# A Multi-Scale Model for the Spread of HIV in a Population Considering the Immune Status of People

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Population Model: Sexual Contact Network

- 1.
- Infection dynamics by the virus in the immunological scale.
- 2.
- Defines the virus propagation in the population scale, considering aspects from the immunological scale.

#### 2.2. Model for HIV Infection in the Immunological Scale

## 3. Results

#### 3.1. Local Stability Analysis of the Immunological Model

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

#### 3.2. Simulation of the Immunological Model

#### 3.3. Coupling Algorithm of the Immunological and Population Scales

- 1.
- The network was constructed by beginning with the random values [38,39,40] shown in the following, which seeks for each individual i, for $i=1,\cdots ,{N}_{T}$, from the network to have a different immune system. Additionally, for the values of the other parameters used in the system of differential equations that describe the immunological scale, those described in Table 2 were used.
- (a)
- Initial number of healthy CD4 T cells in the i-th person per mm${}^{3}$: ${T}_{i}\left(0\right)\in [200,1000]$.
- (b)
- Initial number of inactive CD8 T cells in the i-th person per mm${}^{3}$: ${M}_{i}\left(0\right)\in [70,980]$.
- (c)
- Values of infected CD4 T cells, active CD8 T cells, infectious and non-infectious viral particles will be null, given that it is considered that the $i-$th person is still not infected by the virus, that is, ${T}_{i}^{*}\left(0\right)={M}_{i}^{*}\left(0\right)={V}_{i}\left(0\right)={W}_{i}\left(0\right)=0$.
- (d)
- Production rate of healthy cells of the i-th person: ${\sigma}_{i}\in [5,20]$.
- (e)
- Infection probability of the healthy cells of the i-th person: ${\beta}_{i}\in [0,2.5\times {10}^{-5}]$.
- (f)
- Production rate of viral particles of the i-th person: ${N}_{i}\in [100,1500]$.

Then, a vertex from the network is chosen randomly as the first infected and is assigned a random viral load between 1 and 1000 vir/mm${}^{3}$[41]. With the aforementioned, the network is created, obtaining the initial state at $\tau =0$. - 2.
- From the corresponding initial condition, we solve the system of differential equations of the immunological system (7) for each vertex i during one unit of time, saving the numerical solutions ${T}_{i}\left(t\right)$, ${T}_{i}^{*}\left(t\right)$, ${M}_{i}\left(t\right)$, ${M}_{i}^{*}\left(t\right)$, ${V}_{i}\left(t\right)$ and ${W}_{i}\left(t\right)$; therefore, we have the immunological status of each individual in the population during that unit of time. Additionally, the final value of each of the numerical solutions will be used as the initial condition in the following iteration. Furthermore, we saved the number of susceptible subjects, carriers with ART and carriers without ART at the end of the iteration.
- 3.
- Based on the numerical solutions, the following decisions were made:
- (a)
- If the i-th person is an HIV carrier and presents a number of healthy CD4 T cells ${T}_{i}$ below 350 cell/mm${}^{3}$, a random probability of accessing to ART is assigned, if it is >$0.6$, it is established to use ART, so that a random value is assigned for ${u}_{1}$ and ${u}_{2}$ between $(0,1)$.
- (b)
- If the i-th person is an HIV carrier already using ART, a random probability of therapy abandonment is assigned, if it is <$0.2$, it is established that the individual leaves ART, that is ${u}_{1}={u}_{2}=0$.

This is why in the simulations that will be presented ahead in different iterations, the state of the vertices will vary from carriers with therapy to carriers without therapy, and vice versa. - 4.
- The transmission process considers different factors. If the i-th person is an HIV carrier, the decision is made randomly whether to have a sexual encounter with only one of their susceptible “neighbors” (another susceptible vertex sharing an edge). If the random decision was affirmative, the infected person is partnered with one of their susceptible sex partners, then, the probability of transmitting the infection from the infected person i to the susceptible person j is given by ${\lambda}_{ij}=\frac{ij{\lambda}_{i}}{\langle k\rangle}$ (taken from [16]), where $\langle k\rangle $ represents the average degree (associated with the potential number of sex partners) and ${\lambda}_{i}$ is the infection probability through sexual contact; the probability ${\lambda}_{i}$ is taken from [40] and is shown in Table 3 according to the values of vir/mm${}^{3}$.Finally, if person j becomes infected, an initial viral load is assigned with 10% of the load owned by the person i who caused the infection.
- 5.
- At the end of each iteration, the prevalence was calculated, as the rate between the number of HIV carriers with or without therapy and the total number of people from the network, that is,$${P}_{f}=\frac{\sum {y}_{i}+\sum {z}_{i}}{\sum {x}_{i}+\sum {y}_{i}+\sum {z}_{i}}.$$

#### 3.4. Simulation of the Coupled Model

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

HIV | Human Immunodeficiency Virus |

AIDS | Acquired immune deficiency syndrome |

ART | Antiretroviral therapy |

RTI | Reverse transcriptase inhibitors |

PI | Protease Inhibitors |

LAE | locally asymptotically stable |

Sym | Symbol |

## Appendix A. Demonstration of Propositions and Basic Reproduction Number

**Proof**

**of**

**Proposition 1.**

**Deduction of the basic reproduction number.**To find the ${R}_{0}$ of model (7), we define vectors h and g, corresponding to the equations of the populations responsible for the infection:

**Proof**

**of**

**Proposition 2.**

**Proof**

**of**

**Proposition 3.**

- 1.
- There are two positive solutions, that is, two endemic equilibriums if ${\overline{T}}_{1}^{*}$ and ${\overline{T}}_{2}^{*}$ satisfy the conditions to obtain ${\overline{M}}^{*}>0$ (see (8)); that is:$$0<\overline{{T}_{1}^{*}}<\overline{{T}_{2}^{*}}<\frac{\rho}{\alpha}.$$
- 2.
- There are no positive solutions, that is, there is no endemic equilibrium if $\overline{{T}_{1}^{*}}$ and $\overline{{T}_{2}^{*}}$ fulfill:$$\frac{\rho}{\alpha}<\overline{{T}_{1}^{*}}<\overline{{T}_{2}^{*}}.$$
- 3.
- There is a single endemic equilibrium point if $\overline{{T}_{1}^{*}}$ and $\overline{{T}_{2}^{*}}$ fulfill:$$0<\overline{{T}_{1}^{*}}<\frac{\rho}{\alpha}<\overline{{T}_{2}^{*}}.$$

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**Figure 1.**Graphic of P in space $(k,\gamma ,P)$, for $k\in [0,2]$ and $\gamma \in [2,4]$. The orange curve represents the fixed value $\gamma =3.4$ used in the simulations in this study.

**Figure 2.**Numerical solutions of system (7), considering the parameters in Table 2, simulating 300 days, varying the CD4 T cell infection probability $\beta $, taking as fixed the value of the probability of the active CD8 T cells eliminating the infected CD4 T cells, $\gamma =2\times {10}^{-3}$ and considering that ART is not used, that is, the RTI and PI are in null values, ${u}_{i}=0$ with $i=1,2$. When $\beta =1.5\times {10}^{-5}$ is the red curve with ${R}_{0}=1.2168$, $\beta =2.5\times {10}^{-5}$ is the black curve with ${R}_{0}=2.0280$, $\beta =4.5\times {10}^{-5}$ is the magenta curve with ${R}_{0}=3.6504$, and $\beta =5.5\times {10}^{-5}$ is the green curve with ${R}_{0}=4.4616$.

**Figure 3.**Simulations performed for the parameters according to the values from Table 2, varying the probability of active CD8 T cells eliminating the infected CD4 T cells $\gamma $, where $\gamma =9\times {10}^{-2}$ (red), $\gamma =6\times {10}^{-2}$ (black), $\gamma =5\times {10}^{-3}$ (magenta) and $\gamma =2\times {10}^{-3}$ (green). Fixed values were taken for the CD4 T cell infection probability, $\beta =2.5\times {10}^{-5}$, and ART use was not considered, thus, ${u}_{i}=0$, for $i=1,2$. ${R}_{0}=2.0280$ in all the scenarios illustrated.

**Figure 4.**Dynamics of the model (7), fixed values are used for the CD4 T cell infection probability, $\beta =2.5\times {10}^{-5}$, the probability of the active CD8 T cells eliminating the infected CD4 T cells, $\gamma =2\times {10}^{-3}$, and no use of PI is considered, ${u}_{2}=0$. The RTI are varied, thus ${u}_{1}=(0.2,0.4,0.6,0.8)$, corresponding to the colored curves (red, black, magenta, and green) with their respective values of ${R}_{0}=(1.6224,1.3168,1.0112,0.7056)$. Simulations were performed in the interval of 300 days.

**Figure 5.**System simulation (7), taking the CD4 T cell infection probability, $\beta =2.5\times {10}^{-5}$, the probability of active CD8 T cells eliminating the infected CD4 T cells, $\gamma =2\times {10}^{-3}$ and use of RTI is not considered, ${u}_{1}=0$. The PI are varied, ${u}_{2}=(0.2,0.4,0.6,0.8)$ that correspond to the colored curves (red, black, magenta, and green) with their respective values of ${R}_{0}=(1.6224,1.3168,1.0112,0.7056)$ in a simulation time of $t=300$ days.

**Figure 6.**Numerical solutions of system (7) considering the parameters from Table 2, $\gamma =2\times {10}^{-3}$, $\beta =2.5\times {10}^{-5}$, and varying the values for ART effectiveness, under the assumption of being ${u}_{1}={u}_{2}$. The red curve represents ${u}_{i}=0.2$, where ${R}_{0}=1.2979$, the black curve corresponds to ${u}_{i}=0.4$, with ${R}_{0}=0.7301$, the magenta curve to ${u}_{i}=0.6$ and ${R}_{0}=0.3245$, and the green curve is ${u}_{i}=0.8$ with ${R}_{0}=0.0811$, for $i=1,2$.

**Figure 7.**Schematic representation of the network showing the evolution with respect to time for ${N}_{T}=25$, $\tau =20$, $\gamma =3.4$, and $k=2$. The blue vertices represent people susceptible of acquiring the infection; those in red, carriers without ART; those in green, the carriers who use ART. The scenario displayed does not seek to represent reality, it only aims to illustrate the operation of the coupled model.

**Figure 8.**Immunological state of individuals 1, 8, 19, and 24 of the complex network simulated with ${N}_{T}=25$, $\tau =20$, $\gamma =3.4$, and $k=2$. (Black: person susceptible throughout the simulation time. Red: HIV carrier who in the final time did not take ART. Green: HIV carrier who in the final time takes therapy).

**Figure 9.**

**Uper**: Simulation of the complex network for ${N}_{T}=1000$, $\tau =200$, $\gamma =3.4$, and $k=4$ (blue: susceptible people, red: carriers without ART, green: carriers with ART).

**Lower left**: Graphic showing the relation between the HIV carrier population with ART (green) and the carrier population without ART (red).

**Lower right**: Graphic of virus prevalence in the population. The scenario shown does not seek to represent reality, it only aims to illustrate the operation of the coupled model.

**Figure 10.**

**Upper**: Simulation of the complex network for ${N}_{T}=1000$, $\tau =200$, $\gamma =3.4$, and $k=10$ (Blue: susceptible individuals. Red: carriers without ART. Green: carriers with ART).

**Lower left**: Graphic showing the relation between people infected with and without ART.

**Lower right**: Graphic of virus prevalence over time $\tau $. The scenario shown does not seek to represent reality, it only aims to illustrate the operation of the coupled model.

Sym. | Description | Value |
---|---|---|

k | Average number of sex partners | 2, 4 and 10 |

$\gamma $ | Distribution of sex partners | 3.4 |

${N}_{T}$ | Number of susceptible young people, carriers with ART, and carriers without ART | 25 and 1000 |

$\tau $ | Discrete simulation time in weeks | 200 |

${t}_{f}$ | Number of iterations on the network | Varies |

${x}_{i}$ | Number of young people susceptible of acquiring the virus | Varies |

${y}_{i}$ | Number of young people carriers of the virus with ART | Varies |

${z}_{i}$ | Number of young people carriers of the virus without ART | Varies |

**Table 2.**Description of parameters for model (7) and baseline values used in the simulation, with their respective source.

Sym. | Description | Value | Ref. |
---|---|---|---|

$\sigma $ | Production rate of healthy CD4 T cells | 10 mm${}^{3}$ d${}^{-1}$ | [32,33] |

$\beta $ | CD4 T cells infection probability | 2.5 × 10${}^{-5}$ mm${}^{3}$ d${}^{-1}$ | [33] |

$\mu $ | Death rate of the uninfected CD4 T cells | 1 × 10${}^{-2}$ d${}^{-1}$ | [32] |

$\delta $ | Death rate of infected CD4 T cells | 0.26 d${}^{-1}$ | [32,33] |

N | Production rate of viral particles | 500 d${}^{-1}$ | [32] |

c | Rate of virus elimination | 2.4 d${}^{-1}$ | [33] |

$\psi $ | Activation probability of CD8 T cells | 2 × 10${}^{-3}$ mm${}^{3}$ d${}^{-1}$ | - |

$\alpha $ | Replication probability of the active CD8 T cells | 5 × 10${}^{-5}$ mm${}^{3}$ d${}^{-1}$ | [33] |

$\rho $ | Death rate of CD8 T cells | 0.1 mm${}^{3}$ d${}^{-1}$ | [32,33] |

$\gamma $ | Probability of the active CD8 T cells eliminating the infected CD4 T cells | 2 × 10${}^{-3}$ mm${}^{3}$ d${}^{-1}$ | [33] |

$\lambda $ | Production rate of inactive CD8 T cells | 5 mm${}^{3}$ d${}^{-1}$ | [33] |

${u}_{1}$ | Effectiveness of RTI | - | - |

${u}_{2}$ | Effectiveness of PI | - | - |

${T}_{0}$ | Initial value of the healthy CD4 T cells | 1000 mm${}^{3}$ | [33] |

${T}_{0}^{*}$ | Initial value of infected CD4 T cells | 0 mm${}^{3}$ | [33] |

${M}_{0}$ | Initial value of inactive CD8 T cells | 0 mm${}^{3}$ | [33] |

${M}_{0}^{*}$ | Initial value of active CD8 T cells | 1 mm${}^{3}$ | [33] |

${V}_{0}$ | Initial infectious viral load | 1 × 10${}^{3}$ mm${}^{3}$ | [33] |

${W}_{0}$ | Initial non-infectious viral load | - | - |

**Table 3.**Values for the infection probability ${\lambda}_{i}$ according to the number of vir/mm${}^{3}$ found in the immune system [40].

Type of Risk | Viral Load Range | Infection Probability |
---|---|---|

Low risk | ${V}_{i}<200$ | ${\lambda}_{i}\in (0.0006,0.0011)$ |

Medium risk | $200\le {V}_{i}<2000$ | ${\lambda}_{i}\in (0.0007,0.0168)$ |

High risk | ${V}_{i}\ge 2000$ | ${\lambda}_{i}\in (0.002,0.025)$ |

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**MDPI and ACS Style**

Vásquez-Quintero, S.d.A.; Toro-Zapata, H.D.; Prieto-Medellín, D.A. A Multi-Scale Model for the Spread of HIV in a Population Considering the Immune Status of People. *Processes* **2021**, *9*, 1924.
https://doi.org/10.3390/pr9111924

**AMA Style**

Vásquez-Quintero SdA, Toro-Zapata HD, Prieto-Medellín DA. A Multi-Scale Model for the Spread of HIV in a Population Considering the Immune Status of People. *Processes*. 2021; 9(11):1924.
https://doi.org/10.3390/pr9111924

**Chicago/Turabian Style**

Vásquez-Quintero, Sol de Amor, Hernán Darío Toro-Zapata, and Dennis Alexánder Prieto-Medellín. 2021. "A Multi-Scale Model for the Spread of HIV in a Population Considering the Immune Status of People" *Processes* 9, no. 11: 1924.
https://doi.org/10.3390/pr9111924