# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021278
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## A delayed dynamical model for COVID-19 therapy with defective interfering particles and artificial antibodies

 Department of Applied Mathematics, Shanghai Normal University, Road Guilin No.100, 200234, Shanghai, China

* Corresponding author: Yepeng Xing

Received  June 2021 Revised  October 2021 Early access November 2021

Fund Project: The authors were supported by National Natural Science Foundation of China (No.12071297, No.12171320)

In this paper, we use delay differential equations to propose a mathematical model for COVID-19 therapy with both defective interfering particles and artificial antibodies. For this model, the basic reproduction number $\mathcal{R}_0$ is given and its threshold properties are discussed. When $\mathcal{R}_0<1$, the disease-free equilibrium $E_0$ is globally asymptotically stable. When $\mathcal{R}_0>1$, $E_0$ becomes unstable and the infectious equilibrium without defective interfering particles $E_1$ comes into existence. There exists a positive constant $R_1$ such that $E_1$ is globally asymptotically stable when $R_1<1<\mathcal{R}_0$. Further, when $R_1>1$, $E_1$ loses its stability and infectious equilibrium with defective interfering particles $E_2$ occurs. There exists a constant $R_2$ such that $E_2$ is asymptotically stable without time delay if $1<R_1<\mathcal{R}_0<R_2$ and it loses its stability via Hopf bifurcation as the time delay increases. Numerical simulation is also presented to demonstrate the applicability of the theoretical predictions.

Citation: Yanfei Zhao, Yepeng Xing. A delayed dynamical model for COVID-19 therapy with defective interfering particles and artificial antibodies. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021278
##### References:
 [1] D. Adam, What scientists know about new, fast-spreading coronavirus variants, Nature, 594 (2021), 19-20. [2] C. M. Bangham and T. B. L. Kirkwood, Defective interfering particles: Effects in modulating virus growth and persistence, Virology, 179 (1990), 821-826.  doi: 10.1016/0042-6822(90)90150-P. [3] A. Baum, D. Ajithdoss, R. Copin et al., REGN-COV2 antibodies prevent and treat SARSCoV-2 infection in rhesus macaques and hamsters, Science, 370 (2020), 1110-1115. doi: 10.1126/science.abe2402. [4] F. Campbell, B. Archer, H. Laurenson-Schafer et al., Increased transmissibility and global spread of SARS-CoV-2 variants of concern as at June 2021, Eurosurveillance, 26 (2021), 2100509. [5] X. Cao, COVID-19: Immunopathology and its implications for therapy, Nat. Rev. Immunol., 20 (2020), 269-270. [6] P. Chen, A. Nirula, B. Heller et al., SARS-CoV-2 neutralizing antibody LY-CoV555 in outpatients with Covid-19, N. Engl. J. Med., 384 (2021), 229–237. [7] N. J. Dimmock and A. J. Easton, Defective interfering influenza virus RNAs: Time to reevaluate their clinical potential as broad-spectrum antivirals, J. Virol., 88 (2014), 5217-5227.  doi: 10.1128/JVI.03193-13. [8] S. A. Frank, Within-host spatial dynamics of viruses and defective interfering particles, J. Theoret. Biol., 206 (2000), 279-290.  doi: 10.1006/jtbi.2000.2120. [9] T. Frensing, F. S. Heldt, A. Pflugmacher et al., Continuous influenza virus production in cell culture shows a periodic accumulation of defective interfering particles, Plos One, 8 (2013), e72288. doi: 10.1371/journal.pone.0072288. [10] S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biol. Dyn., 2 (2008), 140-153.  doi: 10.1080/17513750701769873. [11] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [12] A. S. Huang and D. 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Virol., 20 (2010), 51-62.  doi: 10.1002/rmv.641. [24] G. W. Nelson and A. S.Perelson, Modeling defective interfering virus therapy for AIDS: Conditions for DIV survival, Math. Biosci., 125 (1995), 127-153.  doi: 10.1016/0025-5564(94)00021-Q. [25] Y. Pan, J. Du, J. Liu et al., Screening of potent neutralizing antibodies against SARS-CoV-2 using convalescent patients-derived phage-display libraries, Cell Discov., 7 (2021), 1–19. [26] 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. [27] U. Rand, S. Y. Kupke, H. Shkarlet et al., Antiviral activity of influenza A virus defective interfering particles against SARS-CoV-2 replication in vitro through stimulation of innate immunity, Cells, 10 (2021), 1756. [28] S. Ruan and J. Wei, On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, Math. Med. 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Microbiol., 4 (2019), 1075-1087.  doi: 10.1038/s41564-019-0465-y. [38] D. Wang, B. Hu, C. Hu et al., Clinical characteristics of 138 hospitalized patients with 2019 novel coronavirus–infected pneumonia in Wuhan, China, JAMA, 323 (2020), 1061–1069. [39] 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. [40] Y. Wu, C. Chen and Y. Chan, The outbreak of COVID-19: An overview, J. Chin. Med. Assoc., 83 (2020), 217-220.  doi: 10.1097/JCMA.0000000000000270. [41] S. Yao, A. Narayanan, S. A. Majowicz, J. Jose and M. Archetti, A synthetic defective interfering SARS-CoV-2, PeerJ, 9 (2021), e11686. [42] M. Zhang, J. Xiao, A. Deng et al., Transmission dynamics of an outbreak of the COVID-19 delta variant B. 1.617. 2-Guangdong Province, China, May–June 2021, CCDC Weekly, 3 (2021), 584–586. [43] T. Zhang, Y. Song, Z. Jiang and J. Wang, Dynamical analysis of a delayed HIV virus dynamic model with cell-to-cell transmission and apoptosis of bystander cells, Complexity, 2 (2020), 126-144. [44] H. Zhao, K. K. To, H. Chu et al., Dual-functional peptide with defective interfering genes effectively protects mice against avian and seasonal influenza, Nat. Commun., 9 (2018), 1–14. doi: 10.1038/s41467-018-04792-7. [45] X. Zhou, X. Song and X. Shi, A differential equation model of HIV infection of CD4+ T-cells with cure rate, J. Math. Anal. Appl., 342 (2008), 1342-1355.  doi: 10.1016/j.jmaa.2008.01.008. [46] H. Zhu and X. Zou, Dynamics of HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 511-524.  doi: 10.3934/dcdsb.2009.12.511.

show all references

##### References:
 [1] D. Adam, What scientists know about new, fast-spreading coronavirus variants, Nature, 594 (2021), 19-20. [2] C. M. Bangham and T. B. L. Kirkwood, Defective interfering particles: Effects in modulating virus growth and persistence, Virology, 179 (1990), 821-826.  doi: 10.1016/0042-6822(90)90150-P. [3] A. Baum, D. Ajithdoss, R. Copin et al., REGN-COV2 antibodies prevent and treat SARSCoV-2 infection in rhesus macaques and hamsters, Science, 370 (2020), 1110-1115. doi: 10.1126/science.abe2402. [4] F. Campbell, B. Archer, H. Laurenson-Schafer et al., Increased transmissibility and global spread of SARS-CoV-2 variants of concern as at June 2021, Eurosurveillance, 26 (2021), 2100509. [5] X. Cao, COVID-19: Immunopathology and its implications for therapy, Nat. Rev. Immunol., 20 (2020), 269-270. [6] P. Chen, A. Nirula, B. Heller et al., SARS-CoV-2 neutralizing antibody LY-CoV555 in outpatients with Covid-19, N. Engl. J. Med., 384 (2021), 229–237. [7] N. J. Dimmock and A. J. Easton, Defective interfering influenza virus RNAs: Time to reevaluate their clinical potential as broad-spectrum antivirals, J. Virol., 88 (2014), 5217-5227.  doi: 10.1128/JVI.03193-13. [8] S. A. Frank, Within-host spatial dynamics of viruses and defective interfering particles, J. Theoret. Biol., 206 (2000), 279-290.  doi: 10.1006/jtbi.2000.2120. [9] T. Frensing, F. S. Heldt, A. Pflugmacher et al., Continuous influenza virus production in cell culture shows a periodic accumulation of defective interfering particles, Plos One, 8 (2013), e72288. doi: 10.1371/journal.pone.0072288. [10] S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biol. Dyn., 2 (2008), 140-153.  doi: 10.1080/17513750701769873. [11] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [12] A. S. Huang and D. Baltimore, Defective viral particles and viral disease processes, Nature, 226 (1970), 325-327.  doi: 10.1038/226325a0. [13] C. Huang, Y. Wang, X. Li et al., Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China, Lancet, 395 (2020), 497–506. [14] T. Kajiwara, T. Sasaki and Y. Takeuchi, Construction of Lyapunov functionals for delay differential equations in virology and epidemiology, Nonlinear Anal. Real World Appl., 13 (2012), 1802-1826.  doi: 10.1016/j.nonrwa.2011.12.011. [15] T. B. Kirkwood and C. R. Bangham, Cycles, chaos, and evolution in virus cultures: A model of defective interfering particles, Proc. Natl. Acad. Sci., 91 (1994), 8685-8689.  doi: 10.1073/pnas.91.18.8685. [16] R. L. Kruse, Therapeutic strategies in an outbreak scenario to treat the novel coronavirus originating in Wuhan, China, F1000Research, 9 (2020), 72. [17] J. P. La Salle, The Stability of Dynamical Systems, Society for Industrial and Applied Mathematics, 1976. [18] Q. Li, X. Guan, P. Wu et al., Early transmission dynamics in Wuhan, China, of novel coronavirus–infected pneumonia, N. Engl. J. Med., 382 (2020), 1199–1207. doi: 10.1056/NEJMoa2001316. [19] W. Li, M. J. Moore, N. Vasilieva et al., Angiotensin-converting enzyme 2 is a functional receptor for the SARS coronavirus, Nature, 426 (2003), 450–454. doi: 10.1038/nature02145. [20] X. Li and J. Wei, On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays, Chaos Solitons Fractals, 26 (2005), 519-526.  doi: 10.1016/j.chaos.2005.01.019. [21] C. C. MacDuffee, The Theory of Matrices, Springer, New York, 2012. [22] T. B. Manzoni and C. B. López, Defective (interfering) viral genomes re-explored: Impact on antiviral immunity and virus persistence, Future Virol., 13 (2018), 493-503.  doi: 10.2217/fvl-2018-0021. [23] A. C. Marriott and N. J. Dimmock, Defective interfering viruses and their potential as antiviral agents, Rev. Med. Virol., 20 (2010), 51-62.  doi: 10.1002/rmv.641. [24] G. W. Nelson and A. S.Perelson, Modeling defective interfering virus therapy for AIDS: Conditions for DIV survival, Math. Biosci., 125 (1995), 127-153.  doi: 10.1016/0025-5564(94)00021-Q. [25] Y. Pan, J. Du, J. Liu et al., Screening of potent neutralizing antibodies against SARS-CoV-2 using convalescent patients-derived phage-display libraries, Cell Discov., 7 (2021), 1–19. [26] 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. [27] U. Rand, S. Y. Kupke, H. Shkarlet et al., Antiviral activity of influenza A virus defective interfering particles against SARS-CoV-2 replication in vitro through stimulation of innate immunity, Cells, 10 (2021), 1756. [28] S. Ruan and J. Wei, On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, Math. Med. Biol., 18 (2001), 41-52.  doi: 10.1093/imammb/18.1.41. [29] H. L. Smith, Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems, Bull. Amer. Math. Soc., 33 (1996), 203-209. [30] X. Sun and J. Wei, Stability and bifurcation analysis in a viral infection model with delays, Adv. Differential Equations, 2015 (2015), Article number: 332, 22 pp. doi: 10.1186/s13662-015-0664-7. [31] Y. Sun, D. Jain, C. J. Koziol-White et al., Immunostimulatory defective viral genomes from respiratory syncytial virus promote a strong innate antiviral response during infection in mice and humans, Plos Pathog., 11 (2015), e1005122. doi: 10.1371/journal.ppat.1005122. [32] E. Szathmáry, Co-operation and defection: Playing the field in virus dynamics, J. Theoret. Biol., 165 (1993), 341-356. [33] F. Tapia, T. Laske, M. A. Wasik et al., Production of defective interfering particles of influenza a virus in parallel continuous cultures at two residence times-insights from qPCR measurements and viral dynamics modeling, Front. Bioeng. Biotech., 7 (2019), 275. doi: 10.3389/fbioe.2019.00275. [34] Y. Tian, Y. Bai and P. Yu, Impact of delay on HIV-1 dynamics of fighting a virus with another virus, Math. Biosci. Eng., 11 (2014), 1181-1198.  doi: 10.3934/mbe.2014.11.1181. [35] M. N. Tortorici and D. Veesler, Structural insights into coronavirus entry, Adv. Virus. Res., 105 (2019), 93-116.  doi: 10.1016/bs.aivir.2019.08.002. [36] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [37] M. Vignuzzi and C. B. López, Defective viral genomes are key drivers of the virus–host interaction, Nat. Microbiol., 4 (2019), 1075-1087.  doi: 10.1038/s41564-019-0465-y. [38] D. Wang, B. Hu, C. Hu et al., Clinical characteristics of 138 hospitalized patients with 2019 novel coronavirus–infected pneumonia in Wuhan, China, JAMA, 323 (2020), 1061–1069. [39] 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. [40] Y. Wu, C. Chen and Y. Chan, The outbreak of COVID-19: An overview, J. Chin. Med. Assoc., 83 (2020), 217-220.  doi: 10.1097/JCMA.0000000000000270. [41] S. Yao, A. Narayanan, S. A. Majowicz, J. Jose and M. Archetti, A synthetic defective interfering SARS-CoV-2, PeerJ, 9 (2021), e11686. [42] M. Zhang, J. Xiao, A. Deng et al., Transmission dynamics of an outbreak of the COVID-19 delta variant B. 1.617. 2-Guangdong Province, China, May–June 2021, CCDC Weekly, 3 (2021), 584–586. [43] T. Zhang, Y. Song, Z. Jiang and J. Wang, Dynamical analysis of a delayed HIV virus dynamic model with cell-to-cell transmission and apoptosis of bystander cells, Complexity, 2 (2020), 126-144. [44] H. Zhao, K. K. To, H. Chu et al., Dual-functional peptide with defective interfering genes effectively protects mice against avian and seasonal influenza, Nat. Commun., 9 (2018), 1–14. doi: 10.1038/s41467-018-04792-7. [45] X. Zhou, X. Song and X. Shi, A differential equation model of HIV infection of CD4+ T-cells with cure rate, J. Math. Anal. Appl., 342 (2008), 1342-1355.  doi: 10.1016/j.jmaa.2008.01.008. [46] H. Zhu and X. Zou, Dynamics of HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 511-524.  doi: 10.3934/dcdsb.2009.12.511.
Artificial antibodies block SARS-CoV-2 from infecting cells
Pathogen viral particles $V$ infect normal cells $T$ producing infected cells $I$; $W$ can produce in infected cells; artificial antibodies $F$ bind to virus, infected cells are able to produce virus $V$ and defective interfering particles $W$
When $\mathcal{R}_0<1$, $\tau = 1$, the disease-free equilibrium $E_0$ is globally asymptotically stable
When $R_1<1<\mathcal{R}_0$, $\tau = 0.8, 1,1.5$, the infectious equilibrium without defective interfering particles $E_1$ is globally asymptotically stable
When $1<R_1<\mathcal{R}_0$, $\tau = 1.6$, the infectious equilibrium with defective intefering particles $E_2$ is locally asymptotically stable
When $1<R_1<\mathcal{R}_0$, $\tau = 1.6$, the infectious equilibrium with defective interfering particles $E_2$ showing bifurcation to a stable limit cycle
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