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

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

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**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**).

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**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