# Nonlocal Reaction–Diffusion Model of Viral Evolution: Emergence of Virus Strains

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Bifurcations of Periodic Structures

#### 2.1. Single Nonlocal Term

#### 2.2. Examples

#### 2.3. Double Nonlocal Equation

**Proposition**

**1.**

#### 2.4. Delay Equation

#### 2.5. Nonlocal Delay Equation

## 3. Emergence of Strains as Periodic Wave Propagation

#### 3.1. Propagation Of Waves

#### 3.1.1. Nonlocal Equation

#### 3.1.2. Bifurcations of Waves and Pulses

#### 3.2. Emergence of Strains

#### 3.2.1. Initiation of Periodic Waves

#### 3.2.2. the Influence of Immune Response

#### 3.2.3. Effect of the Delay of the Antiviral Immune Response

#### 3.2.4. the Influence of Genotype-Dependent Mortality

## 4. Discussion

#### 4.1. Virus Quasi-Species

#### 4.2. Emergence of New Quasi-Species: Summary of the Results

#### 4.3. Biological Interpretations

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Additional Simulations

**Figure A1.**Numerical simulations of Equation (14) for the linear function $f\left(u\right)={k}_{1}u$. Level lines of the solution $u(x,t)$ on the plane $(x,t)$ (

**left**). Two snapshots of solution (

**right**). The value of parameters: $r=1,q=1,{k}_{1}=0.95,N=0.1,\tau =4$, $D=0.0001$ (

**left**), $D=0.00001$ (

**right**), $t=150$.

**Figure A2.**Level lines of the solution $u(x,t)$ of Equation (17) on the $(x,t)$-plane. Values of parameters: $r=1,q=1,N=0.2,\tau =0$, $f\left(u\right)=0$ (left and middle), the maximum of the initial condition $0.9$, $D=0.0001,t=35$ (

**left**) and $D=0.0005,t=20$ (

**right**).

**Figure A3.**Level lines of the solution $u(x,t)$ of Equation (17) on the $(x,t)$-plane. Values of parameters: $r=1,q=1,N=0.1,\tau =0$, $f\left(u\right)=0$ (left and middle), the maximum of the initial condition $0.9$, $D=0.00001,t=130$ (

**left**) and $D=0.0001,t=75$ (

**right**).

**Figure A4.**Level lines of the solution $u(x,t)$ of Equation (17). The values of parameters: $r=1,q=1$, $L=2,D={10}^{-5},{k}_{1}=0.9,\tau =4,N=0.05$ (

**left**), $N=0.1$ (

**right**), $t=150$.

## References

- Keele, B.H.; Giorgi, E.E.; Salazar-Gonzalez, J.F.; Decker, J.M.; Pham, K.T.; Salazar, M.G.; Sun, C.; Grayson, T.; Wang, S.; Li, H.; et al. Identication and characterization of transmitted and early founder virus envelopes in primary HIV-1 infection. Proc. Natl. Acad. Sci. USA
**2008**, 105, 7552–7557. [Google Scholar] [CrossRef] [PubMed][Green Version] - Plikat, U.; Nieselt-Struwe, K.; Meyerhans, A. Genetic drift can dominate short-term human immunodeficiency virus type 1 nef quasispecies evolution in vivo. J. Virol.
**1997**, 71, 4233–4240. [Google Scholar] [CrossRef] [PubMed][Green Version] - Biebricher, C.K.; Eigen, M. What is a quasispecies? Curr. Top. Microbiol. Immunol.
**2006**, 299, 1–31. [Google Scholar] [PubMed] - Domingo, E.; Perales, C. Viral quasispecies. PLoS Genet.
**2019**, 15, e1008271. [Google Scholar] [CrossRef] [PubMed][Green Version] - Goodenow, M.; Huet, T.; Saurin, W.; Kwok, S.; Sninsky, J.; Wain-Hobson, S. HIV-1 isolates are rapidly evolving quasispecies: Evidence for viral mixtures and preferred nucleotide substitutions. J. Acquir. Immune Defic. Syndr.
**1989**, 2, 344–352. [Google Scholar] [PubMed] - Holland, J.J.; De La Torre, J.C.; Steinhauer, D.A. RNA virus populations as quasispecies. Curr. Top. Microbiol. Immunol.
**1992**, 176, 1–20. [Google Scholar] - Meyerhans, A.; Cheynier, R.; Albert, J.; Seth, M.; Kwok, S.; Sninsky, J.; Morfeldt-Manson, L.; Asjo, B.; Wain-Hobson, S. Temporal fluctuations in HIV quasispecies in vivo are not reflected by sequential HIV isolations. Cell
**1989**, 58, 901–910. [Google Scholar] [CrossRef] - Phillips, R.E.; Rowland-Jones, S.; Nixon, D.F.; Gotch, F.M.; Edwards, J.P.; Ogunlesi, A.O.; Elvin, J.G.; Rothbard, J.A.; Bangham, C.R.; Rizza, C.R.; et al. Human immunodeficiency virus genetic variation that can escape cytotoxic T cell recognition. Nature
**1991**, 354, 453–459. [Google Scholar] [CrossRef] - Larder, B.A.; Kemp, S.D. Multiple mutations in HIV-1 reverse transcriptase confer high-level resistance to zidovudine (AZT). Science
**1989**, 246, 1155–1158. [Google Scholar] [CrossRef] - Vignuzzi, M.; Stone, J.K.; Arnold, J.J.; Cameron, C.E.; Andino, R. Quasispecies diversity determines pathogenesis through cooperative interactions in a viral population. Nature
**2006**, 439, 344–348. [Google Scholar] [CrossRef] - Collier, D.A.; Monit, C.; Gupta, R.K. The Impact of HIV-1 drug escape on the global treatment landscape. Cell Host Microbe
**2019**, 26, 48–60. [Google Scholar] [CrossRef] [PubMed] - Esposito, I.; Trinks, J.; Soriano, V. Hepatitis C virus resistance to the new direct-acting antivirals. Expert Opin. Drug Metab. Toxicol.
**2016**, 12, 1197–1209. [Google Scholar] [CrossRef] [PubMed] - Kiepiela, P.; Leslie, A.J.; Honeyborne, I.; Ramduth, D.; Thobakgale, C.; Chetty, S.; Rathnavalu, P.; Moore, C.; Pfaerott, K.J.; Hilton, L. Dominant inuence of HLA-B in mediating the potential co-evolution of HIV and HLA. Nature
**2004**, 432, 769–775. [Google Scholar] [CrossRef] [PubMed] - Haas, G.; Plikat, U.; Debré, P.; Lucchiari, M.; Katlama, C.; Dudoit, Y.; Bonduelle, O.; Bauer, M.; Ihlenfeldt, H.G.; Jung, G.; et al. Dynamics of viral variants in HIV-1 Nef and specific cytotoxic T lymphocytes in vivo. J. Immunol.
**1996**, 157, 4212–4221. [Google Scholar] [PubMed] - Ganusov, V.V.; Goonetilleke, N.; Liu, M.K.; Ferrari, G.; Shaw, G.M.; McMichael, A.J.; Borrow, P.; Korber, B.T.; Perelson, A.S. Fitness costs and diversity of the cytotoxic T lymphocyte (CTL) response determine the rate of CTL escape during acute and chronic phases of HIV infection. J. Virol.
**2011**, 85, 10518–10528. [Google Scholar] [CrossRef] [PubMed][Green Version] - Turnbull, E.L.; Wong, M.; Wang, S.; Wei, X.; Jones, N.A.; Conrod, K.E.; Aldam, D.; Turner, J.; Pellegrino, P.; Keele, B.F.; et al. Kinetics of expansion of epitope-specific T cell responses during primary HIV-1 infection. J. Immunol.
**2009**, 182, 7131–7145. [Google Scholar] [CrossRef][Green Version] - Ganusov, V.V.; Neher, R.A.; Perelson, A.S. Mathematical modeling of escape of HIV from cytotoxic T lymphocyte responses. J. Stat. Mech.
**2013**, 2013, P01010. [Google Scholar] [CrossRef] - Bocharov, G.A.; Ford, N.J.; Edwards, J.; Breinig, T.; Wain-Hobson, S.; Meyerhans, A. A genetic algorithm approach to simulating human immunodeficiency virus evolution reveals the strong impact of multiply infected cells and recombination. J. Gen. Virol.
**2005**, 86, 3109–3118. [Google Scholar] [CrossRef] - Swanstrom, R.; Coffin, J. HIV-1 pathogenesis: The virus. Cold Spring Harb. Perspect. Med.
**2012**, 2, a007443. [Google Scholar] [CrossRef][Green Version] - Bessonov, N.; Reinberg, N.; Volpert, V. Mathematics of Darwin’s diagram. Math. Model. Nat. Phenom.
**2014**, 9, 5–25. [Google Scholar] [CrossRef][Green Version] - Bessonov, N.; Reinberg, N.; Banerjee, M.; Volpert, V. The origin of species by means of mathematical modelling. Acta Bioteoretica
**2018**, 66, 333–344. [Google Scholar] - Genieys, S.; Volpert, V.; Auger, P. Adaptive dynamics: Modelling Darwin’s divergence principle. Comptes Rendus Biol.
**2006**, 329, 876–879. [Google Scholar] [CrossRef] [PubMed] - Bessonov, N.; Bocharov, G.; Meyerhans, A.; Popov, V.; Volpert, V. Nonlocal reaction-diffusion model of viral evolution. Existence and dynamics of strains. Preprints
**2019**. [Google Scholar] [CrossRef][Green Version] - Volpert, V. Elliptic Partial Differential Equations. Volume 2. Reaction-Diffusion Equations; Birkhäuser: Basel, Switzerland, 2014. [Google Scholar]
- Bessonov, N.; Bocharov, G.; Meyerhans, A.; Volpert, V. Interplay between reaction and diffusion processes in governing the dynamics of virus infections. J. Theor. Biol.
**2018**, 457, 221–236. [Google Scholar] - Bocharov, G.; Volpert, V.; Ludewig, B.; Meyerhans, A. Mathematical Immunology of Virus Infections; Springer: Cham, Switzerland, 2018. [Google Scholar]
- Bocharov, G.; Meyerhans, A.; Bessonov, N.; Trofimchuk, S.; Volpert, V. Spatiotemporal dynamics of virus infection spreading in tissues. PLoS ONE
**2006**. [Google Scholar] [CrossRef] - Bocharov, G.; Meyerhans, A.; Bessonov, N.; Trofimchuk, S.; Volpert, V. Modelling the dynamics of virus infection and immune response in space and time. Int. J. Parallel Emerg. Distrib. Syst.
**2019**, 34, 341–355. [Google Scholar] [CrossRef][Green Version] - Perthame, B.; Genieys, S. Concentration in the nonlocal Fisher equation: The Hamilton-Jacobi limit. Math. Model. Nat. Phenom.
**2007**, 4, 135–151. [Google Scholar] [CrossRef][Green Version] - Banerjee, M.; Volpert, V. Spatio-temporal pattern formation in Rosenzweig–Macarthur model: Effect of nonlocal interactions. Ecol. Complex.
**2017**, 30, 2–10. [Google Scholar] [CrossRef] - Lorz, A.; Lorenzi, T.; Clairambault, J.; Escargueil, A.; Perthame, B. Modeling the effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors. Bull. Math. Biol.
**2015**, 77, 1–22. [Google Scholar] [CrossRef][Green Version] - Gourley, S.A.; Chaplain, M.A.J.; Davidson, F.A. Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation. Dyn. Syst.
**2001**, 16, 173–192. [Google Scholar] [CrossRef] - Alfaro, M.; Ducrot, A.; Giletti, T. Travelling waves for a non-monotone bistable equation with delay: Existence and oscillations. Proc. Lond. Math. Soc.
**2018**, 116, 729–759. [Google Scholar] [CrossRef][Green Version] - Alfaro, M.; Coville, J.; Raoul, G. Bistable travelling waves for nonlocal reaction diffusion equations. Discret. Contin. Dyn. Syst.
**2014**, 34, 1775–1791. [Google Scholar] [CrossRef] - Apreutesei, N.; Bessonov, N.; Volpert, V.; Vougalter, V. Spatial structures and generalized travelling waves for an integro-differential equation. Discret. Contin. Dyn. Syst. Ser. B
**2010**, 13, 537–557. [Google Scholar] [CrossRef] - Nadin, G.; Rossi, L.; Ryzhik, L.; Perthame, B. Wave-like solutions for nonlocal reaction-diffusion equations: A toy model. Math. Model. Nat. Phenom.
**2013**, 8, 33–41. [Google Scholar] [CrossRef][Green Version] - Bessonov, N.; Bocharov, G.; Touaoula, T.M.; Trofimchuk, S.; Volpert, V. Delay reaction-diffusion equation for infection dynamics. Discret. Contin. Dyn. Syst. B
**2019**, 24, 2073–2091. [Google Scholar] [CrossRef][Green Version] - Volpert, V. Pulses and waves for a bistable nonlocal reaction-diffusion equation. Appl. Math. Lett.
**2015**, 44, 21–25. [Google Scholar] [CrossRef] - Britton, N.F. Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model. SIAM J. Appl. Math.
**1990**, 6, 1663–1688. [Google Scholar] [CrossRef] - Volpert, V. Asymptotic behavior of solutions of a nonlinear diffusion equation with a source term of general form. Sib. Math. J.
**1989**, 30, 25–36. [Google Scholar] [CrossRef] - Trofimchuk, S.; Volpert, V. Traveling waves for a bistable reaction-diffusion equation with delay. SIAM J. Math. Anal.
**2018**, 50, 1175–1199. [Google Scholar] [CrossRef][Green Version] - Trofimchuk, S.; Volpert, V. Global continuation of monotone waves for a unimodal bistable reaction-diffusion equation with delay. Nonlinearity
**2019**, 32, 2593. [Google Scholar] [CrossRef][Green Version] - Coyne, J.A.; Orr, H.A. Speciation; Sinauer Associates: Sunderland, MA, USA, 2004. [Google Scholar]
- Volpert, V. Branching and aggregation in self-reproducing systems. ESAIM Proc. Surv.
**2014**, 47, 116–129. [Google Scholar] [CrossRef]

**Figure 1.**(

**Left**) Graphical solution of Equation (9): functions $(sin(\xi N\left)\right)/\left(\xi N\right)$ and $-{f}^{\prime}\left({u}_{0}\right)-D{\xi}^{2}/{u}_{0}$, where $N=0.1$, ${f}^{\prime}\left({u}_{0}\right)=-0.34$, $D=0.00033,{u}_{0}=0.93$. (

**Right**) Graphical solution of Equation (12): the function $\varphi \left(\xi \right)+b\tilde{\psi}\left(\xi \right)$ for the values of parameters $b=1$, ${N}_{1}=3,{N}_{2}=5$ (solid line), ${N}_{1}=3,{N}_{2}=3$ (dashed line), and the function $-D/{u}_{-}{\xi}^{2}$ with $D/{u}_{-}=0.1508$ (point line).

**Figure 2.**Numerical simulations of Equation (14) for the linear function $f\left(u\right)={k}_{1}u$. Spatial and temporal perturbation decay if the solution ${u}_{-}$ is stable (

**left**). The spatial perturbation at the center of the interval leads to the emergence of a spatiotemporal pattern propagating from the center and gradually filling the whole spatial domain (

**right**). The value of parameters: $r=1,q=1,{k}_{1}=1.5,D={10}^{-5},\tau =3$, ${N}_{1}=0.01$ (

**left**) and ${N}_{1}=0.1$ (

**right**), $t=50$. Here and in all figures below, $L=1$, unless another value is indicated.

**Figure 3.**Snapshots of different regimes of wave propagation in numerical simulations of Equation (19) in the monostable–bistable case. The speed of the monostable wave is greater than the speed of the bistable wave, and the distance between them grows (

**upper row, left**). The intermediate equilibrium between the wave becomes unstable, and the monostable wave is space periodic (

**upper row, middle**). This periodic wave can be followed by complex spatiotemporal oscillations (

**upper row, right**). The lower row shows the position of local maxima of the same solutions on the $(x,t)$-plane. Reprinted from [37] with permission.

**Figure 4.**Numerical simulations of Equation (17) show the waves propagating from the center of the interval towards its boundaries in the monostable case. In the first monostable case (

**left**) the periodic perturbation propagates slower than the $[{u}_{+},{u}_{-}]$-wave, and the distance between them grows. In the second monostable case (

**right**), the periodic perturbation propagates faster, it merges with the wave, and they form a single periodic wave. The values of parameters: $D={10}^{-5},r=1,q=1$, ${k}_{1}=5,{k}_{3}=3$, $N=0.035$ (

**left**), $N=0.1$ (

**right**); $f\left(u\right)={k}_{1}u{e}^{-{k}_{3}u}$, $\tau =0$.

**Figure 5.**Numerical simulations of Equation (17) with $f\left(u\right)={k}_{1}u$. If the solution ${u}_{-}$ is stable, then there is a $[{w}_{+},{w}_{-}]$-wave propagating with a constant speed and profile with possible spatial oscillations independent of time (

**left**). If this solution is unstable, then this wave is followed by spatiotemporal oscillations (

**right**). The values of parameters: $r=1,q=1$, $L=2,D={10}^{-5},{k}_{1}=0.9,\tau =3,N=0.25$, $\tau =2$ (

**left**), $\tau =4$ (

**right**), $t=150$.

**Figure 6.**Emergence of a periodic wave in numerical simulations of Equation (17). (

**a**) At the first stage, solution growth remaining localized at the center of the interval. (

**b**) Then it decreases and widens, and after some time, other peaks of solution appear. (

**c**) Another representation of the same solution as in (

**b**). Values of parameters: $D={10}^{-5},r=1,q=1,N=0.2,\tau =0$, $f\left(u\right)=0$, the maximum of the initial condition $0.9$, $t=75$.

**Figure 7.**Numerical simulations of Equation (17) with two different initial conditions and the same values of parameters: $D={10}^{-5},r=1,q=1,N=0.2,\tau =0$, $f\left(u\right)={k}_{1}{e}^{-{k}_{3}u}$, ${k}_{1}=1,{k}_{3}=0.6$, the maximum of the initial condition equals $0.1$ (

**left**) and $0.9$ (

**right**), ${x}_{1}=0.48,{x}_{2}=0.52$, $t=75$.

**Figure 8.**Numerical simulations of Equation (17). Virus evolution with time delay in the term describing the immune response represented as level lines of the solution $u(x,t)$ on the $(x,t)$-plane. Different regimes coexist for the same values of parameters depending on the initial conditions, with high initial viral load (

**left**) and low initial viral load (

**middle**). Values of parameters: $D={10}^{-4},r=1,q=1$, $N\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.1$, $f\left(u\right)={k}_{1}{e}^{-{k}_{3}u}$, ${k}_{1}=8$, ${k}_{3}=3$, $t=80$ (

**left**and

**middle**), ${k}_{3}=6$, $t=50$ (

**right**); the maximum of the initial condition $0.9$ (

**left**), $0.1$ (

**middle**and

**right**).

**Figure 9.**Numerical simulations of Equation (17). Virus evolution without immune response and with the genotype-dependent mortality $\sigma \left(x\right)$ represented as level lines of the solution $u(x,t)$ on the $(x,t)$-plane. Values of parameters: $D={10}^{-5},r=1,q=1,N=0.09$ (

**left**), $N=0.08$ and $0.09$ (

**middle**), $N=0.2$ (

**right**).

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bessonov, N.; Bocharov, G.; Meyerhans, A.; Popov, V.; Volpert, V.
Nonlocal Reaction–Diffusion Model of Viral Evolution: Emergence of Virus Strains. *Mathematics* **2020**, *8*, 117.
https://doi.org/10.3390/math8010117

**AMA Style**

Bessonov N, Bocharov G, Meyerhans A, Popov V, Volpert V.
Nonlocal Reaction–Diffusion Model of Viral Evolution: Emergence of Virus Strains. *Mathematics*. 2020; 8(1):117.
https://doi.org/10.3390/math8010117

**Chicago/Turabian Style**

Bessonov, Nikolai, Gennady Bocharov, Andreas Meyerhans, Vladimir Popov, and Vitaly Volpert.
2020. "Nonlocal Reaction–Diffusion Model of Viral Evolution: Emergence of Virus Strains" *Mathematics* 8, no. 1: 117.
https://doi.org/10.3390/math8010117