# Within-Host Phenotypic Evolution and the Population-Level Control of Chronic Viral Infections by Treatment and Prophylaxis

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## Abstract

**:**

## 1. Introduction

## 2. Description of the Models and Their Structural Properties

#### 2.1. A Baseline Model of a Chronic Multi-Strain Virus Infection

#### 2.2. A Generalized Model with Differentially Effective Control, Variable Transmissibility and Prophylaxis

- The efficacy of the treatment program depends on the viral strain. That is, the treatment program fails with certain probability, which varies depending on the virus strain, causing the treated individuals to thus revert back to active chronic infection.
- Virus strains have different levels of contagiousness.
- The efficacy of prophylactic measures depends on the viral strain. While on prophylaxis, an individual acquires protection against the virus depending on the specific viral strain.

**Notation.**We let $\mathbf{0}$, $\mathbf{1}$, and $\phantom{\rule{0.166667em}{0ex}}\mathrm{E}$ denote the matrices of zeros, ones, and the identity matrix (the use of notation $\phantom{\rule{0.166667em}{0ex}}\mathrm{E}$ instead of $\phantom{\rule{0.166667em}{0ex}}\mathrm{I}$ for the identity matrix is common in German and Russian mathematical texts (Germ., Einheitsmatrix); here we use it to avoid confusing notation $\phantom{\rule{0.166667em}{0ex}}\mathrm{I}$ with the letter I used for infected compartments). The sizes of the respective matrices are indicated as subscripts. A single subscript, for example, as in $\phantom{\rule{0.166667em}{0ex}}{\mathrm{E}}_{n}$, denotes a square $[n\times n]$ matrix of respective type. Furthermore, $\phantom{\rule{0.166667em}{0ex}}{\mathrm{I}}_{A}$ and $\phantom{\rule{0.166667em}{0ex}}{\mathrm{I}}_{C}$ denote the column vectors of respective variables and $\phantom{\rule{0.166667em}{0ex}}\mathrm{A}$ denotes the matrix of $\alpha $’s:

#### 2.3. Structural Analysis

**Non-negativity of the solutions.**The Equation (1) can be written as

**Boundedness of solutions.**Observe that the m-simplex ${\Delta}_{m}$, formed as the convex hull of m unit vectors ${\mathbf{e}}_{j}$, $j=1,\phantom{\rule{3.33333pt}{0ex}}\dots ,\phantom{\rule{3.33333pt}{0ex}}m$, is invariant with respect to Equation (1):

## 3. Local Analysis at a Disease-Free Equilibrium

#### 3.1. Basic Reproduction Number for the Baseline Model

**Theorem**

**1.**

**Proof.**

**Sensitivity analysis.**When devising an intervention strategy, the main question to be answered is whether the proposed treatment or prophylaxis scheme is capable of eliminating the infection, i.e., driving the basic reproduction number below 1. To address this issue we introduce the sensitivity parameter(s) ${R}_{1}$ that quantify the efficiency of sufficiently small controls in reducing the value of ${R}_{0}$, [16]. In particular, the controlled basic reproduction number ${R}_{0}^{\alpha}\left({u}_{T}\right)$ is expanded as

**Definition**

**1.**

- 1.
- Locally efficient if the respective sensitivity parameter is negative, i.e., ${R}_{1}<0$;
- 2.
- (Globally) efficient if there exists a non-negative value ${u}^{*}$ such that ${R}_{0}\left({u}^{*}\right)=1$.

**Lemma**

**1.**

**Proof.**

**Remark**

**1.**

#### 3.2. Basic Reproduction Number for the Extended Model

**Theorem**

**2.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Sensitivity analysis.**We begin this paragraph by writing down an expansion of ${R}_{0}^{\beta}({u}_{T},{u}_{P})$.

**Theorem**

**3.**

**Proof.**

**Lemma**

**3.**

## 4. Endemic Equilibrium

**Theorem**

**4.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Theorem**

**5.**

#### Structure of the Matrix $\phantom{\rule{0.166667em}{0ex}}\mathrm{A}$: Uniform within Host Mutations

## 5. Numerical Simulation for Different Scenarios and Illustration of Results

**Case 1.**- ${v}_{j}/{v}_{j+1}=3$, $j=1,2,3$. Assuming that ${\pi}_{4}=0.25$ one can compute the remaining probabilities using Equation (22): ${\pi}_{1}\approx 0.97$, ${\pi}_{2}\approx 0.83$, and ${\pi}_{3}=0.5$. Finally, the endemic frequencies are $[{v}_{1},\phantom{\rule{0.166667em}{0ex}}{v}_{2},\phantom{\rule{0.166667em}{0ex}}{v}_{3},\phantom{\rule{0.166667em}{0ex}}{v}_{4}]=[0.675,\phantom{\rule{0.166667em}{0ex}}0.225,\phantom{\rule{0.166667em}{0ex}}0.075,\phantom{\rule{0.166667em}{0ex}}0.025]$.
**Case 2.**- ${v}_{j}/{v}_{j+1}=7$, $j=1,2,3$. Similarly to the previous case, we fix ${\pi}_{4}=0.25$ and compute the remaining probabilities ${\pi}_{1}=0.9985$, ${\pi}_{2}=0.9796$, and ${\pi}_{3}=0.8571$. The respective endemic frequencies are $[{v}_{1},\phantom{\rule{0.166667em}{0ex}}{v}_{2},\phantom{\rule{0.166667em}{0ex}}{v}_{3},\phantom{\rule{0.166667em}{0ex}}{v}_{4}]=[0.8575,\phantom{\rule{0.166667em}{0ex}}0.1225,\phantom{\rule{0.166667em}{0ex}}0.0175,\phantom{\rule{0.166667em}{0ex}}0.0025]$.

#### 5.1. Controlled Basic Reproduction Number

#### 5.2. Endemic Distribution with Variable Transmissibility

#### 5.3. Endemic Distribution with Variable Prophylaxis Effects

#### 5.4. Endemic Distribution with Imperfect Treatment

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Necessary Ingredients from Matrix Algebra

#### Appendix A.1. Non-Negative Matrices

#### Appendix A.2. Stochastic Matrices

**Theorem**

**A1.**

**Proof.**

## Appendix B. Proofs

**Proof of Theorem 1.**

**Proof of Theorem 2.**

**Proof of Theorem 3.**

**Proof of Theorem 4.**

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**Figure 1.**The panel shows the values of ${R}_{0}({u}_{T},{u}_{P})$ as a function of two controls for two cases described above. The red color corresponds to the case ${R}_{0}\le 1$. We assume a uniform rate of transmission, i.e., ${\beta}_{i}=\beta =0.3$ for all $i=1,\phantom{\rule{3.33333pt}{0ex}}\dots ,\phantom{\rule{3.33333pt}{0ex}}4$ and fully efficient treatment, i.e., ${\zeta}_{i}=0$, $i=1,\phantom{\rule{3.33333pt}{0ex}}\dots ,\phantom{\rule{3.33333pt}{0ex}}4$. Remaining parameters are: $\xi =5$; $\gamma =3$; $\mu =0.025$; and $\delta =0$. The subfigures forming the panel correspond to the following values of prophylaxis efficiency coefficients: left, $\psi =[1,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}1]$; central, $\psi =[1,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}0]$; right, $\psi =[0,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}1]$.

**Figure 2.**The relative endemic distribution of infected individuals for different values of the transmissibility rate of the 4th strain, parametrized with a: ${\beta}_{C,4}=a{\beta}_{C}$. The values at $a=1$ (marked by a red dashed line) correspond to the baseline case, where all transmissibility rates are equal. Subfigures (

**a**) and (

**b**) correspond to different values of mutation probabilities ${\pi}_{i}$.

**Figure 3.**The relative endemic distribution of infected individuals for different values of ${u}_{P}$. Subfigures (

**a**) and (

**b**) correspond to different values of mutation probabilities ${\pi}_{i}$.

**Figure 4.**The relative endemic distribution of infected individuals for different values of ${\zeta}_{4}$. Subfigures (

**a**) and (

**b**) correspond to different values of mutation probabilities ${\pi}_{i}$.

**Table 1.**Model parameters. Parameters indicated with an asterisk are used only in the extended model Equation (4).

State Variable | Range | Description |
---|---|---|

${I}_{Ai}$ | $[0,1]$ | Fraction of acutely infected individuals infected by the virus of type i. |

${I}_{Ci}$ | $[0,1]$ | Fraction of chronically infected individuals infected by the virus of type i. |

S | $[0,1]$ | Fraction of susceptible individuals |

T | $[0,1]$ | Fraction of patients involved in treatment |

${{T}_{i}}^{*}$ | $[0,1]$ | Fraction of patients infected by the virus of type i that are involved in treatment |

${P}^{*}$ | $[0,1]$ | Fraction of patients involved in prophylaxis |

Parameter | Range | Description |

${u}_{T}$ | Rate at which chronically infected are enrolled into treatment (controlled parameter) | |

${{u}_{P}}^{*}$ | Rate at which susceptible individuals are enrolled into prophylaxis (controlled parameter) | |

$\gamma $ | Inverse duration of the acute phase | |

$\mu $ | Mortality rate | |

${\alpha}_{ij}$ | [0, 1] | Fraction of type i viruses in the viral population of an individual initially infected by the type j virus. |

${\beta}_{A}$, ${\beta}_{C}$ | Transmissibility rates of acute and chronically infected individuals. | |

$\xi $ | Proportionality coefficient of the transmissibility in acute and chronic stages | |

${{\zeta}_{i}}^{*}$ | Failure rate of treatment for individuals infected by the virus of type i | |

${\delta}^{*}$ | Failure rate of prophylaxis | |

${{\psi}_{i}}^{*}$ | [0, 1] | The level of protection against the virus strain i, which is conferred by prophylaxis; ${\psi}_{i}=1$ corresponds to full protection |

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**MDPI and ACS Style**

Gromov, D.; Romero-Severson, E.O.
Within-Host Phenotypic Evolution and the Population-Level Control of Chronic Viral Infections by Treatment and Prophylaxis. *Mathematics* **2020**, *8*, 1500.
https://doi.org/10.3390/math8091500

**AMA Style**

Gromov D, Romero-Severson EO.
Within-Host Phenotypic Evolution and the Population-Level Control of Chronic Viral Infections by Treatment and Prophylaxis. *Mathematics*. 2020; 8(9):1500.
https://doi.org/10.3390/math8091500

**Chicago/Turabian Style**

Gromov, Dmitry, and Ethan O. Romero-Severson.
2020. "Within-Host Phenotypic Evolution and the Population-Level Control of Chronic Viral Infections by Treatment and Prophylaxis" *Mathematics* 8, no. 9: 1500.
https://doi.org/10.3390/math8091500