• Previous Article
    Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis
  • DCDS-B Home
  • This Issue
  • Next Article
    Outer synchronization of delayed coupled systems on networks without strong connectedness: A hierarchical method
March  2018, 23(2): 861-885. doi: 10.3934/dcdsb.2018046

Hopf bifurcation of an age-structured virus infection model

1. 

Faculty of Mathematics, K. N. Toosi University of Technology, P. O. Box:16315-1618, Tehran, Iran

2. 

Department of Mathematics, University of Louisiana, Lafayette, LA, USA

3. 

Department of Mathematical Sciences, Sharif University of Technology, P. O. Box: 11155-9415, Tehran, Iran

* Corresponding author

The first author is supported by The Department of Iranian Student Affairs

Received  December 2016 Revised  September 2017 Published  December 2017

In this paper, we introduce and analyze a mathematical model of a viral infection with explicit age-since infection structure for infected cells. We extend previous age-structured within-host virus models by including logistic growth of target cells and allowing for absorption of multiple virus particles by infected cells. The persistence of the virus is shown to depend on the basic reproduction number $R_{0}$. In particular, when $R_{0}≤1$, the infection free equilibrium is globally asymptotically stable, and conversely if $R_{0}> 1$, then the infection free equilibrium is unstable, the system is uniformly persistent and there exists a unique positive equilibrium. We show that our system undergoes a Hopf bifurcation through which the infection equilibrium loses the stability and periodic solutions appear.

Citation: Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046
References:
[1]

R. Adams and J. Fournier, Sobolev spaces, "Second edition", Pure Appl. Math., 140, Elsevier/Academic Press, Amsterdam, 2003. Google Scholar

[2]

I. Rob De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, Journal of Theoretical Biology, 190 (1998), 201-214.   Google Scholar

[3]

B. Brandenburg, L. Y. Lee, M. Lakadamyali, M. J. Rust, X. Zhuang and J. M. Hogle, Imaging poliovirus entry in live cells, PLoS Biology, 5 (2007), e183, http://doi.org/10.1371/journal.pbio.0050183. Google Scholar

[4]

C. J. Browne, A multi-strain virus model with infected cell age structure: Application to HIV, Nonlinear Analysis: Real World Applications, 22 (2015), 354-372.  doi: 10.1016/j.nonrwa.2014.10.004.  Google Scholar

[5]

C. J. Browne, Immune response in virus model structured by cell infection-age, Mathematical Biosciences and Engineering, 13 (2016), 887-909.  doi: 10.3934/mbe.2016022.  Google Scholar

[6]

C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, DCDS-B, 18 (2013), 1999-2017.  doi: 10.3934/dcdsb.2013.18.1999.  Google Scholar

[7]

R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27-39.  doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[8]

P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327.  doi: 10.1137/S0036139902406905.  Google Scholar

[9]

M. N. Dixit and A. S. Perelson, Multiplicity of human immunodeficiency virus infections in lymphoid tissue, Journal of Virology, 78 (2004), 8942-8945.   Google Scholar

[10]

P. Dustin and D. Wodarz, Modeling multiple infection of cells by viruses: Challenges and insights, Mathematical biosciences, 264 (2015), 21-28.  doi: 10.1016/j.mbs.2015.03.001.  Google Scholar

[11]

J. K. HaleJ. P. Lasalle and M. Slemrod, Theory of a general class of dissipative processes, J. Math. Anal. Appl., 39 (1972), 177-191.  doi: 10.1016/0022-247X(72)90233-8.  Google Scholar

[12]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematics Surveys and Monographs, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[13]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.  doi: 10.1137/0520025.  Google Scholar

[14]

S. HongyingL. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM Journal on Applied Mathematics, 73 (2013), 1280-1302.  doi: 10.1137/120896463.  Google Scholar

[15]

G. HuangX. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-58.  doi: 10.1137/110826588.  Google Scholar

[16]

L. Josefsson, M. S. King, B. Makitalo, J. Brännström, W. Shao, F. Maldarelli and M. F. Kearney et al. Majority of CD4+ T cells from peripheral blood of HIV-1 infected individuals contain only one HIV DNA molecule, Proceedings of the National Academy of Sciences, 108 (2011), 11199-11204. Google Scholar

[17]

D. Kirschner and G. F. Webb, A model for treatment strategy in the chemotherapy of AIDS, Bull. Math. Biol., 58 (1996), 367-390.  doi: 10.1007/BF02458312.  Google Scholar

[18]

P. Magal and X. Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[19]

C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng, 9 (2012), 819-841.  doi: 10.3934/mbe.2012.9.819.  Google Scholar

[20]

P. W. NelsonM. A. GilchristD. CoombsJ. M. Hyman and A. S. Perelson, An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288.  doi: 10.3934/mbe.2004.1.267.  Google Scholar

[21]

M. A. Nowak and R. M. May, Virus dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, 2000.  Google Scholar

[22]

A. S. PerelsonA. U. NeumannM. MarkowitzJ. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span and viral generation time, Science, 271 (1996), 1582-1586.  doi: 10.1126/science.271.5255.1582.  Google Scholar

[23]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.  doi: 10.1137/S0036144598335107.  Google Scholar

[24]

H. PourbashashS. S. PilyuginP. De Leenheer and C. C. McCluskey, Global analysis of within host virus models with cell-to-cell viral transmission, DCDS-B, 19 (2014), 3341-3357.  doi: 10.3934/dcdsb.2014.19.3341.  Google Scholar

[25]

L. RongZ. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy, SIAM. J. Appl. Math., 67 (2007), 731-756.  doi: 10.1137/060663945.  Google Scholar

[26]

Libin Rong and A. S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, Journal of Theoretical Biology, 260 (2009), 308-331.  doi: 10.1016/j.jtbi.2009.06.011.  Google Scholar

[27]

M. A. StaffordL. CoreyY. CaoE. S. DaarD. D. Ho and A. S. Perelson, Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 23 (2000), 285-301.  doi: 10.1006/jtbi.2000.1076.  Google Scholar

[28]

R. V. UrsacheY. E. ThomassenG. Van EikenhorstP. J. T. Verheijen and W. A. M. Bakker, Mathematical model of adherent Vero cell growth and poliovirus production in animal component free medium, Bioprocess Biosyst Eng., 38 (2015), 543-555.  doi: 10.1007/s00449-014-1294-2.  Google Scholar

[29]

Y. Wang, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, Journal of Mathematical Biology, 67 (2013), 901-934.  doi: 10.1007/s00285-012-0580-3.  Google Scholar

[30]

G. F. Webb and C. J. Browne. A model of the Ebola epidemics in West Africa incorporating age of infection, A model of the Ebola epidemics in West Africa incorporating age of infection, Journal of Biological Dynamics, 10 (2016), 18-30.  doi: 10.1080/17513758.2015.1090632.  Google Scholar

[31]

Y. YangS. Ruan and D. Xiao, Global stability of an age-structured virus dynamics model with Bedington-Deangelis infection function, Math. Biosci. Eng., 12 (2015), 859-877.  doi: 10.3934/mbe.2015.12.859.  Google Scholar

[32]

J. A. ZackS. J. ArrigoS. R. WeitsmanA. S. GoA. Haislip and I. S. Chen, HIV-1 entry into quiescent primary lymphocytes: Molecular analysis reveals a labile latent viralstructure, Cell, 61 (1990), 213-222.  doi: 10.1016/0092-8674(90)90802-L.  Google Scholar

show all references

References:
[1]

R. Adams and J. Fournier, Sobolev spaces, "Second edition", Pure Appl. Math., 140, Elsevier/Academic Press, Amsterdam, 2003. Google Scholar

[2]

I. Rob De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, Journal of Theoretical Biology, 190 (1998), 201-214.   Google Scholar

[3]

B. Brandenburg, L. Y. Lee, M. Lakadamyali, M. J. Rust, X. Zhuang and J. M. Hogle, Imaging poliovirus entry in live cells, PLoS Biology, 5 (2007), e183, http://doi.org/10.1371/journal.pbio.0050183. Google Scholar

[4]

C. J. Browne, A multi-strain virus model with infected cell age structure: Application to HIV, Nonlinear Analysis: Real World Applications, 22 (2015), 354-372.  doi: 10.1016/j.nonrwa.2014.10.004.  Google Scholar

[5]

C. J. Browne, Immune response in virus model structured by cell infection-age, Mathematical Biosciences and Engineering, 13 (2016), 887-909.  doi: 10.3934/mbe.2016022.  Google Scholar

[6]

C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, DCDS-B, 18 (2013), 1999-2017.  doi: 10.3934/dcdsb.2013.18.1999.  Google Scholar

[7]

R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27-39.  doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[8]

P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327.  doi: 10.1137/S0036139902406905.  Google Scholar

[9]

M. N. Dixit and A. S. Perelson, Multiplicity of human immunodeficiency virus infections in lymphoid tissue, Journal of Virology, 78 (2004), 8942-8945.   Google Scholar

[10]

P. Dustin and D. Wodarz, Modeling multiple infection of cells by viruses: Challenges and insights, Mathematical biosciences, 264 (2015), 21-28.  doi: 10.1016/j.mbs.2015.03.001.  Google Scholar

[11]

J. K. HaleJ. P. Lasalle and M. Slemrod, Theory of a general class of dissipative processes, J. Math. Anal. Appl., 39 (1972), 177-191.  doi: 10.1016/0022-247X(72)90233-8.  Google Scholar

[12]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematics Surveys and Monographs, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[13]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.  doi: 10.1137/0520025.  Google Scholar

[14]

S. HongyingL. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM Journal on Applied Mathematics, 73 (2013), 1280-1302.  doi: 10.1137/120896463.  Google Scholar

[15]

G. HuangX. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-58.  doi: 10.1137/110826588.  Google Scholar

[16]

L. Josefsson, M. S. King, B. Makitalo, J. Brännström, W. Shao, F. Maldarelli and M. F. Kearney et al. Majority of CD4+ T cells from peripheral blood of HIV-1 infected individuals contain only one HIV DNA molecule, Proceedings of the National Academy of Sciences, 108 (2011), 11199-11204. Google Scholar

[17]

D. Kirschner and G. F. Webb, A model for treatment strategy in the chemotherapy of AIDS, Bull. Math. Biol., 58 (1996), 367-390.  doi: 10.1007/BF02458312.  Google Scholar

[18]

P. Magal and X. Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

[19]

C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng, 9 (2012), 819-841.  doi: 10.3934/mbe.2012.9.819.  Google Scholar

[20]

P. W. NelsonM. A. GilchristD. CoombsJ. M. Hyman and A. S. Perelson, An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288.  doi: 10.3934/mbe.2004.1.267.  Google Scholar

[21]

M. A. Nowak and R. M. May, Virus dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, 2000.  Google Scholar

[22]

A. S. PerelsonA. U. NeumannM. MarkowitzJ. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span and viral generation time, Science, 271 (1996), 1582-1586.  doi: 10.1126/science.271.5255.1582.  Google Scholar

[23]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.  doi: 10.1137/S0036144598335107.  Google Scholar

[24]

H. PourbashashS. S. PilyuginP. De Leenheer and C. C. McCluskey, Global analysis of within host virus models with cell-to-cell viral transmission, DCDS-B, 19 (2014), 3341-3357.  doi: 10.3934/dcdsb.2014.19.3341.  Google Scholar

[25]

L. RongZ. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy, SIAM. J. Appl. Math., 67 (2007), 731-756.  doi: 10.1137/060663945.  Google Scholar

[26]

Libin Rong and A. S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, Journal of Theoretical Biology, 260 (2009), 308-331.  doi: 10.1016/j.jtbi.2009.06.011.  Google Scholar

[27]

M. A. StaffordL. CoreyY. CaoE. S. DaarD. D. Ho and A. S. Perelson, Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 23 (2000), 285-301.  doi: 10.1006/jtbi.2000.1076.  Google Scholar

[28]

R. V. UrsacheY. E. ThomassenG. Van EikenhorstP. J. T. Verheijen and W. A. M. Bakker, Mathematical model of adherent Vero cell growth and poliovirus production in animal component free medium, Bioprocess Biosyst Eng., 38 (2015), 543-555.  doi: 10.1007/s00449-014-1294-2.  Google Scholar

[29]

Y. Wang, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, Journal of Mathematical Biology, 67 (2013), 901-934.  doi: 10.1007/s00285-012-0580-3.  Google Scholar

[30]

G. F. Webb and C. J. Browne. A model of the Ebola epidemics in West Africa incorporating age of infection, A model of the Ebola epidemics in West Africa incorporating age of infection, Journal of Biological Dynamics, 10 (2016), 18-30.  doi: 10.1080/17513758.2015.1090632.  Google Scholar

[31]

Y. YangS. Ruan and D. Xiao, Global stability of an age-structured virus dynamics model with Bedington-Deangelis infection function, Math. Biosci. Eng., 12 (2015), 859-877.  doi: 10.3934/mbe.2015.12.859.  Google Scholar

[32]

J. A. ZackS. J. ArrigoS. R. WeitsmanA. S. GoA. Haislip and I. S. Chen, HIV-1 entry into quiescent primary lymphocytes: Molecular analysis reveals a labile latent viralstructure, Cell, 61 (1990), 213-222.  doi: 10.1016/0092-8674(90)90802-L.  Google Scholar

Figure 1.  A numerical solution of system (40) tends to the infection-free equilibrium $E_0$, as time tends to infinity, wherein parameter values are $[s, k, T_{0}, p, d, \gamma, \delta] = [0.038, 0.0045, 4.6137, 1, 0.093, 0.4, 0.028]$. In this case $R_0 = 0.6518$. (A) Time series of $T$, $T^*$ and $V$. (B) An orbit in the $TVT^*$ space. Eigenvalues of linearized matrix about $\overline{E}$ are $\lambda_1 =-0.0779 + 0.0000i, \lambda_2 = 0.0004 - 0.0209i, \lambda_3 =0.0004 + 0.0209i.$
Figure 2.  A numerical solution of system (40) approaches to $\overline{E}$, as time tends to infinity, and $\bar{E}$ is stable, wherein parameter values are $[s, k, T_{0}, p, d, \gamma, \delta] = $ $[0.03285, 0.01, 4.6137, 1.3, 0.045, 0.1, 0.0351]$. In this case $R_0 = 16.0999$, $\bar{E} = (\bar{T},\bar{V},\bar{I}) =[0.2212, 3.1275, 0.1971]$. (A) Time series of $T$, $T^*$ and $V$. (B) An orbit in the $TVT^*$ space.
Figure 3.  A numerical solution of system (40) tends to the limit cycle, as time tends to infinity, and $\bar{E}$ is unstable, wherein parameter values are $[s, k, T_{0}, p, d, \gamma, \delta] = $ $[0.03285, 0.01, 4.6137, 1.3, 0.03, 0.1, 0.0351]$. In this case $R_0 = 22.4437$, $(\bar{T},\bar{V},\bar{I}) =[ 0.1115, 3.2056, 0.1018]$. (A) Time series of $T$, $T^*$ and $V$. (B) An orbit in the $TVT^*$ space. Eigenvalues of linearized matrix about $\overline{E}$ are $\lambda_1 =-0.0779 + 0.0000i, \lambda_2 = 0.0004 - 0.0209i, \lambda_3 =0.0004 + 0.0209i.$
Figure 4.  A numerical solution of system (41)-(44) tends to the DFE, as time tends to infinity, wherein parameter values are $[s,T0,k,\rho,d,\gamma,\tau,\mu,\nu] = $ $[.1,100000,0.0000005,200,13,0.000003,2,0.05,0.7]$. In this case $R_0 =0.9905$ and $(\bar{T},\bar{V},\bar{I}) =[ 10^5, 0 , 0]$. (A) Time series of $T$, $T^* = J+I$ and $V$. (B) An orbit in the $TVT^*$ space.
Figure 5.  A numerical solution of system (41)-(44) tends to the $\bar{E}$, as time tends to infinity, and $\bar{E}$ is stable, wherein parameter values are $[s,T0,k,\rho,d,\gamma,\tau,\mu,\nu] = $ $[.1,100000,0.0000008,200,13,0.000003,2,0.05,0.7]$. In this case $R_0 =1.5812$, $(\bar{T},\bar{V},\bar{J}+\bar{I}) =[ 6.3209\times 10^4, 4.5989\times 10^4, 7.4321\times 10^3]$. The probability of re-infection of infected cells during eclipse phase (during age $0\leq a \leq \tau$) calculated at $\bar{E}$ is $\pi(\tau) = 0.23$. (A) Time series of $T$, $T^* = J+I$ and $V$. (B) An orbit in the $TVT^*$ space.
Figure 6.  A numerical solution of system (41)-(44) tends to the limit cycle, as time tends to infinity, and $\bar{E}$ is unstable, wherein parameter values are $[s,T0,k,\rho,d,\gamma,\tau,\mu,\nu] = $ $[1,100000, 0.000005,200, 13, 0.000001, 2, 0.05, 0.7]$. In this case $R_0 =9.5750$, $(\bar{T},\bar{V},\bar{J}+\bar{I}) =[ 1.0119\times 10^4, 1.7976\times 10^5, 2.9066\times 10^4]$. The probability of re-infection of infected cells during eclipse phase calculated at $\bar{E}$ is $\pi(\tau) = 0.2882$. (A) Time series of $T$, $T^* = J+I$ and $V$. (B) An orbit in the $TVT^*$ space.
Table 1.  Parameter definition and values from literatures.
Parameter Value Description Reference
$e$ day$^{-1}$ Maximum proliferation rate See text
$g$ 0.008 day$^{-1}$ Death rate of uninfected cells [27]
$T_{\text{max}}$ mm$^{-3}$ Density of $T$ cell at which proliferation shouts off See text
$k$ $5 \times 10^{-7}$ ml virion day$^{-1}$ Infection rate of target cells by virus [27]
$\delta$ 0.8 day$^{-1}$ Death rate of infected cells [32]
p Varied Virion production rate of an infected cell See text
$d$ 3 day$^{-1}$ clearance rate of free virus [22]
$\gamma$ day$^{-1}$ Reinfection rate of infected cells by virus See text
Parameter Value Description Reference
$e$ day$^{-1}$ Maximum proliferation rate See text
$g$ 0.008 day$^{-1}$ Death rate of uninfected cells [27]
$T_{\text{max}}$ mm$^{-3}$ Density of $T$ cell at which proliferation shouts off See text
$k$ $5 \times 10^{-7}$ ml virion day$^{-1}$ Infection rate of target cells by virus [27]
$\delta$ 0.8 day$^{-1}$ Death rate of infected cells [32]
p Varied Virion production rate of an infected cell See text
$d$ 3 day$^{-1}$ clearance rate of free virus [22]
$\gamma$ day$^{-1}$ Reinfection rate of infected cells by virus See text
[1]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[2]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[3]

Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446

[4]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342

[5]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084

[6]

Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021013

[7]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219

[8]

Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331

[9]

Sarra Nouaoura, Radhouane Fekih-Salem, Nahla Abdellatif, Tewfik Sari. Mathematical analysis of a three-tiered food-web in the chemostat. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020369

[10]

Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391

[11]

Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156

[12]

Alexander Dabrowski, Ahcene Ghandriche, Mourad Sini. Mathematical analysis of the acoustic imaging modality using bubbles as contrast agents at nearly resonating frequencies. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021005

[13]

Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173

[14]

Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010

[15]

Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034

[16]

Philippe G. Ciarlet, Liliana Gratie, Cristinel Mardare. Intrinsic methods in elasticity: a mathematical survey. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 133-164. doi: 10.3934/dcds.2009.23.133

[17]

M. Dambrine, B. Puig, G. Vallet. A mathematical model for marine dinoflagellates blooms. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 615-633. doi: 10.3934/dcdss.2020424

[18]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[19]

Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455

[20]

Jean-Paul Chehab. Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021002

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (273)
  • HTML views (478)
  • Cited by (0)

[Back to Top]