# American Institute of Mathematical Sciences

January  2022, 27(1): 141-165. doi: 10.3934/dcdsb.2021035

## Monotonic and nonmonotonic immune responses in viral infection systems

 1 School of Mathematics and Statistics, Bioinformatics Center of Henan University, Kaifeng 475001, Henan, China 2 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, Jiangsu, China 3 Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada

* Corresponding author: Shaoli Wang

Received  September 2019 Revised  November 2020 Published  January 2022 Early access  January 2021

Fund Project: This work is supported by Science and Foundation of Technology Department of Henan Province (No.192102310089), Foundation of Henan Educational Committee (No.19A110009), Natural Science Foundation of Henan (No. 202300410045) and Grant of Bioinformatics Center of Henan University (No. 2019YLXKJC02)

In this paper, we study two-dimensional, three-dimensional monotonic and nonmonotonic immune responses in viral infection systems. Our results show that the viral infection systems with monotonic immune response has no bistability appear. However, the systems with nonmonotonic immune response has bistability appear under some conditions. For immune intensity, we got two important thresholds, post-treatment control threshold and elite control threshold. When immune intensity is less than post-treatment control threshold, the virus will be rebound. The virus will be under control when immune intensity is larger than elite control threshold. While between the two thresholds is a bistable interval. When immune intensity is in the bistable interval, the system can have bistability appear. Select the rate of immune cells stimulated by the viruses as a bifurcation parameter for nonmonotonic immune responses, we prove that the system exhibits saddle-node bifurcation and transcritical bifurcation.

Citation: Shaoli Wang, Huixia Li, Fei Xu. Monotonic and nonmonotonic immune responses in viral infection systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 141-165. doi: 10.3934/dcdsb.2021035
##### References:
 [1] J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng., 10 (1968), 707-723.  doi: 10.1002/bit.260100602. [2] C. Bartholdy, J. P. Christensen, D. Wodarz and A. R. Thomsen, Persistent virus infection despite chronic cytotoxic T-lymphocyte activation in Gamma interferon-deficient mice infected with lymphocytic chroriomeningitis virus, J. Virol., 74 (2000), 10304-10311.  doi: 10.1128/JVI.74.22.10304-10311.2000. [3] S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971-6976.  doi: 10.1073/pnas.94.13.6971. [4] S. Bonhoeffer, M. Rembiszewski, G. M. Ortiz and D. Nixon, Risks and benefits of structured antiretroviral drug therapy interruptions in HIV-1 infection, AIDS, 14 (2000), 2313-2322.  doi: 10.1097/00002030-200010200-00012. [5] J. M. Conway and A. S. Perelson, Post-treatment control of HIV infection, Proc. Natl. Acad. Sci. USA, 112 (2015), 5467-5472.  doi: 10.1073/pnas.1419162112. [6] R. Culshaw, S. Ruan and R. J. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545-562.  doi: 10.1007/s00285-003-0245-3. [7] S. Debroy, B. M. Bolker and M. Martcheva, Bistability and long-term cure in a within-host model of hepatitis C, J. Biol. Systems, 19 (2011), 533-550.  doi: 10.1142/S0218339011004135. [8] M. Haque, Ratio-dependent predator-prey models of interacting populations, Bull. Math. Biol., 71 (2009), 430-452.  doi: 10.1007/s11538-008-9368-4. [9] A. V. M. Herz, S. Bonhoeffer and R. M. Anderson, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247-7251.  doi: 10.1073/pnas.93.14.7247. [10] J. Huang and D. Dong, Analyses of bifurcations and stability in a predator-prey system with Holling Type-IV functional response, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 167-178.  doi: 10.1007/s10255-004-0159-x. [11] Y. Iwasa, F. Michor and M. Nowak, Some basic properties of immune selection, J. Theoret. Biol., 229 (2004), 179-188.  doi: 10.1016/j.jtbi.2004.03.013. [12] H. K. Khalil, Nonlinear System, Prentice-Hall, 1996. [13] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.  doi: 10.1016/j.bulm.2004.02.001. [14] J. P. La Salle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. [15] P. D. Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327.  doi: 10.1137/S0036139902406905. [16] F. Li, W. Ma, Z. Jiang and D Li, Stability and Hopf bifurcation in a delayed HIV infection model with general incidence rate and immune impairment, Comput. Math. Methods Med., 2015 (2015), 206205. doi: 10.1155/2015/206205. [17] W. Liu, Nonlinear oscillations in models of immune responses to persistent viruses, Theoret. Population Biol., 52 (1997), 224-230.  doi: 10.1006/tpbi.1997.1334. [18] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.  doi: 10.1126/science.272.5258.74. [19] M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas and H. McDade, Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci. USA, 93 (1996), 4398-4402.  doi: 10.1073/pnas.93.9.4398. [20] L. Perko, Differential Equation and Dynamical System, Speinger-Verlag, New York, 7 2001. doi: 10.1007/978-1-4613-0003-8. [21] R. R. Regoes, D. Wodarz and M. A. Nowak, Virus dynamics: The effect of target cell limitation and immune responses on virus evolution, J. Theoret. Biol., 191 (1998), 451-462.  doi: 10.1006/jtbi.1997.0617. [22] F. Rothe and D. S. Shafer, Multiple bifurcation in a predator-prey system with non-monotonic predator response, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 313-347.  doi: 10.1017/S0308210500032169. [23] S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic function response, SIAM. J. Appl. Math., 61 (2000/01), 1445-1472.  doi: 10.1137/S0036139999361896. [24] W. Sokol and J. A. Howell, Kinetics of phenol oxidation by washed cells, Biotechnol. Bioeng., 23 (1980), 2039-2049.  doi: 10.1002/bit.260230909. [25] J. Sotomayor, Generic bifurcation of dynamical system, Dynam. Syst., Academic Press, New York, 1973,561–582. [26] K. Wang, Y. Jin and A. Fan, The effect of immune responses in viral infections: A mathematical model view, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3379-3396.  doi: 10.3934/dcdsb.2014.19.3379. [27] K. Wang and Y. Kuang, Fluctuation and extinction dynamics in host-microparasite systems, Commun. Pure Appl. Anal., 10 (2011), 1537-1548.  doi: 10.3934/cpaa.2011.10.1537. [28] Z. Wang and X. Liu, A chronic viral infection model with immune impairment, J. Theoret. Biol., 249 (2007), 532-542.  doi: 10.1016/j.jtbi.2007.08.017. [29] K. Wang, Z. Qiu and G. Deng, Study on a population dynamic model of virus infection, J. Systems Sci. Math. Sci., 23 (2003), 433-443. [30] S. Wang and F. Xu, Analysis of an HIV model with post-treatment control, J. Appl. Anal. Comput., 10 (2020), 667-685.  doi: 10.11948/20190081. [31] S. Wang and F. Xu, Thresholds and bistability in virus-immune dynamics, Appl. Math. Lett., 78 (2018), 105-111.  doi: 10.1016/j.aml.2017.11.002. [32] S. Wang, F. Xu and L. Rong, Bistability analysis of an HIV model with immune response, J. Biol. Systems, Vol 25 (2017), 677–695. doi: 10.1142/S021833901740006X. [33] S. Wang, F. Xu and X. Song, Threshold and bistability in HIV infection models with oxidative stress, arXiv: 1808.02276 (2018). [34] D. Wodarz, J. P. Christensen and A. R. Thomsen, The importance of lytic and nonlytie immune responses in viral infections, Trends Immunol., 23 (2002), 194-200.  doi: 10.1016/S1471-4906(02)02189-0. [35] D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses, J. Gen. Virol., 84 (2003), 1743-1750.  doi: 10.1099/vir.0.19118-0.

show all references

##### References:
 [1] J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng., 10 (1968), 707-723.  doi: 10.1002/bit.260100602. [2] C. Bartholdy, J. P. Christensen, D. Wodarz and A. R. Thomsen, Persistent virus infection despite chronic cytotoxic T-lymphocyte activation in Gamma interferon-deficient mice infected with lymphocytic chroriomeningitis virus, J. Virol., 74 (2000), 10304-10311.  doi: 10.1128/JVI.74.22.10304-10311.2000. [3] S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971-6976.  doi: 10.1073/pnas.94.13.6971. [4] S. Bonhoeffer, M. Rembiszewski, G. M. Ortiz and D. Nixon, Risks and benefits of structured antiretroviral drug therapy interruptions in HIV-1 infection, AIDS, 14 (2000), 2313-2322.  doi: 10.1097/00002030-200010200-00012. [5] J. M. Conway and A. S. Perelson, Post-treatment control of HIV infection, Proc. Natl. Acad. Sci. USA, 112 (2015), 5467-5472.  doi: 10.1073/pnas.1419162112. [6] R. Culshaw, S. Ruan and R. J. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545-562.  doi: 10.1007/s00285-003-0245-3. [7] S. Debroy, B. M. Bolker and M. Martcheva, Bistability and long-term cure in a within-host model of hepatitis C, J. Biol. Systems, 19 (2011), 533-550.  doi: 10.1142/S0218339011004135. [8] M. Haque, Ratio-dependent predator-prey models of interacting populations, Bull. Math. Biol., 71 (2009), 430-452.  doi: 10.1007/s11538-008-9368-4. [9] A. V. M. Herz, S. Bonhoeffer and R. M. Anderson, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247-7251.  doi: 10.1073/pnas.93.14.7247. [10] J. Huang and D. Dong, Analyses of bifurcations and stability in a predator-prey system with Holling Type-IV functional response, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 167-178.  doi: 10.1007/s10255-004-0159-x. [11] Y. Iwasa, F. Michor and M. Nowak, Some basic properties of immune selection, J. Theoret. Biol., 229 (2004), 179-188.  doi: 10.1016/j.jtbi.2004.03.013. [12] H. K. Khalil, Nonlinear System, Prentice-Hall, 1996. [13] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.  doi: 10.1016/j.bulm.2004.02.001. [14] J. P. La Salle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. [15] P. D. Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327.  doi: 10.1137/S0036139902406905. [16] F. Li, W. Ma, Z. Jiang and D Li, Stability and Hopf bifurcation in a delayed HIV infection model with general incidence rate and immune impairment, Comput. Math. Methods Med., 2015 (2015), 206205. doi: 10.1155/2015/206205. [17] W. Liu, Nonlinear oscillations in models of immune responses to persistent viruses, Theoret. Population Biol., 52 (1997), 224-230.  doi: 10.1006/tpbi.1997.1334. [18] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.  doi: 10.1126/science.272.5258.74. [19] M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas and H. McDade, Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci. USA, 93 (1996), 4398-4402.  doi: 10.1073/pnas.93.9.4398. [20] L. Perko, Differential Equation and Dynamical System, Speinger-Verlag, New York, 7 2001. doi: 10.1007/978-1-4613-0003-8. [21] R. R. Regoes, D. Wodarz and M. A. Nowak, Virus dynamics: The effect of target cell limitation and immune responses on virus evolution, J. Theoret. Biol., 191 (1998), 451-462.  doi: 10.1006/jtbi.1997.0617. [22] F. Rothe and D. S. Shafer, Multiple bifurcation in a predator-prey system with non-monotonic predator response, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 313-347.  doi: 10.1017/S0308210500032169. [23] S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic function response, SIAM. J. Appl. Math., 61 (2000/01), 1445-1472.  doi: 10.1137/S0036139999361896. [24] W. Sokol and J. A. Howell, Kinetics of phenol oxidation by washed cells, Biotechnol. Bioeng., 23 (1980), 2039-2049.  doi: 10.1002/bit.260230909. [25] J. Sotomayor, Generic bifurcation of dynamical system, Dynam. Syst., Academic Press, New York, 1973,561–582. [26] K. Wang, Y. Jin and A. Fan, The effect of immune responses in viral infections: A mathematical model view, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3379-3396.  doi: 10.3934/dcdsb.2014.19.3379. [27] K. Wang and Y. Kuang, Fluctuation and extinction dynamics in host-microparasite systems, Commun. Pure Appl. Anal., 10 (2011), 1537-1548.  doi: 10.3934/cpaa.2011.10.1537. [28] Z. Wang and X. Liu, A chronic viral infection model with immune impairment, J. Theoret. Biol., 249 (2007), 532-542.  doi: 10.1016/j.jtbi.2007.08.017. [29] K. Wang, Z. Qiu and G. Deng, Study on a population dynamic model of virus infection, J. Systems Sci. Math. Sci., 23 (2003), 433-443. [30] S. Wang and F. Xu, Analysis of an HIV model with post-treatment control, J. Appl. Anal. Comput., 10 (2020), 667-685.  doi: 10.11948/20190081. [31] S. Wang and F. Xu, Thresholds and bistability in virus-immune dynamics, Appl. Math. Lett., 78 (2018), 105-111.  doi: 10.1016/j.aml.2017.11.002. [32] S. Wang, F. Xu and L. Rong, Bistability analysis of an HIV model with immune response, J. Biol. Systems, Vol 25 (2017), 677–695. doi: 10.1142/S021833901740006X. [33] S. Wang, F. Xu and X. Song, Threshold and bistability in HIV infection models with oxidative stress, arXiv: 1808.02276 (2018). [34] D. Wodarz, J. P. Christensen and A. R. Thomsen, The importance of lytic and nonlytie immune responses in viral infections, Trends Immunol., 23 (2002), 194-200.  doi: 10.1016/S1471-4906(02)02189-0. [35] D. Wodarz, Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses, J. Gen. Virol., 84 (2003), 1743-1750.  doi: 10.1099/vir.0.19118-0.
Bifurcation diagram of system (3). The solid line represents the stable equilibrium of infected CD4+ T cells and the dashed line represents the unstable equilibrium of infected CD4+ T cells. The post-treatment control threshold is $c_{2} = 0.2500$, the elite control threshold is $c^{**}_1\approx0.6505$ and the bistable interval is $(0.2500, 0.6505).$ Here, $c = 0.37\; \; \mbox{day}^{-1}$ and the values of other parameters are listed in (4)
Time histories and trajectories of system (3) with different initial conditions. The system has a stable equilibria $E^{(2)}_{1}$. Here, $c = 0.2\; \; \mbox{day}^{-1}$ is less than the post-treatment control threshold $P_I$ and other parameter values are listed in (4)
Time histories and trajectories of system (3) with different initial conditions. Here, system (3) has two different stable equilibria $E^{(2)}_{1}$ and $E_{-}^{2*}$ with $c = 0.37\; \; \mbox{day}^{-1}$. Other parameter values are listed in (4)
Time histories and trajectories of system (3) with different initial conditions. System (3) only has the positive equilibrium $E_{-}^{2*}$, which is stable with $c = 0.65\; \; \mbox{day}^{-1}$. Other parameter values are listed in (4)
Bifurcation diagram of system (6). The solid line is the stable equilibrium and the dashed line denotes the unstable equilibrium. The post-treatment control threshold is $c_{2} = 2.5000$, the elite control threshold is $c^{**}_2\approx3.5278$ and the bistable interval is $(2.5000, 3.5278).$ Here, $c = 3\; \; \mbox{day}^{-1}$ and other parameter values are listed in (7)
Time histories and phase portraits of system (6). System (6) has two different stable equilibria $E^{(4)}_{1}$ and $E_{*}^{4-}$. Here, $c = 3\; \; \mbox{day}^{-1}$ and other parameter values are listed in (7). We choose different initial values
Phase portraits of system (6). (A) Choosing $c = 2\; \; \mbox{day}^{-1}$, which is less than the post-treatment control threshold $c_{2} = 2.5000$, system (6) only has a stable equilibrium $E^{(4)}_{1}$; (B) Choosing $c = 4\; \; \mbox{day}^{-1}$, which is greater than the elite control threshold $c^{**}_2\approx3.5278$, system (6) only has the stable equilibria $E_{*}^{4-}$. Other parameter values are listed in (7)
The stabilities of the equilibria and the behaviors of system (3) in the case $1<\mathcal {R}^{(2)}_{0}<\mathcal {R}^{(1)}_{c}$
 $E^{(2)}_{0}$ $E^{(2)}_{1}$ $E_{*}^{2-}$ $E_{*}^{2+}$ System (3) $R^{(2)}_{0}<1$ GAS — — — Converges to $E^{(2)}_{0}$ $1 $ E^{(2)}_{0}  E^{(2)}_{1}  E_{*}^{2-}  E_{*}^{2+} $System (3)$ R^{(2)}_{0}<1 $GAS — — — Converges to$ E^{(2)}_{0}  1
The stabilities of the equilibria and the behaviors of system (3) in the case $\mathcal {R}^{(2)}_{0}>\mathcal {R}^{(1)}_{c}$
 $E^{(2)}_{0}$ $E^{(2)}_{1}$ $E_{*}^{2-}$ $E_{*}^{2+}$ System (3) $R^{(2)}_{0}<1$ GAS — — — Converges to $E^{(2)}_{0}$ $R^{(2)}_{0}>1$, $0R^{(1)}_{c}>1,$ $c_{2}R^{(1)}_{c}>1,$ $c>c^{**}_1$ US US LAS — Converges to $E_{*}^{2-}$
 $E^{(2)}_{0}$ $E^{(2)}_{1}$ $E_{*}^{2-}$ $E_{*}^{2+}$ System (3) $R^{(2)}_{0}<1$ GAS — — — Converges to $E^{(2)}_{0}$ $R^{(2)}_{0}>1$, $0R^{(1)}_{c}>1,$ $c_{2}R^{(1)}_{c}>1,$ $c>c^{**}_1$ US US LAS — Converges to $E_{*}^{2-}$
The stabilities of the equilibria and the behaviors of system (6) in the case $1<\mathcal {R}^{(4)}_{0}<\mathcal {R}^{(2)}_{c}$
 $E^{(4)}_{0}$ $E^{(4)}_{1}$ $E_{*}^{4-}$ $E_{*}^{4+}$ System (6) $R^{(4)}_{0}<1$ GAS — — — Converges to $E^{(4)}_{0}$ $1 $ E^{(4)}_{0}  E^{(4)}_{1}  E_{*}^{4-}  E_{*}^{4+} $System (6)$ R^{(4)}_{0}<1 $GAS — — — Converges to$ E^{(4)}_{0}  1
The stabilities of the equilibria and the behaviors of system (6) in the case $\mathcal {R}^{(4)}_{0}>\mathcal {R}^{(2)}_{c}$
 $E^{(4)}_{0}$ $E^{(4)}_{1}$ $E_{*}^{4-}$ $E_{*}^{4+}$ System (6) $R^{(4)}_{0}<1$ GAS — — — Converges to $E^{(4)}_{0}$ $R^{(4)}_{0}>1$, $0R^{(2)}_{c}>1,$ $c_{2}R^{(2)}_{c}>1,$ $c>c^{**}_2$ US US GAS — Converges to $E_{*}^{4-}$
 $E^{(4)}_{0}$ $E^{(4)}_{1}$ $E_{*}^{4-}$ $E_{*}^{4+}$ System (6) $R^{(4)}_{0}<1$ GAS — — — Converges to $E^{(4)}_{0}$ $R^{(4)}_{0}>1$, $0R^{(2)}_{c}>1,$ $c_{2}R^{(2)}_{c}>1,$ $c>c^{**}_2$ US US GAS — Converges to $E_{*}^{4-}$

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