American Institute of Mathematical Sciences

Advanced
January  2021, 26(1): 515-539. doi: 10.3934/dcdsb.2020261

Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus

Tin Phan 1, , Bruce Pell 2, , Amy E. Kendig 3, , Elizabeth T. Borer 4, and  Yang Kuang 1,,

1. 

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
2. 

Department of Mathematics and Computer Science, Lawrence Technological University, Southfield, MI 48075, USA
3. 

Agronomy Department, University of Florida, Gainesville, FL 32611, USA
4. 

Department of Ecology, Evolution, and Behavior, University of Minnesota, St. Paul, MN 55108, USA

* Corresponding author: Yang Kuang

Received  March 2020 Revised  July 2020 Published  January 2021 Early access  August 2020

Fund Project: TP and YK are supported by NSF grants DMS-1615879 and DEB-1930728. YK is also partially supported by the NIGMS of the National Institutes of Health (NIH) under award number R01GM131405

Viral dynamics within plant hosts can be important for understanding plant disease prevalence and impacts. However, few mathematical modeling efforts aim to characterize within-plant viral dynamics. In this paper, we derive a simple system of delay differential equations that describes the spread of infection throughout the plant by barley and cereal yellow dwarf viruses via the cell-to-cell mechanism. By incorporating ratio-dependent incidence function and logistic growth of the healthy cells, the model can capture a wide range of biologically relevant phenomena via the disease-free, endemic, mutual extinction steady states, and a stable periodic orbit. We show that when the basic reproduction number is less than $ 1 $ ($ R_0 < 1 $), the disease-free steady state is asymptotically stable. When $ R_0>1 $, the dynamics either converge to the endemic equilibrium or enter a periodic orbit. Using a ratio-dependent transformation, we show that if the infection rate is very high relative to the growth rate of healthy cells, then the system collapses to the mutual extinction steady state. Numerical and bifurcation simulations are provided to demonstrate our theoretical results. Finally, we carry out parameter estimation using experimental data to characterize the effects of varying nutrients on the dynamics of the system. Our parameter estimates suggest that varying the nutrient supply of nitrogen and phosphorous can alter the dynamics of the infection in plants, specifically reducing the rate of viral production and the rate of infection in certain cases.

Keywords: plant virus, Barley and cereal yellow dwarf viruses, logistic growth, ratio-dependent, cell-to-cell transmission, resource modeling, delay differential equation, stability analysis, Lotka-Volterra.
Mathematics Subject Classification: Primary: 34K20, 92C80; Secondary: 92D25, 92D40.
Citation: Tin Phan, Bruce Pell, Amy E. Kendig, Elizabeth T. Borer, Yang Kuang. Rich dynamics of a simple delay host-pathogen model of cell-to-cell infection for plant virus. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 515-539. doi: 10.3934/dcdsb.2020261
References:
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M. Ali, S. Hameed and M. Tahir, Luteovirus: Insights into pathogenicity, Archives of Virology, 159 (2014), 2853-2860.  doi: 10.1007/s00705-014-2172-6.

[2]

R. Antia, B. R. Levin and R. M. May, Within-host population dynamics and the evolution and maintenance of microparasite virulence, The American Naturalist, 144 (1994), 457-472.  doi: 10.1086/285686.

[3]

F. Atkinson and J. Haddock, Criteria for asymptotic constancy of solutions of functional differential equations, Journal of Mathematical Analysis and Applications, 91 (1983), 410-423.  doi: 10.1016/0022-247X(83)90161-0.

[4]

J. Bak, D. J. Newman and D. J. Newman, Complex Analysis, Springer, 2010. doi: 10.1007/978-1-4419-7288-0.

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Y. M. Bar-On, R. Phillips and R. Milo, The biomass distribution on earth, Proceedings of the National Academy of Sciences, 115 (2018), 6506-6511.  doi: 10.1073/pnas.1711842115.

[6]

M. Begon, M. Bennett, R. G. Bowers, N. P. French, S. Hazel and J. Turner, A clarification of transmission terms in host-microparasite models: numbers, densities and areas, Epidemiology & Infection, 129 (2002), 147-153.  doi: 10.1017/S0950268802007148.

[7]

C. Bendix and J. D. Lewis, The enemy within: Phloem-limited pathogens, Molecular Plant Pathology, 19 (2018), 238-254.  doi: 10.1111/mpp.12526.

[8]

E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection, Mathematical Biosciences, 149 (1998), 57-76.  doi: 10.1016/S0025-5564(97)10015-3.

[9]

E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear Analysis. Real World Applications, 2 (2001), 35-74.  doi: 10.1016/S0362-546X(99)00285-0.

[10]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM Journal on Mathematical Analysis, 33 (2002), 1144-1165.  doi: 10.1137/S0036141000376086.

[11]

P. Bernardo, T. Charles-Dominique, M. Barakat, P. Ortet, E. Fernandez, D. Filloux, P. Hartnady, T. A. Rebelo, S. R. Cousins, F. Mesleard et al., Geometagenomics illuminates the impact of agriculture on the distribution and prevalence of plant viruses at the ecosystem scale, The ISME Journal, 12 (2018), 173-184. doi: 10.1038/ismej.2017.155.

[12]

E. T. Borer, A.-L. Laine and E. W. Seabloom, A multiscale approach to plant disease using the metacommunity concept, Annual Review of Phytopathology, 54 (2016), 397-418.  doi: 10.1146/annurev-phyto-080615-095959.

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J. C. Carrington, K. D. Kasschau, S. K. Mahajan and M. C. Schaad, Cell-to-cell and long-distance transport of viruses in plants., The Plant Cell, 8 (1996), 1669.

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R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell spread of hiv-1 that includes a time delay, Journal of Mathematical Biology, 46 (2003), 425-444.  doi: 10.1007/s00285-002-0191-5.

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C. J. D'Arcy and P. A. Burnett, Barley Yellow Dwarf: 40 Years of Progress, 1995.

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[17]

S. Eikenberry, S. Hews, J. D. Nagy and Y. Kuang, The dynamics of a delay model of hbv infection with logistic hepatocyte growth, Math. Biosc. Eng, 6 (2009), 283-299.  doi: 10.3934/mbe.2009.6.283.

[18]

G. F. Gause, The Struggle for Existence: A Classic of Mathematical Biology and Ecology, Courier Dover Publications, 2019.

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M. A. Gilchrist, D. Coombs and A. S. Perelson, Optimizing within-host viral fitness: Infected cell lifespan and virion production rate, Journal of theoretical biology, 229 (2004), 281-288.  doi: 10.1016/j.jtbi.2004.04.015.

[20]

C. Gill and J. Chong, Cytopathological evidence for the division of barley yellow dwarf virus isolates into two subgroups, Virology, 95 (1979), 59-69.  doi: 10.1016/0042-6822(79)90401-X.

[21]

S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis b virus infection, Journal of Biological Dynamics, 2 (2008), 140-153.  doi: 10.1080/17513750701769873.

[22]

Z. Grossman, M. B. Feinberg and W. E. Paul, Multiple modes of cellular activation and virus transmission in hiv infection: a role for chronically and latently infected cells in sustaining viral replication, Proceedings of the National Academy of Sciences, 95 (1998), 6314-6319.  doi: 10.1073/pnas.95.11.6314.

[23]

S. Hews, S. Eikenberry, J. D. Nagy and Y. Kuang, Rich dynamics of a hepatitis b viral infection model with logistic hepatocyte growth, Journal of Mathematical Biology, 60 (2010), 573-590.  doi: 10.1007/s00285-009-0278-3.

[24]

S.-B. Hsu, T.-W. Hwang and Y. Kuang, Global analysis of the michaelis–menten-type ratio-dependent predator-prey system, Journal of Mathematical Biology, 42 (2001), 489-506.  doi: 10.1007/s002850100079.

[25]

M. Jackson and B. M. Chen-Charpentier, Modeling plant virus propagation with delays, Journal of Computational and Applied Mathematics, 309 (2017), 611-621.  doi: 10.1016/j.cam.2016.04.024.

[26]

A. E. Kendig, E. T. Borer, E. N. Boak, T. C. Picard and E. W. Seabloom, Soil nitrogen and phosphorus effects on plant virus density, transmission, and species interactions, URLhttps://doi.org/10.6073/pasta/01e7bf593676a942f262623710acba13.

[27]

D. A. Kennedy, V. Dukic and G. Dwyer, Pathogen growth in insect hosts: Inferring the importance of different mechanisms using stochastic models and response-time data, The American Naturalist, 184 (2014), 407-423.  doi: 10.1086/677308.

[28]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator–prey system, Journal of Mathematical Biology, 36 (1998), 389-406.  doi: 10.1007/s002850050105.

[29]

P. Kumberger, K. Durso-Cain, S. Uprichard, H. Dahari and F. Graw, Accounting for space–quantification of cell-to-cell transmission kinetics using virus dynamics models, Viruses, 10 (2018), 200. doi: 10.3390/v10040200.

[30]

C. Lacroix, E. W. Seabloom and E. T. Borer, Environmental nutrient supply alters prevalence and weakens competitive interactions among coinfecting viruses, New Phytologist, 204 (2014), 424-433.  doi: 10.1111/nph.12909.

[31]

C. Lacroix, E. W. Seabloom and E. T. Borer, Environmental nutrient supply directly alters plant traits but indirectly determines virus growth rate, Frontiers in Microbiology, 8 (2017), 2116. doi: 10.3389/fmicb.2017.02116.

[32]

P. Lefeuvre, D. P. Martin, S. F. Elena, D. N. Shepherd, P. Roumagnac and A. Varsani, Evolution and ecology of plant viruses, Nature Reviews Microbiology, 17 (2019), 632-644.  doi: 10.1038/s41579-019-0232-3.

[33]

R. F. Luck, Evaluation of natural enemies for biological control: A behavioral approach, Trends in Ecology & Evolution, 5 (1990), 196-199.  doi: 10.1016/0169-5347(90)90210-5.

[34]

G. Neofytou, Y. Kyrychko and K. Blyuss, Mathematical model of plant-virus interactions mediated by rna interference, Journal of Theoretical Biology, 403 (2016), 129-142.  doi: 10.1016/j.jtbi.2016.05.018.

[35]

J. C. Ng and K. L. Perry, Transmission of plant viruses by aphid vectors, Molecular Plant Pathology, 5 (2004), 505-511.  doi: 10.1111/j.1364-3703.2004.00240.x.

[36]

M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas and H. McDade, Viral dynamics in hepatitis b virus infection, Proceedings of the National Academy of Sciences, 93 (1996), 4398-4402.  doi: 10.1073/pnas.93.9.4398.

[37]

B. Pell, A. E. Kendig, E. T. Borer and Y. Kuang, Modeling nutrient and disease dynamics in a plant-pathogen system 2, Mathematical Biosciences and Engineering, 16 (2019), 234-264. 

[38]

M. J. Roossinck and E. R. Bazán, Symbiosis: Viruses as intimate partners, Annual Review of Virology, 4 (2017), 123-139.  doi: 10.1146/annurev-virology-110615-042323.

[39]

M. J. Roossinck, P. Saha, G. B. Wiley, J. Quan, J. D. White, H. Lai, F. Chavarria, G. Shen and B. A. Roe, Ecogenomics: Using massively parallel pyrosequencing to understand virus ecology, Molecular Ecology, 19 (2010), 81-88.  doi: 10.1111/j.1365-294X.2009.04470.x.

[40]

M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time, Science, 171 (1971), 385-387.  doi: 10.1126/science.171.3969.385.

[41]

A. Sigal, J. T. Kim, A. B. Balazs, E. Dekel, A. Mayo, R. Milo and D. Baltimore, Cell-to-cell spread of hiv permits ongoing replication despite antiretroviral therapy, Nature, 477 (2011), 95-98.  doi: 10.1038/nature10347.

[42]

A. L. Vuorinen, J. Kelloniemi and J. P. Valkonen, Why do viruses need phloem for systemic invasion of plants?, Plant Science, 181 (2011), 355-363.  doi: 10.1016/j.plantsci.2011.06.008.

[43]

X. Wang, S. Tang, X. Song and L. Rong, Mathematical analysis of an hiv latent infection model including both virus-to-cell infection and cell-to-cell transmission, Journal of Biological Dynamics, 11 (2017), 455-483.  doi: 10.1080/17513758.2016.1242784.

[44]

Z. Wu, T. Phan, J. Baez, Y. Kuang and E. J. Kostelich, Predictability and identifiability assessment of models for prostate cancer under androgen suppression therapy, Mathematical Biosciences and Engineering, 16 (2019), 3512-3536. 

[45]

Y. Yang, L. Zou and S. Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Mathematical Biosciences, 270 (2015), 183-191.  doi: 10.1016/j.mbs.2015.05.001.

[46]

P. Zhong, L. M. Agosto, J. B. Munro and W. Mothes, Cell-to-cell transmission of viruses, Current Opinion in Virology, 3 (2013), 44-50.  doi: 10.1016/j.coviro.2012.11.004.

show all references

References:
[1]

M. Ali, S. Hameed and M. Tahir, Luteovirus: Insights into pathogenicity, Archives of Virology, 159 (2014), 2853-2860.  doi: 10.1007/s00705-014-2172-6.

[2]

R. Antia, B. R. Levin and R. M. May, Within-host population dynamics and the evolution and maintenance of microparasite virulence, The American Naturalist, 144 (1994), 457-472.  doi: 10.1086/285686.

[3]

F. Atkinson and J. Haddock, Criteria for asymptotic constancy of solutions of functional differential equations, Journal of Mathematical Analysis and Applications, 91 (1983), 410-423.  doi: 10.1016/0022-247X(83)90161-0.

[4]

J. Bak, D. J. Newman and D. J. Newman, Complex Analysis, Springer, 2010. doi: 10.1007/978-1-4419-7288-0.

[5]

Y. M. Bar-On, R. Phillips and R. Milo, The biomass distribution on earth, Proceedings of the National Academy of Sciences, 115 (2018), 6506-6511.  doi: 10.1073/pnas.1711842115.

[6]

M. Begon, M. Bennett, R. G. Bowers, N. P. French, S. Hazel and J. Turner, A clarification of transmission terms in host-microparasite models: numbers, densities and areas, Epidemiology & Infection, 129 (2002), 147-153.  doi: 10.1017/S0950268802007148.

[7]

C. Bendix and J. D. Lewis, The enemy within: Phloem-limited pathogens, Molecular Plant Pathology, 19 (2018), 238-254.  doi: 10.1111/mpp.12526.

[8]

E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection, Mathematical Biosciences, 149 (1998), 57-76.  doi: 10.1016/S0025-5564(97)10015-3.

[9]

E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear Analysis. Real World Applications, 2 (2001), 35-74.  doi: 10.1016/S0362-546X(99)00285-0.

[10]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM Journal on Mathematical Analysis, 33 (2002), 1144-1165.  doi: 10.1137/S0036141000376086.

[11]

P. Bernardo, T. Charles-Dominique, M. Barakat, P. Ortet, E. Fernandez, D. Filloux, P. Hartnady, T. A. Rebelo, S. R. Cousins, F. Mesleard et al., Geometagenomics illuminates the impact of agriculture on the distribution and prevalence of plant viruses at the ecosystem scale, The ISME Journal, 12 (2018), 173-184. doi: 10.1038/ismej.2017.155.

[12]

E. T. Borer, A.-L. Laine and E. W. Seabloom, A multiscale approach to plant disease using the metacommunity concept, Annual Review of Phytopathology, 54 (2016), 397-418.  doi: 10.1146/annurev-phyto-080615-095959.

[13]

J. C. Carrington, K. D. Kasschau, S. K. Mahajan and M. C. Schaad, Cell-to-cell and long-distance transport of viruses in plants., The Plant Cell, 8 (1996), 1669.

[14]

R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell spread of hiv-1 that includes a time delay, Journal of Mathematical Biology, 46 (2003), 425-444.  doi: 10.1007/s00285-002-0191-5.

[15]

C. J. D'Arcy and P. A. Burnett, Barley Yellow Dwarf: 40 Years of Progress, 1995.

[16]

V. Eastop, Worldwide importance of aphids as virus vectors, in Aphids as Virus Vectors, Elsevier, 1977, 3–62. doi: 10.1016/B978-0-12-327550-9.50006-9.

[17]

S. Eikenberry, S. Hews, J. D. Nagy and Y. Kuang, The dynamics of a delay model of hbv infection with logistic hepatocyte growth, Math. Biosc. Eng, 6 (2009), 283-299.  doi: 10.3934/mbe.2009.6.283.

[18]

G. F. Gause, The Struggle for Existence: A Classic of Mathematical Biology and Ecology, Courier Dover Publications, 2019.

[19]

M. A. Gilchrist, D. Coombs and A. S. Perelson, Optimizing within-host viral fitness: Infected cell lifespan and virion production rate, Journal of theoretical biology, 229 (2004), 281-288.  doi: 10.1016/j.jtbi.2004.04.015.

[20]

C. Gill and J. Chong, Cytopathological evidence for the division of barley yellow dwarf virus isolates into two subgroups, Virology, 95 (1979), 59-69.  doi: 10.1016/0042-6822(79)90401-X.

[21]

S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis b virus infection, Journal of Biological Dynamics, 2 (2008), 140-153.  doi: 10.1080/17513750701769873.

[22]

Z. Grossman, M. B. Feinberg and W. E. Paul, Multiple modes of cellular activation and virus transmission in hiv infection: a role for chronically and latently infected cells in sustaining viral replication, Proceedings of the National Academy of Sciences, 95 (1998), 6314-6319.  doi: 10.1073/pnas.95.11.6314.

[23]

S. Hews, S. Eikenberry, J. D. Nagy and Y. Kuang, Rich dynamics of a hepatitis b viral infection model with logistic hepatocyte growth, Journal of Mathematical Biology, 60 (2010), 573-590.  doi: 10.1007/s00285-009-0278-3.

[24]

S.-B. Hsu, T.-W. Hwang and Y. Kuang, Global analysis of the michaelis–menten-type ratio-dependent predator-prey system, Journal of Mathematical Biology, 42 (2001), 489-506.  doi: 10.1007/s002850100079.

[25]

M. Jackson and B. M. Chen-Charpentier, Modeling plant virus propagation with delays, Journal of Computational and Applied Mathematics, 309 (2017), 611-621.  doi: 10.1016/j.cam.2016.04.024.

[26]

A. E. Kendig, E. T. Borer, E. N. Boak, T. C. Picard and E. W. Seabloom, Soil nitrogen and phosphorus effects on plant virus density, transmission, and species interactions, URLhttps://doi.org/10.6073/pasta/01e7bf593676a942f262623710acba13.

[27]

D. A. Kennedy, V. Dukic and G. Dwyer, Pathogen growth in insect hosts: Inferring the importance of different mechanisms using stochastic models and response-time data, The American Naturalist, 184 (2014), 407-423.  doi: 10.1086/677308.

[28]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator–prey system, Journal of Mathematical Biology, 36 (1998), 389-406.  doi: 10.1007/s002850050105.

[29]

P. Kumberger, K. Durso-Cain, S. Uprichard, H. Dahari and F. Graw, Accounting for space–quantification of cell-to-cell transmission kinetics using virus dynamics models, Viruses, 10 (2018), 200. doi: 10.3390/v10040200.

[30]

C. Lacroix, E. W. Seabloom and E. T. Borer, Environmental nutrient supply alters prevalence and weakens competitive interactions among coinfecting viruses, New Phytologist, 204 (2014), 424-433.  doi: 10.1111/nph.12909.

[31]

C. Lacroix, E. W. Seabloom and E. T. Borer, Environmental nutrient supply directly alters plant traits but indirectly determines virus growth rate, Frontiers in Microbiology, 8 (2017), 2116. doi: 10.3389/fmicb.2017.02116.

[32]

P. Lefeuvre, D. P. Martin, S. F. Elena, D. N. Shepherd, P. Roumagnac and A. Varsani, Evolution and ecology of plant viruses, Nature Reviews Microbiology, 17 (2019), 632-644.  doi: 10.1038/s41579-019-0232-3.

[33]

R. F. Luck, Evaluation of natural enemies for biological control: A behavioral approach, Trends in Ecology & Evolution, 5 (1990), 196-199.  doi: 10.1016/0169-5347(90)90210-5.

[34]

G. Neofytou, Y. Kyrychko and K. Blyuss, Mathematical model of plant-virus interactions mediated by rna interference, Journal of Theoretical Biology, 403 (2016), 129-142.  doi: 10.1016/j.jtbi.2016.05.018.

[35]

J. C. Ng and K. L. Perry, Transmission of plant viruses by aphid vectors, Molecular Plant Pathology, 5 (2004), 505-511.  doi: 10.1111/j.1364-3703.2004.00240.x.

[36]

M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas and H. McDade, Viral dynamics in hepatitis b virus infection, Proceedings of the National Academy of Sciences, 93 (1996), 4398-4402.  doi: 10.1073/pnas.93.9.4398.

[37]

B. Pell, A. E. Kendig, E. T. Borer and Y. Kuang, Modeling nutrient and disease dynamics in a plant-pathogen system 2, Mathematical Biosciences and Engineering, 16 (2019), 234-264. 

[38]

M. J. Roossinck and E. R. Bazán, Symbiosis: Viruses as intimate partners, Annual Review of Virology, 4 (2017), 123-139.  doi: 10.1146/annurev-virology-110615-042323.

[39]

M. J. Roossinck, P. Saha, G. B. Wiley, J. Quan, J. D. White, H. Lai, F. Chavarria, G. Shen and B. A. Roe, Ecogenomics: Using massively parallel pyrosequencing to understand virus ecology, Molecular Ecology, 19 (2010), 81-88.  doi: 10.1111/j.1365-294X.2009.04470.x.

[40]

M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time, Science, 171 (1971), 385-387.  doi: 10.1126/science.171.3969.385.

[41]

A. Sigal, J. T. Kim, A. B. Balazs, E. Dekel, A. Mayo, R. Milo and D. Baltimore, Cell-to-cell spread of hiv permits ongoing replication despite antiretroviral therapy, Nature, 477 (2011), 95-98.  doi: 10.1038/nature10347.

[42]

A. L. Vuorinen, J. Kelloniemi and J. P. Valkonen, Why do viruses need phloem for systemic invasion of plants?, Plant Science, 181 (2011), 355-363.  doi: 10.1016/j.plantsci.2011.06.008.

[43]

X. Wang, S. Tang, X. Song and L. Rong, Mathematical analysis of an hiv latent infection model including both virus-to-cell infection and cell-to-cell transmission, Journal of Biological Dynamics, 11 (2017), 455-483.  doi: 10.1080/17513758.2016.1242784.

[44]

Z. Wu, T. Phan, J. Baez, Y. Kuang and E. J. Kostelich, Predictability and identifiability assessment of models for prostate cancer under androgen suppression therapy, Mathematical Biosciences and Engineering, 16 (2019), 3512-3536. 

[45]

Y. Yang, L. Zou and S. Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Mathematical Biosciences, 270 (2015), 183-191.  doi: 10.1016/j.mbs.2015.05.001.

[46]

P. Zhong, L. M. Agosto, J. B. Munro and W. Mothes, Cell-to-cell transmission of viruses, Current Opinion in Virology, 3 (2013), 44-50.  doi: 10.1016/j.coviro.2012.11.004.

Figure 1.  Increasing $ \tau $ changes the stability of the positive equilibrium, which gives rise to a stable orbit. For this simulation, we use $ r = 0.3,K = 10^3,\beta = 0.1,\delta = 0.0001 $ and $ \tau $ varies from $ 1 $ to $ 80 $. We plot $ \tau $ over a viable region. For smaller value of $ \tau $, either the condition for the theorem is not satisfied or the positive steady does not exist. The switching between a stable positive steady state and a stable orbit takes place around $ \tau \approx 50.5 $
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Figure 2.  Corresponding examples for Figure 1. (a) $ \tau = 50 $, the oscillation is damping toward the positive steady state. (b) $ \tau = 51 $, the oscillation is stable
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Figure 3.  For this simulation, we start with the following values $ r = 0.3,K = 10^3,\beta = 0.1,\delta = 0.0001 $ and $ \tau = 51 $. (a) increasing the infection rate $ \beta $ can have a destabilizing effect on the endemic equilibrium; however, as $ \beta $ increases, $ S $ and $ I $ approach closely to 0. (b) Decreasing the death rate $ \delta $ can be destabilizing as well. (c) The growth rate $ r $ can be both stabilizing or destabilizing as it varies. As $ r $ decreases, it can result in mutual extinction. (d) shows additional details of Figure 1 and 2. Note that varying the carrying capacity $ K $ only changes the size but not the stability
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Figure 4.  Parameter fitting result for the cell-to-cell transmission model. The description of each experiment is given in subsection 1.2 and additional details can be found in Kendig et al. [26]
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Figure 5.  Comparison of data fitting between mass action model and ratio-dependent model. The description of each experiment is given in subsection 1.2 and additional details can be found in Kendig et al. [26]
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Table 1.  Estimated parameter for cell-to-cell model. Note that $ \delta $ is fixed to be $ 1/13 $ day$ ^{-1} $. The value of $ R_0 $ is calculated based on the estimated parameters. The description of each experiment is given in subsection 1.2 and additional details can be found in Kendig et al. [26]
Parameter Fitted (CTRL) Fitted (+N) Fitted (+P) Fitted (+NP) Units
$ r $ 0.9000 0.9000 0.9000 0.8860 day$ ^{-1} $
$ K $ 515024 719563 400294 400000 cells
$ \beta $ 0.5387 0.4355 0.8925 0.6710 cells virion$ ^{-1} $ day$ ^{-1} $
$ b $ 65 94 62 80 virions cell$ ^{-1} $ day$ ^{-1} $
$ \tau $ 8.27 12.00 12.00 12.00 days
$ R_0 $ 1.62 2.07 2.30 2.23 unitless
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Parameter Fitted (CTRL) Fitted (+N) Fitted (+P) Fitted (+NP) Units
$ r $ 0.9000 0.9000 0.9000 0.8860 day$ ^{-1} $
$ K $ 515024 719563 400294 400000 cells
$ \beta $ 0.5387 0.4355 0.8925 0.6710 cells virion$ ^{-1} $ day$ ^{-1} $
$ b $ 65 94 62 80 virions cell$ ^{-1} $ day$ ^{-1} $
$ \tau $ 8.27 12.00 12.00 12.00 days
$ R_0 $ 1.62 2.07 2.30 2.23 unitless
Table 2.  Fitting errors for the cell-to-cell transmission model. The description of each experiment is given in subsection 1.2 and additional details can be found in Kendig et al. [26]
Experiment control +N +P +NP
RMSE 4.27e+6 5.97e+6 3.66e+6 8.96e+6
MAPE 8.03e-1 6.35e-1 4.10e-1 7.14e-1
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Experiment control +N +P +NP
RMSE 4.27e+6 5.97e+6 3.66e+6 8.96e+6
MAPE 8.03e-1 6.35e-1 4.10e-1 7.14e-1
Table 3.  Stability results and open questions in terms of $ \beta, \delta, r $ and $ \tau $ $ \left(\text{note: } R_0 = \frac{\beta-\delta}{\beta e^{-\delta\tau}}\right) $
Conditions Results or question
1. $ \beta<\delta $ $ (K,0) $ is globally asymptotically stable
2. $ \beta>\delta $ and $ \frac{\beta-\delta}{\beta e^{-\delta\tau}}<1 $ $ (K,0) $ is locally asymptotically stable
3. $ \beta>\delta $ and $ \frac{\beta-\delta}{\beta-\delta-r}>\frac{\beta-\delta}{\beta e^{-\delta\tau}}>1 $ Open question 1: is $ E^* $ stable?
when does a periodic orbit occurs?
4. $ \beta>\delta+r $ and $ \frac{\beta-\delta}{\beta-\delta-r}<\frac{\beta-\delta}{\beta e^{-\delta\tau}} $ (0, 0) is globally asymptotically stable
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Conditions Results or question
1. $ \beta<\delta $ $ (K,0) $ is globally asymptotically stable
2. $ \beta>\delta $ and $ \frac{\beta-\delta}{\beta e^{-\delta\tau}}<1 $ $ (K,0) $ is locally asymptotically stable
3. $ \beta>\delta $ and $ \frac{\beta-\delta}{\beta-\delta-r}>\frac{\beta-\delta}{\beta e^{-\delta\tau}}>1 $ Open question 1: is $ E^* $ stable?
when does a periodic orbit occurs?
4. $ \beta>\delta+r $ and $ \frac{\beta-\delta}{\beta-\delta-r}<\frac{\beta-\delta}{\beta e^{-\delta\tau}} $ (0, 0) is globally asymptotically stable
Table 4.  Estimated parameter for mass action model. The description of each experiment is given in subsection 1.2 and additional details can be found in Kendig et al. [26]
Parameter Fitted (CTRL) Fitted (+N) Fitted (+P) Fitted (+NP) Units
$ r $ 0.0993 0.0100 0.8579 0.1549 day$ ^{-1} $
$ K $ 4.0000e+5 6.0164e+5 4.0038e+5 1.0987e+6 cells
$ \beta $ 2.0273e-6 2.8651e-7 1.9817e-6 1.9188e-6 cells virion$ ^{-1} $ day$ ^{-1} $
$ d $ 0.7129 0.1001 0.1001 0.1001 day$ ^{-1} $
$ b $ 118.2189 199.9803 60.4637 56.2613 virions cell$ ^{-1} $ day$ ^{-1} $
$ \tau $ 9.6880 21.0000 4.9741 7.4480 days
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Parameter Fitted (CTRL) Fitted (+N) Fitted (+P) Fitted (+NP) Units
$ r $ 0.0993 0.0100 0.8579 0.1549 day$ ^{-1} $
$ K $ 4.0000e+5 6.0164e+5 4.0038e+5 1.0987e+6 cells
$ \beta $ 2.0273e-6 2.8651e-7 1.9817e-6 1.9188e-6 cells virion$ ^{-1} $ day$ ^{-1} $
$ d $ 0.7129 0.1001 0.1001 0.1001 day$ ^{-1} $
$ b $ 118.2189 199.9803 60.4637 56.2613 virions cell$ ^{-1} $ day$ ^{-1} $
$ \tau $ 9.6880 21.0000 4.9741 7.4480 days
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