# Space and Genotype-Dependent Virus Distribution during Infection Progression

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## Abstract

**:**

## 1. Introduction

## 2. Wave Existence and Stability

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

## 3. 1D Problem Depending on the Eigenvalue

#### 3.1. Properties of the Principal Eigenvalue

#### 3.2. Dynamics of 1D Waves

#### 3.3. Dependence on $\lambda $

#### 3.4. Dependence on $f\left(v\right)$

## 4. 2D Wave Dynamics

#### 4.1. Influence of the Genotype Distribution

#### 4.2. Comparison with 1D Problem

#### 4.3. Parameter Dependence

## 5. The Properties of Infection Progression

- Genotype distribution. Virus genotype distribution depends on parameter ${y}_{0}$ characterizing the interval where the virus reproduction rate exceeds its natural mortality rate (without immune response), and on the mutation rate that determines the diffusion coefficient ${D}_{y}$. These two parameters determine the principal eigenvalue ${\lambda}_{0}$ of problem (11). Let us note that proportional decrease of ${D}_{y}$ and ${y}_{0}$ does not change ${\lambda}_{0}$, so that the results presented above are appropriate in a wide range of mutation rates.
- Initial viral load. The initial viral load corresponds to the initial condition for Equations (1) or (10). In 2D simulations, we set ${u}_{0}(x,y)={u}_{0}^{*}$ for $0\le x\le {l}_{x}$ and $\left|y\right|\le {l}_{y}$ with some positive numbers ${l}_{x}$ and ${l}_{y}$, and ${u}_{0}(x,y)=0$ otherwise. In this context, the initial viral load ${V}_{0}$ is the integral of the initial condition, ${V}_{0}=2{l}_{x}{l}_{y}{u}_{0}^{*}$.
- Strength of immune response. Adaptive immune response proceeds by clonal expansion of lymphocytes due to the interaction with antigens-presenting cells (macrophages, dendritic cells). For small viral loads, increase of the level of pathogens in the organism intensifies the immune response. Some viral infections can affect the functioning of lymphocytes by downregulating their proliferation and increasing their death (e.g., HIV, LCMV [41] but not coronavirus). Therefore, the function $f\left(u\right)$ increases for small u and decreases for large u. Stronger immune response corresponds to a larger function $f\left(u\right)$. In modelling, we characterize the strength of immune response by the value of parameter ${a}_{1}$.
- Immunity. Vaccination and previous infections can lead to the appearance of antibodies and memory cells responding to a new antigen. This response can be attenuated by the reduced affinity to the antigen of the immunity mediators (antibodies, T cell receptors). Immunity slows down infection progression and accelerates clonal expansion of immune cells. We model the presence of immunity by the coefficient ${a}_{2}$. If it is positive, then immune response starts from some positive value under the introduction of antigen, $f\left(0\right)>0$. Infection-free equilibrium ${v}_{0}=0$ becomes stable for sufficiently narrow genotype distribution ${y}_{0}$, and infection is eliminated unless the initial viral load is sufficiently large to cause its persistent progression.
- Viral load. The level of infection in the tissue determines the severity of symptoms and the intensity of infection transmission to other individuals. Viral load is determined by all factors presented in the previous paragraphs. In modelling, it corresponds to the virus density distribution after the wave propagation.
- Virus virulence. Virus virulence is characterized by its spreading rate. In modelling, it is determined by the coefficient k in Equation (1). In the absence of immune response, it is related to the minimal speed of the monostable wave ${c}_{0}=2\sqrt{{D}_{x}k}$. Multiplicity of infection tests in cell culture [42] characterize the virulence of infection by the speed of viral plaque growth, that is, by the wave speed. We will also characterize the virulence of infection by the wave speed in the presence of immune response.

## 6. Discussion

#### 6.1. Model

#### 6.2. Method and Results

#### 6.3. Limitations and Perspectives

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Analytical Solution for Nonlocal Equation

**Theorem**

**A1.**

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**Figure 1.**A snapshot of solution $u(x,y,t)$ of Equation (1) without immune response ($f\left(u\right)=0$) and with a piece-wise constant genotype-dependent mortality $\sigma \left(y\right)$. The cross section in the y-direction (genetic variable) shows a stationary distribution describing virus quasi-species. It has a maximum for some most frequent genotype, and it rapidly decays as the genotype moves away. This virus quasi-species progresses in the tissue (variable x).

**Figure 2.**Function ${F}_{\lambda}\left(v\right)=v(1-\lambda -v-f\left(v\right))$ for $\lambda =0$ (upper curve), $\lambda =0.1$ (middle curve), and $\lambda =0.2$ (lower curve), $f\left(v\right)=({a}_{1}v+{a}_{2}){e}^{-{a}_{3}v}$, ${a}_{1}=13,{a}_{2}=0,{a}_{3}=6$.

**Figure 3.**Wave speed for the 1D Equation (10) (dashed line) and for the 2D Equation (1) (solid lines). Curves 1 and 3 correspond to the monostable and bistable waves, respectively, for the same values of parameters, and curve 2 shows the transition from the bistable wave to the monostable wave. The values of parameters are ${D}_{x}={D}_{y}=0.005,k=1,{\sigma}_{0}=1.1$, $f\left(v\right)={a}_{1}v{e}^{-{a}_{3}v}$, ${a}_{3}=6$, ${a}_{1}=15$ (curves 1, 3, and dashed curve), and ${a}_{1}=13$ (curve 2).

**Figure 4.**Snapshots of solutions of Equation (1) with two different initial conditions and the same values of parameters. Two waves with different speeds propagate one after another for a sufficiently large initial condition (

**left**). A faster monostable wave with a small amplitude is followed by a slower bistable wave with a large amplitude. If the initial condition is small enough, then only the fast monostable wave with a small amplitude is observed (

**right**). The values of parameters: ${D}_{x}={D}_{y}=0.005,b=1,k=1,{y}_{0}=0.5$, $f\left(u\right)=({a}_{1}u+{a}_{2}){e}^{-{a}_{3}u}$, ${a}_{1}=15,{a}_{2}=0,{a}_{3}=6$.

**Figure 5.**The critical value of immunity (coefficient ${a}_{2}$) is shown as a function of the width of the genotype distribution (${y}_{0}$). If the value of immunity exceeds the critical value, then infection is eliminated. Otherwise, it progresses. Solid line represents the results of 2D simulations, and dashed line is given by the formula ${a}_{2}=1-\lambda \left({y}_{0}\right)$ for the 1D problem. The values of parameters: ${D}_{x}={D}_{y}=0.005,b=1,k=1$, $f\left(u\right)=({a}_{1}u+{a}_{2}){e}^{-{a}_{3}u}$, ${a}_{1}=15,$ and ${a}_{3}=6$.

**Figure 6.**Dependence of virulence (wave speed) on immunity (${a}_{2}$). The minimal wave speed is found by the formula ${c}_{0}=2\sqrt{{D}_{x}k(1-\lambda \left({y}_{0}\right)-{a}_{2})}$ for the 1D problem. The upper curve corresponds to $\lambda =0$ (limit of large ${y}_{0}$), the middle curve to $\lambda =0.17$ (${y}_{0}=0.1$), and the lower curve to $\lambda =0.41$ (${y}_{0}=0.2$). The values of parameters: ${D}_{x}={D}_{y}=0.005,b=1,k=1$, $f\left(v\right)=({a}_{1}v+{a}_{2}){e}^{-{a}_{3}v}$, ${a}_{1}=15,$ and ${a}_{3}=6$.

**Figure 7.**Dependence of the viral load (wave amplitude) on the strength of immune response (${a}_{1}$) for the 1D problem. The wave amplitude is found as a solution of the equation ${F}_{\lambda}\left(v\right)=0$. The values of parameters: ${D}_{x}={D}_{y}=0.005,b=1,k=1$, $f\left(v\right)=({a}_{1}v+{a}_{2}){e}^{-{a}_{3}v}$, Curve 1: ${a}_{2}=0.7,{a}_{3}=6$, $\lambda =0.41$ (${y}_{0}=0.1$). Curve 2: ${a}_{2}=0,{a}_{3}=6$, $\lambda =0.17$ (${y}_{0}=0.2$).

**Table 1.**Four critical values ${\lambda}_{1},{\lambda}_{2},{\lambda}_{3},{\lambda}_{4}$, determine the intervals of $\lambda $ with different stationary points, waves, and their speeds.

No. | $\mathit{\lambda}$ | Stationary Points | Waves | Speed |
---|---|---|---|---|

1 | ${\lambda}_{1}<\lambda <{\sigma}_{0}$ | ${v}_{0}=0$ | - | - |

2 | ${\lambda}_{2}<\lambda <{\lambda}_{1}$ | ${v}_{0}$, ${v}_{1}$ | monostable $[{v}_{0},{v}_{1}]$-waves | $c\ge {c}_{0}=2\sqrt{{D}_{x}(1-\lambda )}$ |

3 | ${\lambda}_{3}<\lambda <{\lambda}_{2}$ | ${v}_{0}$, ${v}_{1}$, ${v}_{2}$, ${v}_{3}$ | monostable, bistable | $c\ge {c}_{0}$, ${c}_{1}<0$ |

4 | ${\lambda}_{4}<\lambda <{\lambda}_{3}$ | ${v}_{0}$, ${v}_{1}$, ${v}_{2}$, ${v}_{3}$ | monostable, bistable | $c\ge {c}_{0}$, $0<{c}_{1}<{c}_{0}$ |

5 | $0<\lambda <{\lambda}_{4}$ | ${v}_{0}$, ${v}_{3}$ | monostable $[{v}_{0},{v}_{3}]$-waves | $c\ge {c}_{0}$ |

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**MDPI and ACS Style**

Bessonov, N.; Bocharov, G.; Volpert, V.
Space and Genotype-Dependent Virus Distribution during Infection Progression. *Mathematics* **2022**, *10*, 96.
https://doi.org/10.3390/math10010096

**AMA Style**

Bessonov N, Bocharov G, Volpert V.
Space and Genotype-Dependent Virus Distribution during Infection Progression. *Mathematics*. 2022; 10(1):96.
https://doi.org/10.3390/math10010096

**Chicago/Turabian Style**

Bessonov, Nicholas, Gennady Bocharov, and Vitaly Volpert.
2022. "Space and Genotype-Dependent Virus Distribution during Infection Progression" *Mathematics* 10, no. 1: 96.
https://doi.org/10.3390/math10010096