Abstract
This paper analyzes the role of neutral viruses in the phenomenon of local immunodeficiency. We show that, even in the absence of altruistic viruses, neutral viruses can support the existence of persistent viruses and thus local immunodeficiency. However, in all such cases neutral viruses can maintain only bounded (relatively small) concentration of persistent viruses. Moreover, in all such cases the state of local immunodeficiency could only be marginally stable, while it is known that altruistic viruses can maintain stable local immunodeficiency. We also present an absolutely minimal cross-immunoreactivity network where a stable and robust state of local immunodeficiency can be maintained. It is now a challenge to synthetic biology to build such small networks with stable local immunodeficiency. Another important challenge for biology is to understand which types of viruses can play a role of persistent, altruistic and neutral ones and whether a role which a given virus plays depends on the structure (topology) of a given cross-immunoreactivity network.
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Acknowledgements
The authors are indebted to P. Skums for valuable discussions. This work was partially supported by the NSF Grant CCF-BSF-1664836 and by the NIH Grant 1R01EB025022.
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Appendices
Appendix A. Computation of \(P(\lambda )\) from the Characteristic Polynomial of the Jacobian for Fig. 1
Appendix B. Computation of the Characteristic Polynomial of the Jacobian for Fig. 3
Let \(\lambda _1=\frac{cx_1}{\alpha r_2}-b\),
Appendix C. Computation of the Characteristic Polynomial of the Jacobian for Fig. 4
Let \(\lambda _1=\frac{cx_1}{\alpha (r_2+r_3)}-b<\frac{b}{\alpha }-b\), then
By expanding along the fourth row, we get
Expanding now along the second row, we obtain
where
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Bunimovich, L., Shu, L. Local Immunodeficiency: Role of Neutral Viruses. Bull Math Biol 82, 140 (2020). https://doi.org/10.1007/s11538-020-00813-z
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DOI: https://doi.org/10.1007/s11538-020-00813-z