Special Issues

Age-structured cell population model to study the influence of growth factors on cell cycle dynamics

  • Cell proliferation is controlled by many complex regulatory networks. Our purpose is to analyse, through mathematical modeling, the effects of growth factors on the dynamics of the division cycle in cell populations.
        Our work is based on an age-structured PDE model of the cell division cycle within a population of cells in a common tissue. Cell proliferation is at its first stages exponential and is thus characterised by its growth exponent, the first eigenvalue of the linear system we consider here, a growth exponent that we will explicitly evaluate from biological data.Moreover, this study relies on recent and innovative imaging data (fluorescence microscopy) that make us able to experimentally determine the parameters of the model and to validate numerical results.This model has allowed us to study the degree of simultaneity of phase transitions within a proliferating cell population and to analyse the role of an increased growth factor concentration in this process.
        This study thus aims at helping biologists to elicit the impact of growth factor concentration on cell cycle regulation, at making more precise the dynamics of key mechanisms controlling the division cycle in proliferating cell populations, and eventually at establishing theoretical bases for optimised combined anticancer treatments.

    Citation: Frédérique Billy, Jean Clairambault, Franck Delaunay, Céline Feillet, Natalia Robert. Age-structured cell population model to study the influence of growth factors on cell cycle dynamics[J]. Mathematical Biosciences and Engineering, 2013, 10(1): 1-17. doi: 10.3934/mbe.2013.10.1

    Related Papers:

    [1] Cameron Browne . Immune response in virus model structured by cell infection-age. Mathematical Biosciences and Engineering, 2016, 13(5): 887-909. doi: 10.3934/mbe.2016022
    [2] Khalid Hattaf, Noura Yousfi . Dynamics of SARS-CoV-2 infection model with two modes of transmission and immune response. Mathematical Biosciences and Engineering, 2020, 17(5): 5326-5340. doi: 10.3934/mbe.2020288
    [3] Ting Guo, Zhipeng Qiu . The effects of CTL immune response on HIV infection model with potent therapy, latently infected cells and cell-to-cell viral transmission. Mathematical Biosciences and Engineering, 2019, 16(6): 6822-6841. doi: 10.3934/mbe.2019341
    [4] Chentong Li, Jinhu Xu, Jiawei Liu, Yicang Zhou . The within-host viral kinetics of SARS-CoV-2. Mathematical Biosciences and Engineering, 2020, 17(4): 2853-2861. doi: 10.3934/mbe.2020159
    [5] A. M. Elaiw, A. S. Shflot, A. D. Hobiny . Stability analysis of general delayed HTLV-I dynamics model with mitosis and CTL immunity. Mathematical Biosciences and Engineering, 2022, 19(12): 12693-12729. doi: 10.3934/mbe.2022593
    [6] Xuejuan Lu, Lulu Hui, Shengqiang Liu, Jia Li . A mathematical model of HTLV-I infection with two time delays. Mathematical Biosciences and Engineering, 2015, 12(3): 431-449. doi: 10.3934/mbe.2015.12.431
    [7] Jiawei Deng, Ping Jiang, Hongying Shu . Viral infection dynamics with mitosis, intracellular delays and immune response. Mathematical Biosciences and Engineering, 2023, 20(2): 2937-2963. doi: 10.3934/mbe.2023139
    [8] Suqi Ma . Low viral persistence of an immunological model. Mathematical Biosciences and Engineering, 2012, 9(4): 809-817. doi: 10.3934/mbe.2012.9.809
    [9] Abdessamad Tridane, Yang Kuang . Modeling the interaction of cytotoxic T lymphocytes and influenza virus infected epithelial cells. Mathematical Biosciences and Engineering, 2010, 7(1): 171-185. doi: 10.3934/mbe.2010.7.171
    [10] A. M. Elaiw, Raghad S. Alsulami, A. D. Hobiny . Global dynamics of IAV/SARS-CoV-2 coinfection model with eclipse phase and antibody immunity. Mathematical Biosciences and Engineering, 2023, 20(2): 3873-3917. doi: 10.3934/mbe.2023182
  • Cell proliferation is controlled by many complex regulatory networks. Our purpose is to analyse, through mathematical modeling, the effects of growth factors on the dynamics of the division cycle in cell populations.
        Our work is based on an age-structured PDE model of the cell division cycle within a population of cells in a common tissue. Cell proliferation is at its first stages exponential and is thus characterised by its growth exponent, the first eigenvalue of the linear system we consider here, a growth exponent that we will explicitly evaluate from biological data.Moreover, this study relies on recent and innovative imaging data (fluorescence microscopy) that make us able to experimentally determine the parameters of the model and to validate numerical results.This model has allowed us to study the degree of simultaneity of phase transitions within a proliferating cell population and to analyse the role of an increased growth factor concentration in this process.
        This study thus aims at helping biologists to elicit the impact of growth factor concentration on cell cycle regulation, at making more precise the dynamics of key mechanisms controlling the division cycle in proliferating cell populations, and eventually at establishing theoretical bases for optimised combined anticancer treatments.


    Middle East respiratory syndrome (MERS) is a viral respiratory disease caused by Middle East respiratory syndrome coronavirus (MERS-CoV). The intermediate host of MERS-CoV is probably the dromedary camel, a zoonotic virus [1]. Most MERS cases are acquired by human-to-human transmission. There is no vaccine or specific treatment available, and approximately $ 35\% $ of patients with MERS-CoV infection have died [2]. There has been extensive works on infectious disease models and viral infection models associated with MERS that can help in disease control and provide strategies for potential drug treatments [3,4,5,6,7,8].

    Dipeptidyl peptidase-4 (DPP4) plays an important role in viral infection [2]. Based on classic viral infection models developed in [9,10,11], a four-dimensional ordinary differential equation model is proposed and studied in [8]. The model in [8] describes the interaction mechanisms among uninfected cells, infected cells, DPP4 and MERS-CoV.

    Recently, taking into account periodic factors such as diurnal temperature differences and periodic drug treatment, the model in [8] has been further extended a periodic case in [12], and then the existence of positive periodic solutions is studied by using the theorem in [13].

    It is well-known that CTL immune responses play a very critical role in controlling viral load and the concentration of infected cells. Thus, many scholars have considered CTL immune responses in various viral infection models and have achieved many excellent research results [14,15,16,17,18]. CTL cells can kill virus-infected cells and are important for the control and clearance of MERS-CoV infections [19]. Inspired by the above research works, we consider the following periodic MERS-CoV infection model with CTL immune response:

    $ {˙T(t)=λ(t)β(t)D(t)v(t)T(t)d(t)T(t),˙I(t)=β(t)D(t)v(t)T(t)d1(t)I(t)p(t)I(t)Z(t),˙v(t)=d1(t)M(t)I(t)c(t)v(t),˙D(t)=λ1(t)β1(t)β(t)D(t)v(t)T(t)γ(t)D(t),˙Z(t)=q(t)I(t)Z(t)b(t)Z(t).
    $
    (1.1)

    In model (1.1), $ T(t) $, $ I(t) $, $ v(t) $, $ D(t) $ and $ Z(t) $ represent the concentrations of uninfected cells, infected cells, free virus, DPP4 on the surface of uninfected cells and CTL cells at time $ t $, respectively. CTL cells increase at a rate bilinear rate $ q(t)I(t)Z(t) $ by the viral antigen of the infected cells and decay at rate $ b(t)Z(t) $; infected cells are killed by the CTL immune response at rate $ p(t)I(t)Z(t) $. Except for $ p(t) $, $ q(t) $ and $ b(t) $, all the remaining parameters of model (1.1) have the same biological meanings as in [12].

    Throughout the paper, it is assumed that the functions $ \lambda(t) $, $ \beta(t) $, $ d(t) $, $ d_{1}(t) $, $ p(t) $, $ M(t) $, $ c(t) $, $ \lambda_{1}(t) $, $ \gamma(t) $, $ q(t) $ and $ b(t) $ are positive, continuous and $ \omega $ periodic ($ \omega > 0 $); the function $ \beta_{1}(t) $ is non-negative, continuous and $ \omega $ periodic.

    From point of view in both biology and mathematics, it is one of the most significant topics to study the existence of periodic oscillations of a system (see, for example, [12,20,21,22,23,24,25,26] and the references therein).

    In the next section, some sufficient criteria are given for the existence of positive periodic oscillations of model (1.1). It should be mentioned here that, in the proofs of the main results in the following section, a new technique is developed to obtain a lower bound of the state variable $ Z(t) $ characterizing CTL immune response in model (1.1).

    For some function $ f(t) $ which is continuous and $ \omega $-periodic on $ \mathbb{R} $, let us define the following notations:

    $ f^{U} = \max\limits_{t\in[0,\omega]}f(t), \; \; \; f^{l} = \min\limits_{t \in[0,\omega]} f(t), \; \; \; \widehat{f} = \frac{1}{\omega}\int_{0}^{\omega}f(t)dt. $

    Moreover, for convenience, let us give the following parameters:

    $ R=ˆλβlexp{L3+L4}^d1exp{M2}(βlexp{L3+L4}+dU)>1,ω=ˆb2ˆλˆq,δ=^d12ˆp(R1),M1=ln(λUdl),M2=ln(ˆbˆq+2ˆλω),M3=ln(^(d1M)ˆc)+M2+2ˆcω,M4=ln(λU1γl),M5=ln(ˆβexp{M1+M3+M4}ˆpexp{L2})+2ˆbω,L1=ln(λlβUexp{M3+M4}+dU),L2=ln(ˆbˆq2ˆλω),L3=ln((^d1M)ˆc)+L22ˆcω,L4=ln(λl1(β1β)Uexp{M1+M3}+γU),L5=ln(δ)2ˆbω.
    $

    The following theorem is the main result of this paper.

    Theorem 2.1. If $ R^{*} > 1 $ and $ \omega < \omega^{*} $, then model (1.1) has at least one positive $ \omega $-periodic solution.

    Proof. Making the change of variables $ T(t) = \exp\{u_1(t)\} $, $ I(t) = \exp\{u_2(t)\} $, $ v(t) = \exp\{u_3(t)\} $, $ D(t) = \exp\{u_4(t)\} $, $ Z(t) = \exp\{u_5(t)\} $, then model (1.1) can be rewritten as

    $ {˙u1(t)=λ(t)exp{u1(t)}β(t)exp{u3(t)+u4(t)}d(t),˙u2(t)=β(t)exp{u1(t)+u3(t)+u4(t)}exp{u2(t)}d1(t)p(t)exp{u5(t)},˙u3(t)=d1(t)M(t)exp{u2(t)}exp{u3(t)}c(t),˙u4(t)=λ1(t)exp{u4(t)}β1(t)β(t)exp{u1(t)+u3(t)}γ(t),˙u5(t)=q(t)exp{u2(t)}b(t).
    $
    (2.1)

    Thus, we only need to consider model (2.1).

    Let us set

    $ X = Y = \left\{ u = (u_1(t), u_2(t), u_3(t), u_4(t), u_5(t))^{T} \in C(\mathbb{R}, \mathbb{R}^{5})\; |\; u(t) = u(t+\omega) \right\} $

    with the norm

    $ ||u|| = \max\limits_{t\in[0,\omega]}|u_1(t)|+ \max\limits_{t\in[0,\omega]}|u_2(t)|+ \max\limits_{t\in[0,\omega]}|u_3(t)|+ \max\limits_{t\in[0,\omega]}|u_4(t)|+ \max\limits_{t\in[0,\omega]}|u_5(t)|. $

    It can be shown that $ X $ and $ Y $ are Banach spaces. Define

    $ Nu=[λ(t)exp{u1(t)}β(t)exp{u3(t)+u4(t)}d(t)β(t)exp{u1(t)+u3(t)+u4(t)}exp{u2(t)}d1(t)p(t)exp{u5(t)}d1(t)M(t)exp{u2(t)}exp{u3(t)}c(t)λ1(t)exp{u4(t)}β1(t)β(t)exp{u1(t)+u3(t)}γ(t)q(t)exp{u2(t)}b(t)]:=[N1(t)N2(t)N3(t)N4(t)N5(t)](uX),
    $
    $ Lu = \dot{u}\; (u\in {\rm{Dom}}\; L),\; \; \; P u = \frac{1}{\omega}\int_{0}^{\omega}u(t)dt\; (u\in X),\; \; \; Q u = \frac{1}{\omega}\int_{0}^{\omega}u(t)dt\; (u\in Y), $

    here $ {\rm{Dom}}\; L = \{u\in X, \; \dot{u}\in X\} $. It easily has that $ {\rm{Ker}}\; L = \{u\in X\; |\; u\; \in \mathbb{R}^5\} $ and $ {\rm{Im}}\; L = \{u\in Y\; |\; \int_{0}^{\omega}u(t)dt = 0\} $. Further, it is clear that $ {\rm{Im}}\; L $ is closed in $ Y $ and $ {\rm{dim\; Ker}}\; L = {\rm{codim\; Im}}\; L = 5 $. Hence, $ L $ is a Fredholm mapping with index zero.

    For $ \mu\in(0, 1) $, let us consider the equation $ Lu = \mu Nu $, i.e.,

    $ {˙u1(t)=μ[λ(t)exp{u1(t)}β(t)exp{u3(t)+u4(t)}d(t)],˙u2(t)=μ[β(t)exp{u1(t)+u3(t)+u4(t)}exp{u2(t)}d1(t)p(t)exp{u5(t)}],˙u3(t)=μ[d1(t)M(t)exp{u2(t)}exp{u3(t)}c(t)],˙u4(t)=μ[λ1(t)exp{u4(t)}β1(t)β(t)exp{u1(t)+u3(t)}γ(t)],˙u5(t)=μ[q(t)exp{u2(t)}b(t)].
    $
    (2.2)

    For any solution $ u = (u_1(t), u_2(t), u_3(t), u_4(t), u_5(t))^{T}\in X $ of (2.2), it has

    $ {ω0[λ(t)exp{u1(t)}β(t)exp{u3(t)+u4(t)}d(t)]dt=0,ω0[β(t)exp{u1(t)+u3(t)+u4(t)}exp{u2(t)}d1(t)p(t)exp{u5(t)}]dt=0,ω0[d1(t)M(t)exp{u2(t)}exp{u3(t)}c(t)]dt=0,ω0[λ1(t)exp{u4(t)}β1(t)β(t)exp{u1(t)+u3(t)}γ(t)]dt=0,ω0[q(t)exp{u2(t)}b(t)]dt=0.
    $
    (2.3)

    From the first two equations in (2.2), it has

    $ ˙u1(t)exp{u1(t)}=μ[λ(t)β(t)exp{u1(t)+u3(t)+u4(t)}d(t)exp{u1(t)}],
    $

    and

    $ ˙u2(t)exp{u2(t)}=μ[β(t)exp{u1(t)+u3(t)+u4(t)}d1(t)exp{u2(t)}p(t)exp{u2(t)+u5(t)}].
    $

    Hence, by integrating the above two equations on $ [0, \omega] $, it has

    $ ω0[λ(t)β(t)exp{u1(t)+u3(t)+u4(t)}d(t)exp{u1(t)}]dt=0
    $
    (2.4)

    and

    $ ω0[β(t)exp{u1(t)+u3(t)+u4(t)}d1(t)exp{u2(t)}p(t)exp{u2(t)+u5(t)}]dt=0.
    $
    (2.5)

    Note that $ I(t): = \exp\{u_{2}(t)\} $ satisfies

    $ \dot{I}(t) = \dot{u}_{2}(t)\exp\{u_{2}(t)\} = \mu\left[\beta(t)\exp \{u_1(t)+u_3(t)+u_4(t)\}-d_{1}(t)-p(t)\exp\{u_{2}(t)+u_5(t)\}\right]. $

    Then, from (2.4) and (2.5), it has

    $ ω0|˙I(t)|dtμω0[β(t)exp{u1(t)+u3(t)+u4(t)}+d1(t)+p(t)exp{u2(t)+u5(t)}]dt2ω0β(t)exp{u1(t)+u3(t)+u4(t)}dt2ˆλω.
    $
    (2.6)

    From the third and the fifth equations of (2.2), it has

    $ ω0|˙u3(t)|dtμ[ω0d1(t)M(t)exp{u2(t)}exp{u3(t)}dt+ω0c(t)dt]<2ˆcω,ω0|˙u5(t)|dtμ[ω0q(t)exp{u2(t)}dt+ω0b(t)dt]<2ˆbω.
    $
    (2.7)

    Note that $ u\in X $, there exist $ \xi_{i}, \; \eta_{i}\in[0, \omega] $ $ (i = 1, 2, 3, 4, 5) $, such that

    $ u_{i}(\xi_{i}) = \min\limits_{t\in[0,\omega]}u_i(t), \; \; u_i(\eta_{i}) = \max\limits_{t\in[0,\omega]}u_i(t)\; (i = 1,2,3,4,5). $

    From (2.2), $ \dot{u}_{1}(\eta_{1}) = 0 $ and $ \dot{u}_{4}(\eta_{4}) = 0 $, it has

    $ λ(η1)exp{u1(η1)}β(η1)exp{u3(η1)+u4(η1)}d(η1)=0,λ1(η4)exp{u4(η4)}β1(η4)β(η4)exp{u1(η4)+u3(η4)}γ(η4)=0,
    $

    which imply that

    $ u1(t)u1(η1)ln(λ(η1)d(η1))ln(λUdl)=M1,u4(t)u4(η4)ln(λ1(η4)γ(η4))ln(λU1γl)=M4.
    $
    (2.8)

    From the last equation of (2.3), it has

    $ \int_{0}^{\omega}q(t)\exp\{u_{2}(\xi_{2})\}dt\leq\widehat{b}\omega \leq \int_{0}^{\omega}q(t)\exp\{u_{2}(\eta_{2})\}dt, $

    which implies that

    $ I(\xi_{2}) = \exp\{u_{2}(\xi_{2})\}\leq \frac{\widehat{b}}{\widehat{q}} \leq \exp\{u_{2}(\eta_{2})\} = I(\eta_{2}). $

    Then, from (2.6) and $ \omega < \omega^{*} $, it has

    $ I(t)I(ξ2)+ω0|˙I(t)|dtˆbˆq+2ˆλω,I(t)I(η2)ω0|˙I(t)|dtˆbˆq2ˆλω=2ˆλ(ωω)>0.
    $

    Thus, it has

    $ u2(t)ln(ˆbˆq+2ˆλω)=M2,u2(t)ln(ˆbˆq2ˆλω)=L2.
    $
    (2.9)

    From the third equation of (2.3), it has

    $ \int_{0}^{\omega}d_1(t)M(t)\frac{\exp\{M_2\}} {\exp\{u_3(\xi_3)\}}dt \geq \widehat{c}\omega \geq \int_{0}^{\omega}d_1(t)M(t)\frac{\exp\{L_2\}} {\exp\{u_3(\eta_3)\}}dt, $

    which implies that

    $ u_3(\xi_3)\leq\ln\left(\frac{\widehat{(d_{1}M)}}{\widehat{c}}\right)+M_{2}, \; \; \; u_3(\eta_3)\geq \ln\left(\frac{\widehat{(d_{1}M)}}{\widehat{c}}\right)+L_{2}. $

    Then, from (2.7), it has

    $ u3(t)u3(ξ3)+ω0|˙u3(t)|dtln(^(d1M)ˆc)+M2+2ˆcω=M3,u3(t)u3(η3)ω0|˙u3(t)|dtln(^(d1M)ˆc)+L22ˆcω=L3.
    $
    (2.10)

    From the second equation of (2.3), it has

    $ \widehat{p}\exp\{u_5(\xi_{5})\}\omega\leq \int_{0}^{\omega}\left[\beta(t)\frac{\exp\{M_{1}+M_{3}+M_{4}\}} {\exp \{L_{2}\}}-d_1(t)\right]dt\leq \frac{\exp\{M_{1}+M_{3}+M_{4}\}} {\exp \{L_{2}\}}\widehat{\beta}\omega, $

    which implies that

    $ u_5(\xi_{5})\leq \ln\left( \frac{\widehat{\beta}\exp\{M_{1}+M_{3}+M_{4}\}} {\widehat{p}\exp \{L_{2}\}}\right): = l_{5}. $

    Then, from (2.7), it has

    $ u_{5}(t)\leq u_{5}(\xi_{5})+\int_{0}^{\omega}|\dot{u}_{5}(t)| dt \leq l_{5}+2 \widehat{b}\omega = {M}_5. $

    From $ \dot{u}_1(\xi_1) = 0 $, $ \dot{u}_4(\xi_4) = 0 $, (2.8) and (2.10), it has

    $ exp{u1(ξ1)}=λ(ξ1)β(ξ1)exp{u3(ξ1)+u4(ξ1)}+d(ξ1)λlβUexp{M3+M4}+dU,exp{u4(ξ4)}=λ1(ξ4)β1(ξ4)β(ξ4)exp{u1(ξ4)+u3(ξ4)}+γ(ξ4)λl1(β1β)Uexp{M1+M3}+γU.
    $

    Thus, it has

    $ u1(t)u1(ξ1)ln(λlβUexp{M3+M4}+dU)=L1,u4(t)u4(ξ4)=ln(λl1(β1β)Uexp{M1+M3}+γU)=L4.
    $
    (2.11)

    Let us give an estimate of the lower bound of the state variable $ u_5(t) $ related to CTL immune response. It should be mentioned here that a completely different method from that in [12] has been used.

    Claim A If $ R^{*} > 1 $ and $ \omega < \omega^{*} $, then

    $ \exp\{u_5(\eta_{5})\}\geq \delta^{*}. $

    If Claim A is not true, then it has that, for any $ t $, $ \exp\{u_5(t)\} \leq \exp\{u_5(\eta_{5})\} < \delta^{*} $. Hence, it has from (2.3), (2.9)–(2.11) that

    $ 0=ω0[β(t)exp{u1(t)+u3(t)+u4(t)}exp{u2(t)}d1(t)p(t)exp{u5(t)}]dtω0[β(t)exp{u1(t)+L3+L4}exp{M2}d1(t)p(t)exp{u5(η5)}]dtβlexp{L3+L4}exp{M2}ω0exp{u1(t)}dt(ˆd1+ˆpδ)ω,
    $

    which implies that

    $ ω0d(t)exp{u1(t)}dtdUω0exp{u1(t)}dtdU(ˆd1+ˆpδ)exp{M2}βlexp{L3+L4}ω:=Ψω.
    $
    (2.12)

    Adding (2.4) and (2.5) together, it has

    $ ω0[λ(t)d(t)exp{u1(t)}]dt=ω0[d1(t)exp{u2(t)}+p(t)exp{u2(t)+u5(t)}]dtω0exp{M2}[d1(t)+p(t)exp{u5(η5)}]dtexp{M2}(^d1+ˆpδ)ω,
    $

    which implies that

    $ ω0d(t)exp{u1(t)}dt[ˆλexp{M2}(^d1+ˆpδ)]ω=Ψω+[ˆλΨexp{M2}(^d1+ˆpδ)]ω=Ψω+[ˆλexp{M2}(1+dUβlexp{L3+L4})(^d1+ˆpδ)]ω=Ψω+^d1exp{M2}(1+dUβlexp{L3+L4})(R1ˆp^d1δ)ω=Ψω+^d12exp{M2}(1+dUβlexp{L3+L4})(R1)ω>Ψω,
    $

    which is a contradiction to (2.12). Thus, the claim holds.

    From Claim A and (2.7), it has

    $ u5(t)u5(η5)ω0|˙u5(t)|dtln(δ)2ˆbω=L5.
    $
    (2.13)

    Now, for convenience, let us define

    $ \overline{R^{*}} = \left(\widehat{\lambda}-\frac{\widehat{d_{1}}\widehat{b}}{\widehat{q}}\right) \frac{\widehat{\beta}\widehat{(d_{1}M)}}{\widehat{d}\widehat{d_{1}}\widehat{c}} \frac{\widehat{\lambda_{1}}}{\widehat{({\beta_1\beta})} \frac{\widehat{\lambda}\widehat{(d_{1}M)} \widehat{b}}{\widehat{d}\widehat{c}\widehat{q}}+\widehat{\gamma}}, \; \; \; Z_{max} = \frac{\widehat{q}}{\widehat{p}\widehat{b}} \left(\widehat{\lambda}-\frac{\widehat{d_{1}}\widehat{b}}{\widehat{q}}\right). $

    Note that if $ R^{*} > 1 $, then it has

    $ \widetilde{R^{*}}: = \left(\widehat{\lambda}-\widehat{d_{1}}\exp\{M_{2}\}\right) \frac{\beta^{l}\exp\{L_{3}+L_{4}\}}{d^{U}\widehat{d_{1}}\exp\{M_{2}\}} > 1, $

    which implies that

    $ Zmax>ˆqˆpˆb[ˆλ^d1(ˆbˆq+2ˆλω)]=ˆqˆpˆb(ˆλ^d1exp{M2})>0,¯R(ˆλ^d1ˆbˆq)ˆβ^(d1M)ˆd^d1ˆcλl1(β1β)Uexp{M3+M1}+γU~R>1.
    $

    Let $ (u_{1}, u_{2}, u_{3}, u_{4}, u_{5})^{T}\in \mathbb{R}^{5} $ be the solution of the following equations:

    $ {ˆλexp{u1}ˆβexp{u3+u4}ˆd=0,ˆβexp{u1+u3+u4}exp{u2}ˆd1ˆpexp{u5}=0,^(d1M)exp{u2}exp{u3}ˆc=0,ˆλ1exp{u4}^(β1β)exp{u1+u3}ˆγ=0,ˆqexp{u2}ˆb=0..
    $
    (2.14)

    Define $ \Gamma : [0, Z_{max}] \rightarrow \mathbb{R} $, via

    $ \Gamma(x) = \frac{\widehat{\beta}\widehat{(d_{1}M)}}{\widehat{c}} \frac{\widehat{\lambda_{1}}\Gamma_{1}(x)} {\widehat{(\beta_{1}\beta)}\Gamma_{1}(x)\frac{\widehat{(d_{1}M)}\widehat{b}}{\widehat{q}\widehat{c}} +\widehat{\gamma}} -\widehat{d}_1-\widehat{p}x, $

    where

    $ \Gamma_{1}(x) = \frac{\widehat{\lambda}}{\widehat{d}} -\frac{\widehat{d_{1}}\widehat{b}}{\widehat{d}\widehat{q}} -\frac{\widehat{p}\widehat{b}}{\widehat{d}\widehat{q}}x = \frac{\widehat{p}\widehat{b}}{\widehat{d}\widehat{q}}(Z_{max}-x). $

    Equation (2.14) can be rewritten as

    $ exp{u2}=ˆbˆq,exp{u3}=^(d1M)ˆcexp{u2}=^(d1M)ˆbˆqˆc,exp{u1}=ˆλˆd^d1exp{u2}ˆdˆpexp{u2}ˆdexp{u5}=Γ1(exp{u5}),exp{u4}=^λ1^(β1β)exp{u1+u3}+ˆγ=^λ1^(β1β)Γ1(exp{u5})^(d1M)ˆbˆqˆc+ˆγ,ˆβ^(d1M)ˆcexp{u1+u4}ˆd1ˆpexp{u5}=Γ(exp{u5})=0.
    $

    It is obvious that if there is a solution $ (u_{1}, u_{2}, u_{3}, u_{4}, u_{5})^{T}\in \mathbb{R}^{5} $ for (2.14), it must have $ 0 < \exp\{u_5\} < Z_{max} $. In addition, note that $ \Gamma(x) $ is monotonically decreasing with respect to $ x $ on $ [0, Z_{max}] $. It has from $ \Gamma\left(Z_{max}\right) = -\widehat{d_{1}}-\widehat{p}Z_{max} < 0 $ and

    $ \Gamma(0) = \frac{\widehat{\beta}\widehat{(d_{1}M)}}{\widehat{c}} \frac{\widehat{\lambda_{1}}\Gamma_{1}(0)} {\widehat{(\beta_{1}\beta)}\Gamma_{1}(0) \frac{\widehat{(d_{1}M)}\;\widehat{b}}{\widehat{q}\widehat{c}} +\widehat{\gamma}}-\widehat{d_1} > \frac{\widehat{\beta}\widehat{(d_{1}M)}}{\widehat{c}} \frac{\widehat{\lambda_{1}}(\frac{\widehat{\lambda}}{\widehat{d}} -\frac{\widehat{d_{1}}\widehat{b}}{\widehat{d}\widehat{q}})} {\widehat{(\beta_{1}\beta)}\frac{\widehat{\lambda}}{\widehat{d}} \frac{\widehat{(d_{1}M)}\;\widehat{b}}{\widehat{q}\widehat{c}} +\widehat{\gamma}}-\widehat{d_1} = \widehat{d_{1}}(\overline{R^{*}}-1) > 0 $

    that there exists a unique positive constant $ x = Z^{*}\in(0, Z_{max}) $ such that $ \Gamma(Z^{*}) = 0 $.

    The above discussions show that, if $ R^{*} > 1 $, (2.14) has a unique solution $ (u_{1}^{*}, u_{2}^{*}, u_{3}^{*}, u_{4}^{*}, u_{5}^{*})^{T} $, here $ u_i^* = \ln(e_{i}) $ $ (i = 1, 2, 3, 4, 5) $,

    $ e_{1} = \Gamma_{1}(Z^{*}) > 0,\; \; e_{2} = \frac{\widehat{b}}{\widehat{q}} > 0,\; \; e_{3} = \frac{\widehat{(d_{1}M)}\widehat{b}}{\widehat{q}\widehat{c}} > 0,\; \; e_{4} = \frac{\widehat{\lambda_{1}}} {\widehat{(\beta_{1}\beta)}\Gamma_{1}(Z^{*}) \frac{\widehat{(d_{1}M)}\;\widehat{b}}{\widehat{q}\widehat{c}} +\widehat{\gamma}} > 0,\; \; e_{5} = Z^{*} > 0. $

    Let us define the following set

    $ \Omega = \{u\in X\; |\; ||u|| < U_{1} = 1+\sum\limits_{i = 1}^{5} (\max\{|M_{i}|,|L_{i}|\}+|u_{i}^{*}|)\}\subset X. $

    Moreover, by similar arguments as in [12], it has that $ N $ is $ L $-compact on $ \overline{\Omega} $.

    Now, let us compute the Leray-Schauder degree $ {\rm{deg}}\left\{QN, \partial\Omega\cap Ker\; L, (0, 0, 0, 0, 0)^{T}\right\} : = \Delta $ as follows,

    $ Δ=sign|ˆλe10ˆβe3e4ˆβe3e40ˆβe1e3e4e2ˆβe1e3e4e2ˆβe1e3e4e2ˆβe1e3e4e2ˆpe50^(d1M)e2e3^(d1M)e2e300^(β1β)e1e30^(β1β)e1e3^λ1e400ˆqe2000|=sign{^(d1M)ˆpˆqe22e5e3(ˆλ^λ1e1e4ˆβ^(β1β)e1e23e4)}=sign{^(d1M)ˆpˆqe22e5e3e1e4[ˆde1(^(β1β)e1e3e4+ˆγe4)+ˆβˆγe1e3e24]}=10,
    $

    where $ \widehat{\lambda} = \widehat{\beta}e_{1}e_{3}e_{4}+\widehat{d}e_{1} $ and $ \widehat{\lambda_{1}} = \widehat{(\beta_{1}\beta)}e_{1}e_{3}e_{4}+\widehat{\gamma}e_{4} $ are used.

    Finally, it has those all the conditions of the continuation theorem in [13] (also see, for example, Lemma 2.1 in [12]) are satisfied. This proves that, if $ \omega < \omega^{*} $ and $ R^{*} > 1 $, model (2.1) has at least one $ \omega $-periodic solution.

    Let us consider the following classical viral infection dynamic model [9] with CTL immune response:

    $ \left\{ ˙T(t)=λ(t)β(t)v(t)T(t)d(t)T(t),˙I(t)=β(t)v(t)T(t)d1(t)I(t)p(t)I(t)Z(t),˙v(t)=d1(t)M(t)I(t)c(t)v(t),˙Z(t)=q(t)I(t)Z(t)b(t)Z(t),
    \right. \;\;\;\;\;\; ({\rm{A}}) $

    where, all the coefficients are the same with that in model (1.1).

    Define $ R_{1} : [0, \omega^{*}] \rightarrow \mathbb{R} $, via

    $ R_{1}(x) = \frac{\widehat{\lambda}\beta^{l} \left[\frac{\widehat{(d_{1}M)}}{\widehat{c}}\exp\{-2\widehat{c}x\} \left(\frac{\widehat{b}}{\widehat{q}}-2\widehat{\lambda}x\right)\right] }{\widehat{d_{1}}\left(\frac{\widehat{b}}{\widehat{q}}+2\widehat{\lambda}x\right) \left\{d^{U}+\beta^{l} \left[\frac{\widehat{(d_{1}M)}}{\widehat{c}}\exp\{-2\widehat{c}x\} \left(\frac{\widehat{b}}{\widehat{q}}-2\widehat{\lambda}x\right)\right]\right\}}. $

    Obviously, $ R_1(x) $ is monotonically decreasing on $ [0, \omega^{*}] $ and

    $ R_{1}(0) = \frac{\widehat{\lambda}\beta^{l}\widehat{(d_{1}M)}\widehat{q}} { \widehat{d_{1}}(d^{U}\widehat{c}\widehat{q}+ \beta^{l}\widehat{(d_{1}M)}\widehat{b}) },\; \; \; R_{1}(\omega^{*}) = 0. $

    Therefore, if $ R_{1}(0) > 1 $, then there exists a unique constant $ \omega^{**}\in(0, \omega^{*}) $ such that $ R_{1}(\omega^{**}) = 1 $, $ R_{1}(x) > 1 $ for $ 0\leq x < \omega^{**} $ and $ R_{1}(x) < 1 $ for $ \omega^{**} < x\leq\omega^{*} $.

    For model (A), it is not difficult to derive the following result.

    Theorem 2.2. If $ R_{1}(\omega) > 1 $ and $ \omega < \omega^{*} $ (i.e. $ R_{1}(0) > 1 $ and $ \omega < \omega^{**} < \omega^{*} $), then model (A) has at least one positive $ \omega $-periodic solution.

    Remark 2.1. If all the coefficients in model (A) take constants values, i.e., $ \lambda(t)\equiv\lambda > 0 $, $ \beta(t)\equiv\beta > 0 $, $ d(t)\equiv d > 0 $, $ d_{1}(t)\equiv d_{1} > 0 $, $ p(t)\equiv p > 0 $, $ M(t)\equiv M > 0 $, $ c(t)\equiv c > 0 $, $ q(t)\equiv q > 0 $ and $ b(t)\equiv b > 0 $, then model (A) becomes the classical model which is first proposed by Nowak and Bangham in [9]. the condition $ \omega < \omega^{**} $ in Theorem 2.2 is naturally satisfied. Furthermore, it has $ R_{1}(0) = (\lambda\beta M q)/(dcq+\beta d_{1}Mb): = R_{1} $. From \emph{[9]}, it has that the condition $ R_{1} > 1 $ implies the existence of a unique positive equilibrium. This shows that the conditions and conclusion in Theorem 2.2 are reasonable.

    In summary, Theorem 2.1 in the paper successfully extends the main result in [12]) to a MERS-CoV viral infection model with CTL immune response. In the proof of Theorem 2.1, we use a very different method from that in [9] to obtain the lower bound $ (\ln(\delta^{*})-2\widehat{b}\omega) $ of the state variable $ u_5(t) $. Furthermore, as a special case, Theorem 2.2 gives sufficient conditions for the existence of positive periodic solution of model (A). Model (A) is a natural extension of the classical model in [9]. As the end of the paper, let us give a example to summarize the applications of Theorem 2.1. Let us choose the coefficients in model (1.1) as follows (for the values of some parameters, please refer to [7,27] for the case of some autonomous models), $ \lambda(t) = 45(1+0.1\sin(4\pi t)) $, $ \beta(t) = 1.4\times 10^{-8}(1+0.1\cos(4\pi t)) $, $ d(t) = 0.001(1+0.5\cos(4\pi t)) $, $ d_{1}(t) = 0.056(1+0.5\cos(4\pi t)) $, $ p(t) = 0.00092(1+0.5\cos(4\pi t)) $, $ M(t) = 100000 $, $ c(t) = 2.1(1+0.3\cos(4\pi t)) $, $ \lambda_{1}(t) = 10(1+0.1\sin(4\pi t)) $, $ \beta_{1}(t) = 0.001 $, $ \gamma(t) = 0.01(1+0.1\cos(4\pi t)) $, $ q(t) = 0.005(1+0.5\sin(4\pi t)) $, $ b(t) = 0.5(1+0.4\cos(4\pi t)) $. Then, with the help of Maple mathematical software, it has $ \omega = 0.5 < \omega^{*}\approx1.111111 $, $ M_{1}\approx11.502875 $, $ M_2\approx4.976734 $, $ M_3\approx14.965318 $, $ M_{4}\approx7.108426 $, $ M_{5}\approx18.976215 $, $ L_{1}\approx-0.383569 $, $ L_{2} = 4.007333 $, $ L_{3}\approx9.795917 $, $ L_{4}\approx0.623402 $, $ L_{5}\approx1.387881 $, $ R^{*}\approx1.2170332 > 1 $. From Theorem 2.1, it follows that model (1.1) has at least one positive $ \omega $ ($ \omega = 0.5 $)-periodic solution. Figure 1 gives the corresponding numerical simulation, and the initial value is chosen as $ (T(0), I(0), v(0), D(0), Z(0))^T = (12.5,100, 265000,995.4,423)^{T} $.

    Figure 1.  With the increasing of the time $ t $, the evolution form of the solution of model (1.1).

    This paper is supported by National Natural Science Foundation of China (No.11971055) and Beijing Natural Science Foundation (No.1202019).

    The authors declare there is no conflict of interest.

    [1] Acta Biotheor., 43 (1995), 3-25.
    [2] SIAM J. Appl. Math., 53 (1993), 1480-1504.
    [3] J. Theor. Med., 1 (1997), 35-51.
    [4] Math. Biosci., 218 (2009), 1-14.
    [5] Bull Math. Biol., 67 (2005), 815-830.
    [6] Bull Math. Biol., 69 (2007) 1673-1690.
    [7] Mathematical Modelling of Natural Phenomena, 7 (2012) 306-336.
    [8] Math. Comp. Simul., 2012. in press, available on line Apr. 2012.
    [9] Personalized Medicine, 8 (2011), 271-286.
    [10] Mathematical Modelling of Natural Phenomena, 4 (2009), 183-209.
    [11] Mathematical and Computer Modelling, 53 (2011), 1558-1567.
    [12] Technical report, Number 4892, INRIA, Domaine de Voluceau, BP 105, 78153 Rocquencourt, France, 2003.
    [13] PLoS one, 4 (2009), 1-12.
    [14] Cell, 64 Jan. (1991), 271-280.
    [15] Cancer Causes Control, 17 (2006), 539-545.
    [16] J. Biosci. Bioeng., 114 (2012), 220-227.
    [17] Cancer Causes Control, 17 (2006), 509-514.
    [18] Mutat. Res., 680 (2009), 95-105.
    [19] Genes Cancer, 1 Nov. (2010), 1124-1131.
    [20] J. Math. Biol., 28 (1990), 671-694.
    [21] Cancer Causes Control, 17 (2006), 531-537.
    [22] in "Advances on Estimation of Distribution Algorithms" (editors, J. Lozano, P. Larranaga, I. Inza, and E. Bengoetxea), 75-102. Springer, New York, 2006.
    [23] Theor. Biol. Med. Model, 4 (2007), pp.14.
    [24] J. Theor. Biol., 203 (2000), 177-186.
    [25] Oncogene, 19 Nov. (2000), 5558-5567.
    [26] Math. Models Methods Appl. Sci., 16 (2006), 1155-1172.
    [27] Annu. Rev. Pharmacol. Toxicol., 50 (2010), 377-421.
    [28] Annu. Rev. Pharmacol. Toxicol., 47 (2007), 593-628.
    [29] Nat. Rev. Mol. Cell Biol., 1 Dec. (2000), 169-178.
    [30] Cell, 103 Oct. (2000), 295-309.
    [31] Arch. Med. Res., 39 (2008), 743-752.
    [32] Proc. Natl. Acad. Sci. USA, 31 (1997), 814-819.
    [33] Proc. Edinburgh Math. Soc., 54 (1926), 98-130.
    [34] J. Clin. Oncol., 14 (2003), 2787-2799.
    [35] volume 68 of Lecture Notes in Biomathematics. Springer, New York, 1986.
    [36] Proc. Natl. Acad. Sci. USA, 71 (1974), 1286-1290.
    [37] Frontiers in Mathematics series. Birkhäuser, Boston, 2007.
    [38] J. Cell Sci. Suppl., 18 (1994), 69-73.
    [39] Mol. Cell Biol., 14 (1994), 1669-1679.
    [40] J. Theor. Biol., 243 (2006), 532-541.
    [41] Cell, 132 (2008), 487-498.
    [42] Chem. Biol., 15 (2008), 1243-1248.
    [43] Proc. Natl. Acad. Sci. USA, 105 (2008), 17256-17261.
    [44] Biotechnol. Bioeng., 99 (2008), 960-974.
    [45] Nat. Rev. Mol. Cell Biol., 5 Oct. (2004), 836-847.
    [46] Rocky Mountain J. Math., 20 (1990), 1195-1216.
    [47] Curr. Opin. Cell Biol., 7 Dec. (1995), 835-842.
  • This article has been cited by:

    1. Tuersunjiang Keyoumu, Wanbiao Ma, Ke Guo, Global Stability of a MERS-CoV Infection Model with CTL Immune Response and Intracellular Delay, 2023, 11, 2227-7390, 1066, 10.3390/math11041066
    2. Tongqian Zhang, Xinna Xu, Xinzeng Wang, Dynamic analysis of a cytokine-enhanced viral infection model with time delays and CTL immune response, 2023, 170, 09600779, 113357, 10.1016/j.chaos.2023.113357
    3. Jinlong Lv, Wanbiao Ma, Delay induced stability switch in a mathematical model of CD8 T-cell response to SARS-CoV-2 mediated by receptor ACE2, 2024, 34, 1054-1500, 10.1063/5.0187872
    4. Ahmed M. Elaiw, Amani S. Alsulami, Aatef D. Hobiny, Global properties of delayed models for SARS-CoV-2 infection mediated by ACE2 receptor with humoral immunity, 2024, 9, 2473-6988, 1046, 10.3934/math.2024052
    5. Mengfan Tan, Guijie Lan, Chunjin Wei, Dynamic analysis of HIV infection model with CTL immune response and cell-to-cell transmission, 2024, 156, 08939659, 109140, 10.1016/j.aml.2024.109140
  • Reader Comments
  • © 2013 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3807) PDF downloads(657) Cited by(30)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog