0t1),

    and

    dε(t)=Γ(x)Γ(w)Γ(yw)tw1(1t)yw1dt,

    which is a positive measure on [0,1].

    It is clear that (ϝ(z,t))>0 and ϝ(r,t) is real for |z|r<1 and t[0,1]. Also

    (1ϝ(z,t))=(1(1t)z1z)1+(1t)r1+r=1ϝ(r,t)

    for |z|r<1. Therefore, by Lemma 3, we get

    (1/ϝ(z))1/ϝ(r).

    If we let r1, it follows that

    (1/ϝ(z))1/ϝ(1).

    Thus, we deduce that k-Q(α)B(ν0).

    Theorem 4. Let 0α<1 and k11α. If fk-Q(α), then

    f(z)s(z)=1g(z),

    where

    g(z)=2G1(2λ,1,1λ+1;zz1)(λ=1+αkk(1α)).

    Proof. Suppose that f=. From the proof of Theorem 1, we see that

    (z)+z(z)1λ(z)1+(12μ)z1z1+z1z(λ=1+αkk(1α);μ=kαk1k(1α)).

    If we set λ1=1λ, γ2=0, C=1 and D=1 in Lemma 4, then

    (z)s(z)=1g(z)=z1λ(1z)2λ1/λz0t(1/λ)1(1t)2/λdt.

    By putting t=uz, and using Lemma 5, we obtain

    (z)s(z)=1g(z)=11λ(1z)2λ10u(1/λ)1(1uz)2/λdu=[2G1(2λ,1,1λ+1;zz1)]1,

    which is the desired result of Theorem 4.

    The present investigation was supported by the Key Project of Education Department of Hunan Province under Grant no. 19A097 of the P. R. China. The authors would like to thank the referees for their valuable comments and suggestions, which was essential to improve the quality of this paper.

    The authors declare no conflict of interest.



    [1] J. Chan, S. Yuan, K. H. Kok, K. K. Wang, H. Chu, J. Yang, et al., A familial cluster of pneumonia associated with the 2019 novel coronavirus indicating person-to-person transmission: a study of a family cluster, The Lancet, 395 (2020), 514–523. https://doi.org/10.1016/S0140-6736(20)30154-9 doi: 10.1016/S0140-6736(20)30154-9
    [2] W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics-Ⅰ, Bull. Math. Biol., 53 (1991), 33–55. https://doi.org/10.1007/BF02464423 doi: 10.1007/BF02464423
    [3] Q. Yang, D. Q. Jiang, N. Z. Shi, C. Y. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl., 388 (2017), 248–271. https://doi.org/10.1016/j.jmaa.2011.11.072 doi: 10.1016/j.jmaa.2011.11.072
    [4] H. Huo, P. Yang, H. Xiang, Stability and bifurcation for an SEIS epidemic model with the impact of media, Physica A Stat. Mech. Appl., 490 (2018), 702–720. https://doi.org/10.1016/j.physa.2017.08.139 doi: 10.1016/j.physa.2017.08.139
    [5] Y. Zhao, D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 243 (2014), 718–727. https://doi.org/10.1016/j.amc.2014.05.124 doi: 10.1016/j.amc.2014.05.124
    [6] T. Odagaki, Exact properties of SIQR model for COVID-19, Physica A Stat. Mech. Appl., 564 (2021), 125564. https://doi.org/10.1016/j.physa.2020.125564 doi: 10.1016/j.physa.2020.125564
    [7] S. Jain, S. Kumar, Dynamic analysis of the role of innate immunity in SEIS epidemic model, Eur. Phys. J. Plus, 136 (2021), 439. https://doi.org/10.1140/epjp/s13360-021-01390-3 doi: 10.1140/epjp/s13360-021-01390-3
    [8] A. Omar, Y. Alnafisah, R. A. Elbarkouky, H. M. Ahmed, COVID-19 deterministic and stochastic modeling with optimized daily vaccinations in Saudi Arabia, Results Phys., 28 (2021), 104629. https://doi.org/10.1016/j.rinp.2021.104629 doi: 10.1016/j.rinp.2021.104629
    [9] A. omar, R. A. Elbarkouky, H. M. Ahmed, Fractional stochastic modelling of COVID-19 under wide spread of vaccinations: Egyptian case study, Alexandrian Eng. J., 61 (2022), 8595–8609. https://doi.org/10.1016/j.aej.2022.02.002 doi: 10.1016/j.aej.2022.02.002
    [10] R. Din, E. A. Algehyne, Mathematical analysis of COVID-19 by using SIR model with convex incidence rate, Results Phys., 23 (2021), 103970. https://doi.org/10.1016/j.rinp.2021.103970 doi: 10.1016/j.rinp.2021.103970
    [11] O. Nave, U. Shemesh, I. HarTuv, Applizing Laplace Adomain decomposition method (LADM) for solving a model of COVID-19, Comput. Method. Biomec. Biomed. Eng., 24 (2021), 1618–1628. https://doi.org/10.1080/10255842.2021.1904399 doi: 10.1080/10255842.2021.1904399
    [12] World Health Organization, World health organization, contact tracing in the context of COVID-19, 2021. Available from: https://www.who.int/fr/publications-detail/contact tracing in the context of covid-19.
    [13] G. Zhang, Z. Li, A. Din, A stochastic SIQR epidemic model with Leˊvy jumps and three-time delays, Appl. Math. Comput., 431 (2022), 127329. https://doi.org/10.1016/j.amc.2022.127329 doi: 10.1016/j.amc.2022.127329
    [14] Y. Ma, J. Liu, H. Li, Global dynamics of an SIQR model with vaccination and elimination hybrid strategies, Mathematics, 6 (2018), 328. https://doi.org/10.3390/math6120328 doi: 10.3390/math6120328
    [15] X. Zhang, R. Liu, The stationary distribution of a stochastic SIQS epidemic model with varying total population size, Appl. Math. Lett., 116 (2021), 106974. https://doi.org/10.1016/j.aml.2020.106974 doi: 10.1016/j.aml.2020.106974
    [16] X. Zhang, H. Huo, H. Xiang, X. Meng, Dynamics of the deterministic and stochastic SIQS epidemic model with non-linear incidence, Appl. Math. Comput., 243 (2014), 546–558. https://doi.org/10.1016/j.amc.2014.05.136 doi: 10.1016/j.amc.2014.05.136
    [17] Q. Liu, D. Jiang, N. Shi, Threshold behavior in a stochastic SIQR epidemic model with stanadard incidence and regime switching, Appl. Math. Comput., 316 (2018), 310–325. https://doi.org/10.1016/j.amc.2017.08.042 doi: 10.1016/j.amc.2017.08.042
    [18] S. Ruschel, T. Pereira, S. Yanchuk, L. Young, An SIQ delay differential equations model for disease control via isolation, J. Math. Biol., 79 (2019), 249–279. https://doi.org/10.1007/s00285-019-01356-1 doi: 10.1007/s00285-019-01356-1
    [19] Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, Dynamics of a stochastic multigroup SIQR epidemic model with standard incidence rates, J. Franklin Inst., 356 (2019), 2960–2993. https://doi.org/10.1016/j.jfranklin.2019.01.038 doi: 10.1016/j.jfranklin.2019.01.038
    [20] V. Capasso, G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43–61. https://doi.org/10.1016/0025-5564(78)90006-8 doi: 10.1016/0025-5564(78)90006-8
    [21] S. Ruan, W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Equ., 188 (2003), 135–163. https://doi.org/10.1016/S0022-0396(02)00089-X doi: 10.1016/S0022-0396(02)00089-X
    [22] D. Xiao, S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419–429. https://doi.org/10.1016/j.mbs.2006.09.025 doi: 10.1016/j.mbs.2006.09.025
    [23] A. B. Gumel, S. Ruan, T. Day, J. Watmough, F. Brauer, P. van den Driessche, et al., Modelling strategies for controlling SARS outbreaks, Proc. R. Soc. Lond. B., 271 (2004), 2223–2232. https://doi.org/10.1098/rspb.2004.2800 doi: 10.1098/rspb.2004.2800
    [24] D. Li, J. Cui, M. Liu, S. Liu, The evolutionary dynamics of stochastic epidemic model with nonlinear incidence rate, Bull. Math. Biol., 77 (2015), 1705–1743. https://doi.org/10.1007/s11538-015-0101-9 doi: 10.1007/s11538-015-0101-9
    [25] G. Lan, S. Yuan, B. Song, The impact of hospital resources and environmental perturbations to the dynamics of SIRS model, J. Franklin Inst., 358 (2021), 2405–2433. https://doi.org/10.1016/j.jfranklin.2021.01.015 doi: 10.1016/j.jfranklin.2021.01.015
    [26] P. van den Driessche, Reproduction numbers of infectious disease models, Infect. Dis. Model., 2 (2017), 288–303. https://doi.org/10.1016/j.idm.2017.06.002 doi: 10.1016/j.idm.2017.06.002
    [27] X. R. Mao, Stochastic differential equations and applications, Cambridge: Woodhead Publishing, 2011.
    [28] Y. Cai, Y. Kang, W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comput., 305 (2017), 221–240. https://doi.org/10.1016/j.amc.2017.02.003 doi: 10.1016/j.amc.2017.02.003
    [29] R. Khasminskii, Stochastic stability of differential equations, Berlin: Springer, 2012. https://doi.org/10.1007/978-3-642-23280-0
    [30] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525–546. https://doi.org/10.1137/S0036144500378302 doi: 10.1137/S0036144500378302
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