MATERIALES Y HERRAMIENTAS PARA LA COLOCACIÓN DEL

HIV-1 has two predominant infection modes: the classical virus-to-cell infection and cell-to- cell spread. In the classical virus-to-cell infection, viruses released from infected cells ran- domly move around to find a new target cell to infect. Recently, it was revealed that HIV-1

infection may also occur by the transfer of viruses through direct contact between infected cells and uninfected cells via certain structures, for example membrane nanotubes or macromolec- ular adhesive contacts termed virological synapses [37]. During this cell-to-cell transmission, many viral particles can be simultaneously transferred from infected to uninfected CD4+ T cells.

In this chapter, we have considered a mathematical model to describe presence of both of these two transmission modes. By a rigorous analysis, we have shown that the model has a threshold dynamics. Such a threshold dynamics is fully determined by the basic reproduction numberR₀in the sense that the infection-free equilibriumE0is globally asymptotically stable

ifR₀ < 1, and whenR₀ > 1, E0yields to a globally asymptotically stable positive equilibrium

¯

E implying the infection will persist.

Examining the formula for the basic reproduction number R₀, we found that it is larger than that given in existing models that only considered one infection mode. Indeed, note that when β1 = 0, meaning that infection is exclusively through cell-to-cell transmission, which

is the scenario of the work in [4], the basic reproduction number R₀ reduces to R₀₂. This would be the basic reproduction number of the corresponding model that ignores the virus- to-cell infection mode. Similarly, whenβ2 = 0, R0 reduces to R01 which is exactly the basic

reproduction number for the corresponding model that neglects the cell-to-cell transmission mechanism. Therefore, we see that our model not only reveals that the basic reproduction number of the model that neglects either the cell-to-cell spread or virus-to-cell infection is under-evaluated, but also tells precisely by how much it is under-evaluated, reflected by the relationR0 = R01+R02 and the formulas forR01andR02 in (3.11). This formula also reflects

the impact of the infection age through the distribution function f(s).

When applying models only considering cell-to-cell transmission or infection by cell-free viruses to experimental data, parameters are always estimated to be an average of the effect of both modes of transmission. Thus, the estimate ofR₀based on a model neglecting cell-to-cell transmission is not the exact basic reproductive number of the model with infection by cell-free mode, but an average of both modes of infections.

Cell-to-cell spread not only facilitates rapid viral dissemination, but may also promote im- mune invasion and influence disease [24]. Cell-to-cell spread of HIV-1 may also reduce the effectiveness of neutralizing antibodies and viral inhibitors. However, it is unclear whether this mode of viral spread is susceptible or resistant to inhibition by neutralizing antibodies and entry inhibition. There are ongoing controversies in this field of study [3, 25]. Considering the antiretrovial therapy of reverse transcriptase (RT) inhibitor and incorporating the efficacy

of the RT inhibitor in same way as in [29] (see (3.2)), our model (3.4) now reads                            dT(t) dt =h−dTT(t)−(1−n1)β1V(t)T(t)−(1−n2)β2T(t)T ∗ (t), dT∗_(t) dt = R∞ 0 f(s)e −µs_[(1−n 1)β1T(t− s)V(t−s), +(1−n2)β2T(t− s)T∗(t− s)]ds−δT∗(t), dV(t) dt =bT ∗₍ t)−cV(t), (3.28)

wheren1denotes the efficacy of the RT inhibitor inhibiting the virus-to-cell infection;n2repre-

sents the efficacy of the RT inhibitor with respect to the cell-to-cell channel. Comparing (3.28) to (3.4), we see that the basic reproduction number for (3.28) is

ˆ R₀ = (1−n1)β1ηhb dTδc + (1−n2)β2ηh dTδ =: ˆR₀₁+Rˆ₀₂.

It follows that if the RT inhibitor is very effective for inhibition of virus-to-cell infection, then largen1would make ˆR01less than one, meaning that the virus would be eliminated by the

therapy in the absence of cell-to-cell transmission (β2 = 0). However, if cell-to-cell transmis-

sion co-exists (β2 > 0) and is less sensitive to the RT inhibitor, then n2 could be small, such

that ˆR₀₂ > 1. Thus ˆR₀ > 1, meaning the virus would persist. The virus can be cleared if and only if the RT inhibitor is effective for both modes of infections, such that ˆR₀< 1.

In our model, we do not consider multiple infection per cell which may occur by synaptic transmission. However, the high efficiency of infection by large numbers of virions is likely to result in a transfer of multiple virions to a target cell [7, 15]. Komarova et al. [19, 20] considered multiple infection during the cell-to-cell transmission by mathematical modeling and explored the effect of different strategies of the virus (that is, the number of viruses passed per synapse) on the basic reproductive ratio of the virus. They showed that the strategy of single virus transmission per synapse maximizes the reproductive ratio if the synapses can be formed quickly and the process of infection is independent of the number of resident viruses, while strategies with intermediate numbers of viruses transferred correspond to the highest values of the basic reproductive number if the synapse formation is slow or if the multiplicity of infection strongly influences the kinetic of virus production. Multiple infection of the same cell may waste a large number of viruses that could otherwise enter uninfected target cells, hence fewer newly infected cells are generated and the infection eventually cannot be maintained for larger numbers of transferred viruses.

Multiple infections may reduce the sensitivity to antiretroviral therapies. Sigal et al. [39] showed that cell-to-cell spread of HIV-1 is sufficient to reduce the efficacy of antiretroviral therapy. A possible explanation is that the cell-to-cell transmission may play a significant role for multiple infection per target cell which reduced sensitivity to drugs. They found that

virus-to-cell infection was efficiently prevented bytenofovirandefavirenz. In the presence of tenofovir, virus-to-cell infection declined thirty-fold. But once infection became established, cell-to-cell transfer through direct contact between cells become possible (likely dominant), the infection is much less affected by the presence of drugs. Sigalet al. [39] attempted to explain why highly potent regimens that target several different steps in the HIV-1 life cycle cannot shut down replication, despite reducing HIV-1 replication to very low levels, which could be due to cell-to-cell transfer of multiple virions and the drugs’ inability to inhibit replication when virus levels are high.

However, Permanyeret al. [33] argued that the results of Sigal et al. depend on their par- ticular experimental conditions and that the results therefore might not be correct. Permanyer

et al. also pointed out that the conclusion of drug resistent of cell-to-cell transfer by Sigal

et al. was obtained under the incorrect assumption that each virus transferred will lead to a productive infection. They found that antiretroviral drugs, such as the reverse transcriptase inhibitors zidovudineand tenofovir, and the attachment inhibitor IgGb120, are able to block virus replication with similar efficacy to cell-free virus infections. That indicates that cell-to- cell transmission may not allow for ongoing virus replication in the presence of antiretroviral therapy.

Komarova et al. [21] explored the role of synaptic transmission in susceptibility of HIV-1 infection to antiretroviral drugs, using a virus infection dynamical model with multiple infec- tions. They found that multiple infection via synapses does not simply reduce susceptibility to treatment, which depends on the relative probability of individual virions to infect a cell dur- ing cell-free virus and cell-to-cell virus transmission. If this probability is higher for cell-free virus transmission, then susceptibility to antiretroviral drugs is lowest when a single virus is transferred per synapse, which maximizes the release of free virus. On the other hand, if the infection probability is higher for synaptic transmission, then they found that the susceptibil- ity to antiretroviral drugs is minimized for an intermediate number of virions transferred per synapse. It needs further experimental investigations to determine whether the virus persist by synaptic transmission during antiretroviral therapy.

HIV-1 infection can be very effectively suppressed with antiretroviral therapy, a combina- tion of drugs that block various steps in the HIV-1 lifecycle such as the ability of the virus to reversely transcribe its RNA genome to DNA (RT-inhibitor), integrate DNA into the cell genome, or make viable new virions by the cleavage of viral protein precursors (protease in- hibitor). However, these antiretroviral therapies cannot completely eliminate HIV-1 infection, and the infection can re-establish itself within weeks after therapy interruption. The main rea- son is the existence of reservoir of infected cells that are insensitive to drugs, which could be latently infected cells consisting of those that are quiescent in the genomically integrated form,

long-lived infected cells, or those on ongoing transmission cycles called ongoing replication. It is believed that the reservoir of infected cells is enough to cause a huge rebound in viral load within weeks after stopping an antiretroviral treatment. Considering an antiretroviral therapy in the presence of both cell-free and cell-to-cell transmissions seems to be an interesting yet worthy project.

In our model (3.4), we have assumed that target cells T(t) are produced at a constant rate

hand has a constant death ratedT. It would be more reasonable to consider density dependent

production rate. One possibility is to assume a logistic growth for the healthy cells in the absence of infection, as in [5]. We leave this as a future project.

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Modeling cell-to-cell spread of HIV-1 with

logistic target cell growth

4.1

Introduction

HIV-1 has two predominant infection modes, the classical cell-free infection and direct cell- to-cell transfer. In the classical mode, viruses released from infected cells travel some distance to find a new target cell to infect. Recently, it was revealed that HIV-1 can be transferred from infected cells to uninfected cells through direct contact via some structures, for example membrane nanotubes or macromolecular adhesive contacts termed virological synapses [5, 6, 7]. During the cell-to-cell transfer, many viral particles can be simultaneously transferred from infected CD4+T cells to uninfected ones.

In the preceding chapter, we incorporated the two modes of viral transmission into a classic model leading to the following model system

dT(t) dt = H−dTT(t)−β1T(t)V(t)−β2T(t)T ∗ (t), dT∗_(t) dt = Z ∞ 0 [β1T(t−s)V(t−s)+β2T(t− s)T∗(t− s)]e−msf(s)ds−dT∗T∗(t), dV(t) dt =γT ∗ (t)−dVV(t). (4.1)

Here T(t), T∗(t) and V(t) are the concentrations of susceptible CD4+ T cells (target cells), productively infected T cells and free virus particles at time t respectively. A time delay, s, from the time of initial infection until the production of new virions, is considered, and s is assumed to be distributed according to a probability distribution f(s). Target cells are infected by free virus particles and infectious cells at ratesβ1T(t)V(t) andβ2T(t)T∗(t) respectively. e−ms

represents the survival rate of infected cells during the time delays. Target cells are recruited 79

at a constant rate H. Free viruses are released by infected cells at a rate γT∗(t). The loss rate of target cells, productively infected cells and free virus aredTT(t), dT∗T∗(t) andd_VV(t)

respectively. We found that the basic reproduction number was underestimated by some models when only one mode of virus spread was considered. In this model, we have assumed that target T cells have a constant source term and an exponential death rate. This is mainly for the purpose of reducing the difficulty level in analyzing the model, since introduction of delay into the model has already made the model an infinite dimensional system.

It is more realistic to assume that the population of the CD4+T cells have a logistic growth