Extended-Release Pre-exposure Prophylaxis and Drug-Resistant HIV

Abstract

The pharmacologic tail of long-acting cabotegravir (CAB-LA), an injectable pre-exposure prophylaxis (PrEP), allows for months-long intervals between injections, but it may facilitate the emergence of drug-resistant human immunodeficiency virus (HIV) strains during the acute infection stage. In this chapter, we present a within-host, mechanistic ordinary differential equation model of the HIV latency and infection cycle in CD4${ }^+$ T-cells to investigate the impact of CAB-LA on drug-resistant mutations in both humans and macaques. We develop a pharmacokinetic/pharmacodynamic model for CAB-LA to correlate the inhibitory drug response with the drug concentration in plasma. After validating our model against experimental results, we conduct in silico trials. First, we separately administer CAB-LA to the in silico macaque and human patients before and after exposure to simian-human immunodeficiency virus (SHIV)/HIV, to observe SHIV and HIV infectivity dynamics, respectively. Although the model does not incorporate a mechanism for CAB-LA-induced HIV mutations, we analyze the outcomes when mutations occur naturally. Our findings suggest that CAB-LA may enhance the growth of drug-resistant strains over the wild-type strains during the acute stage. The in silico trials demonstrate that the effectiveness of CAB-LA against mutations and the fitness of the drug-resistant strain to infect T-cells determine the course of the mutated strain.

1 Introduction

Human immunodeficiency virus (HIV) is an aggressive virus that attacks the body’s immune system via destruction of CD4${ }^+$ T-cells. More specifically, HIV is a lentivirus, a class of ribonucleic acid (RNA) viruses that converts RNA into deoxyribonucleic acid (DNA). The name lentivirus is derived from the Latin word for slow, “lenti,” referring to the characteristically long incubation period. During this incubation period, which can last for multiple years, the virus appears to be controlled by the immune system, but in actuality it is not [1]. Recent advances in disease progression have asserted three stages: the acute stage, the chronic stage (i.e., the incubation period), and finally the acquired immunodeficiency syndrome (AIDS) stage [2]. The acute stage of HIV begins with the initial infection and continues for approximately 8–12 weeks [3, 4]. During the acute stage, those infected experience a high viral load. After the acute stage, those infected move into a chronic stage in which their viral load goes down, and many individuals are asymptomatic.

The HIV virion, in this case, has a capsid that protects its RNA-filled inner core, while it is free-floating and searching for a host cell. Once the virion attaches to a host CD4${ }^+$ T-cell and injects its RNA, the infection cycle begins. This RNA injection allows HIV to pass through the early most infectious stage without detection by the immune system [5].

Unlike other retroviruses, lentiviruses do not depend on proliferation of the infected cell to integrate into the host genome [6]. After converting their RNA genome into DNA, lentiviruses integrate into the host genome, a step necessary for the expression of viral proteins. Due to constant selection pressure to evade the innate and adaptive immune systems, HIV undergoes frequent mutation as a result from this evasion pressure [7]. In addition, it is well known that RNA viruses are quite unstable and are inherently more prone to mutation than DNA viruses. More specifically, it has been shown through intracellular fidelity assays, which signal either mutation inactivation or reversion, that the mutation rate for HIV is $10^{-5}$ per replication cycle [7], on average, which is similar to that of other retroviruses [8,9,10].

Note that HIV, in general, refers to HIV-1. Although HIV-1 and HIV-2 share many similarities, HIV-2 is characterized by a reduced likelihood of transmission and progression to AIDS. In terms of epidemiology, HIV-2 remains largely confined to West Africa, whereas HIV-1 is found worldwide.

Since 1987, different forms of antiretroviral drugs were developed and began to transform disease management for HIV-infected individuals as well as susceptible individuals [11]. With reliable life-long adherence, antiretroviral therapies (ART), including combination pills, give HIV-positive individuals a lifespan comparable to that of disease-free individuals [12,13,14,15]. In addition, those individuals who have started ART may be virally suppressed, which also decreases the chance of HIV transmission [16,17,18,19,20]. While ART has helped to reduce HIV fatality rates, allowed infected individuals to live with minimal symptoms, and even protected children from infection during natural birth [21], it is still not the final answer for HIV control. Some of the mutated HIV strains may still have the ability to replicate in the presence of drugs. These HIV mutations can develop while undergoing ART, making finding an effective treatment much more difficult for the individual, as a treatment that once worked will no longer prevent the drug-resistant (DR) mutated strain from replicating.

In addition to managing the viral loads of HIV-positive individuals, ART may be taken to prevent infection after exposure. In 1990, the Centers for Disease Control and Prevention (CDC) recommended post-exposure prophylaxis (PEP) for individuals with occupational HIV exposures [22]. Today, PEP involves a 28-day course of ART within 72 hours of possible exposure and is only used in emergency situations to prevent HIV infection [23].

In 2012, the US Food and Drug Administration (FDA) approved the first medication for high-risk individuals to prevent infection, a strategy known as pre-exposure prophylaxis (PrEP) [24]. By 2016, there were two approved daily pills used for PrEP: Truvada, a combination of emtricitabine and tenofovir disoproxil fumarate, and Descovy, a combination of emtricitabine and tenofovir alafenamide [25]. While daily microdoses of PrEP are widely used and extremely effective in preventing HIV infection even with common exposure, daily adherence can be challenging, especially when mild to extreme side effects such as trouble breathing, fever, tiredness, muscle aches, blisters of the mouth, and swelling of the eyes, face, and tongue occur [26].

In December 2021, the FDA approved an injectable PrEP medication, Apretude, generically known as cabotegravir long-acting (CAB-LA), for use in at-risk adults and adolescents to reduce their risk of sexually acquired HIV [27]. The CAB extended-release injectable suspension (CAB-LA) is first administered as two injections 1 month apart and then administered continually every 2 months [28].

In the ECLAIR [29] trial, the long half-life of CAB-LA meant that the drug was detectable 52 weeks after the last injection in $14\%$ of the trial participants. While this long half-life is beneficial in allowing long intervals between PrEP injections, the long pharmacologic drug tail also means that there can be a long period when the CAB level is too low to prevent an HIV infection but high enough to give mutations an advantage. The wild-type strain of HIV signifies the unaltered version of the virus that has not acquired mutation to an antiretroviral drug, i.e., drug-sensitive. Since the mutated strain is inherently less pressured by PrEP, it may not be effectively suppressed, while the wild-type strain is. This is a concern considering that any PrEP drug is just one component of the drug cocktails used in antiretroviral treatment, and the emergence of mutations to PrEP should be investigated. We note that in January 2021, oral CAB (Vocabria) and a combination injectable CAB drug (Cabenuva) were approved by the FDA [27] for people living with HIV. Hence, modeling the development of drug resistance to CAB-LA as PrEP is necessary.

Continued experimental efforts for PrEP, especially injectable PrEP, and mutated HIV strains in humans can be difficult to find and fund. Thus studies with simian-human immunodeficiency virus (SHIV) in macaques have been used as a proxy to further understand short-term and long-term protection from both wild-type and mutated infections. Previous macaque studies with SHIV prior to seroconversion, i.e., before having enough virions to test positive for SHIV, show that long-acting CAB may encourage rather than inhibit mutations [30]. It has been shown that, in patients receiving ART that does not provide sufficient HIV suppression, many mutants can develop within days, thus decreasing the likelihood of drug efficacy [7]. The growth of mutated strains from the macaque trials and insufficient HIV suppression through ART have informed the FDA decision to require HIV tests prior to each PrEP injection in order to further protect an individual from mutated strains [31]. However, this FDA decision has not been tested using human experiments.

The main goal of mathematical modeling is to serve as a predictive tool that can mechanistically describe highly complex systems. Once a system is understood, mathematical models can be used to test specific parameter regimes or treatment schedules to save time, money, and effort in a clinical setting. Previous mathematical models have been developed to understand wild-type HIV in humans [32,33,34,35]. Similarly, mathematical models have been built to better understand mutation in HIV to particular drugs [36, 37]. However, a model that mechanistically describes both wild-type and mutation in HIV, as well as SHIV in macaques and its response to CAB PrEP treatment campaigns, has not been built.

In this manuscript, we have built a within-host, mechanistic, ordinary differential equation (ODE) model of the HIV latency and infection cycle in CD4${ }^+$ T-cells. Our model incorporates a pharmacokinetic and pharmacodynamic (PK-PD) model to establish the relationship between the inhibitory drug response of CAB and its concentration in the plasma as well as rectal, cervical, and vaginal fluids and tissue. We then verify our model with viral load data extracted from Reeves et al. [38] and Vaidya et al. [39] for humans and macaques, respectively. Once our model is calibrated, we build in silico experiments that involve SHIV and CAB-LA PrEP to replicate behaviors found in literature and observe new phenomena [30, 40, 41]. First, we administer CAB-LA PrEP separately to in silico macaque and human patients, both before and after exposure to SHIV or HIV, respectively, to observe SHIV and HIV infectivity dynamics. We present the drug concentrations and inhibitory response of these protocols in Sect. 4. We then study dynamics of these in silico experiments with mutations occurring at an observed rate [42]. While we do not include a mechanism for PrEP to cause mutations in the model, we can observe what occurs when mutations naturally enter the system. We can see under what conditions exposure to PrEP may encourage the mutant strain to grow. With the results of these in silico trials, we show that the level of mutation, the effectiveness of CAB-LA against the mutant strain, and the aggressiveness with which the mutant strain of virions infects healthy T-cells determine whether the mutant strain grows to a significant level in the acute stage of infection. In particular, we have found that the primary factor determining whether the resistant strain can grow or even overwhelm the wild-type is the degree of fitness, or the infectivity, for the drug-resistant strain of virions to infect healthy T-cells.

The results are presented in the following order. In Sect. 2 we present a schematic of the HIV life cycle along with the various ARTs targeting different stages of the life cycle along with notes about CAB-LA drug resistance. In Sect. 3 we discuss the within-host viral dynamics and the T-cell model. In Sect. 4 we introduce the model for the inhibitory function of CAB-LA drug. In Sect. 5 we calculate the effective reproduction number. Next, we present the parameters with a discussion on estimation, acquisitions, and sensitivity in Sect. 6. We present our numerical results and comparison to experiments for both humans and macaques in Sect. 7 before discussing the connections between human and macaque in silico experiments and outlining future directions for this research.

2 HIV Replication and Antiretroviral Drugs

HIV attacks and destroys the CD4${ }^+$ T-cells of the immune system. CD4${ }^+$ T-cells are a type of white blood cell that plays a major role in protecting the body from infection. HIV uses the machinery of the CD4${ }^+$ T-cells to multiply and spread throughout the body. This process, which is carried out in seven steps or stages, is called the HIV life cycle [43]. The HIV life cycle refers to the series of steps the virus takes to infect cells, reproduce, and spread throughout the body. As shown in Fig. 1, HIV attaches to CD4${ }^+$ T-cells (stage 1), the main target of the virus, using its envelope proteins (GP 120). The virus then fuses with the cell membrane and enters the cell, where the viral RNA is converted into DNA by the viral reverse transcriptase enzyme (stage 2). The viral DNA is integrated into the host cell’s genome (stage 3) and transcribed into RNA, which is then translated into viral proteins by the host cell machinery (stage 4). These viral proteins and RNA come together to form new virions (stage 5), which bud off from the host cell (stage 6) and are released into the bloodstream to infect other cells (stage 7). Several stages of the HIV life cycle are crucial targets for antiretroviral drugs (ARVs), which aim to interrupt the cycle and prevent the virus from replicating and spreading. In the following list, we group the six classes of ARVs used to treat HIV [44] by HIV life cycle stages as illustrated in Fig. 1.

Fig. 1

Schematic of HIV life cycle. The stage numbers of the life cycle are used to group the targets of the antiretroviral drugs

• Stage 1 targets

  CCR5 antagonists: These drugs block virus entry into host cells by binding to the viral envelope or to cellular receptors. By blocking the CCR5 co-receptor, CCR5 antagonists prevent the virus from replicating and spreading within the body, helping to slow the progression of the disease; examples: maraviroc (Selzentry) and vicriviroc (VRC01).

  Fusion inhibitors: These drugs interfere with the initial stages of virus entry into host cells by blocking the fusion of the viral and cellular membranes; example: enfuvirtide (Fuzeon).

• Stage 2 targets

  Nucleoside/nucleotide reverse transcriptase inhibitors (NRTIs): These drugs mimic the building blocks of DNA and are incorporated into the growing viral DNA chain by reverse transcriptase, causing termination of the chain; examples: zidovudine (AZT) and tenofovir (Viread).

  Non-nucleoside reverse transcriptase inhibitors (NNRTIs): These drugs bind directly to reverse transcriptase, blocking its activity; examples: nevirapine (Viramune), efavirenz (Sustiva), and riplpivirine (Edurant).

• Stage 3 targets

  Integrase strand transfer inhibitors (INSTIs): These drugs block the integrase enzyme, which is needed for the integration of the viral DNA into the host cell genome; examples: dolutegravir (Tivicay), raltegravir (Isentress), elitegravir (Biktary), cabotegravir (Vocabria), and long-acting cabotegravir (CAB-LA, Apretude).

• Stage 6 targets

  Protease inhibitors (PIs): These drugs block the protease enzyme, which is needed for the processing and maturation of the viral polyprotein. This prevents the formation of functional virions and the spread of infection; examples: lopinavir (Kaletra) and atazanavir (Reyataz).

Combination antiretroviral therapy (cART) for people living with HIV typically involves using multiple drugs from different classes to target the virus at multiple stages of the life cycle, increasing the chances of blocking replication and reducing the development of drug resistance. One current cART of concern is oral CAB (Vocabria), to be used with other ARV, and a combination injectable CAB drug (Cabenuva). In a recent study [45], Engelman and Engelman reviewed CAB-LA and the technical aspects of integrase inhibitors and resistance. These same authors note that, in general, INSTIs’ resistance occurs through substitution of amino acid residues near the integrase active sites [46, 47]. Cook et al. [48] reports that the INSTI resistance mutations destabilize the magnesium ion cluster, which restricts CAB-LA’s ability to effectively increase its rate of dissociation from the integrase active site. First-generation INSTI compounds raltegravir [49], approved by the FDA in 2007, and elvitegravir [50], approved by the FDA in 2012, select for drug resistance more easily [45]. This does not mean that raltegravir and elvitegravir cause mutations in HIV, only that naturally occurring mutations in HIV can evade these drugs easily. Second-generation INSTI drugs dolutegravir [51] and bictegravir [52], licensed by the FDA in 2013 and 2018, have been proven to be less able to select for drug resistance. CAB is chemically similar to dolutegravir [45]. Parikh et al. [53] note that multiple INSTI mutations are required for extensive CAB drug resistance. However, the macaque studies by Radzio-Basu et al. [30] indicate that these mutations are selected readily when CAB-LA is given to macaques previously infected with SHIV. The alarm is that these mutations may cause resistance to another INSTI drug, such as dolutegravir, a first-line ART in low- and middle-income countries, or other second-line integrase inhibitors, such as bictegravir. WHO recommends monitoring INSTI drug resistance, and the introduction of CAB-LA as PrEP reinforces this need [54, 55].

3 Within-Host Viral Dynamics and T-Cell Model

We model the within-host system with a deterministic system of ODEs to capture the infection of CD4${ }^+$ T-cells by free virus particles, i.e., virions, in plasma. In this system, T represents the concentration of healthy CD4${ }^+$ T-cells. In the absence of disease, the number of T-cells in blood is relatively constant. Thus, we use a logistic term to maintain this balance

$$\displaystyle \begin{aligned} \frac{dT}{dt} = \gamma \Big(1-\frac{T}{K_T}\Big) T - \mu T, {} \end{aligned} $$

(1)

where $\gamma$ is the proliferation of healthy T-cells, $K_T$ is the carrying capacity, and $\mu$ is the natural death rate.

We consider virions that are carrying a wild-type strain of HIV, i.e., drug-sensitive (DS) strain, as well as those carrying a mutant strain of HIV, i.e., drug-resistant (DR) strain. They are denoted by $V_s$ and $V_r$, respectively. Once a virion, or a number of virions, enters a healthy T-cell, we consider the CD4${ }^+$ cell to be infected. These infected cells are divided into two categories, latently infected T-cells, L, and actively infected cells, I. The actively infected T-cell proceeds along the HIV replication path described in Sect. 2. We further classify these infected T-cells as $L_s$ and $I_s$ for those infected with the DS strain and $L_r$ and $I_r$ for those infected with the DR strain. In the latent cells, $L_s$ and $L_r$, HIV hides in an inactive state. In these resting memory T-cells ($L_s$ and $L_r$), HIV evades immune clearance. When the long-lived latently infected cells activate and once their intravirion levels reach some threshold, we consider the T-cell to be actively infected. Once the virions inside the actively infected T-cell cause the cell to burst, those virions are released back into the population of virions in the plasma, $V_s$ and $V_r$. A table of state variables and their descriptions are given in Table 1. Our T-L-I-V model is illustrated in Fig. 2.

Fig. 2

Schematic of HIV latency and infection in CD4${ }^+$ T-cells. Each square node represents a state variable corresponding to either a state of a CD4${ }^+$ T-cell ($T, L_j,$ and $I_j$ for healthy, latently infected, and actively infected, respectively) or an HIV virion ($V_j$), where $j=s$ (representing DS) or r (representing DR). The label DS indicates the drug-sensitive strain, and DR indicates the drug-resistant strain. The solid black arrows represent a movement from one state to another, one state to itself, or a decay. Dashed arrows represent a release of HIV virions into the plasma from lysed CD4${ }^+$ T-cells

Table 1 Symbols and definitions of state variables used in the model. The label DS indicates the drug-sensitive strain, and DR indicates the drug-resistant strain

Healthy CD4${ }^+$ T-cells are recruited at the rate $\beta$ and die at the natural death rate $\mu$. These healthy cells are proliferated at the rate $\gamma (1-W/K_T) T$, which represents the reproduction total of T-cells ($W = T + L_s+L_r+I_s+I_r$) through mitosis up to a carrying capacity $K_T$. The healthy cells are infected by DS virions and DR virions at the rates $k_s$ and $k_r$, respectively. A portion, $\sigma$, of these cells become latently infected, and the rest, $(1-\sigma )$, move directly into the actively infected state. The latent cells die at a slower rate, $\hat {\mu }$, than the actively infected cells. This slower rate implicitly incorporates the self-proliferation of latent cells that make it appear as if the latent cells exit at a lower rate than $\mu$. The latent cells reactivate at the rate $\chi _j$, which describes the transition rate of target cells from latently infected $L_j$ to actively infected $I_j$, where $j=s$ (representing DS) or r (representing DR). The concentration of actively infected T-cells decreases due to natural death, $\mu$, and viral-induced death, d. We are modeling the acute stage of the HIV infection and disregarding the rate at which the immune system attacks the infected cell $I_s$ or $I_r$. CAB-LA, as an INSTI, interferes with the HIV replication stage when the viral DNA is integrated into the host cell’s genome (in stage 3) and transcribed into RNA. Thus, the viral-induced death for the DS-infected T-cells would be reduced to $f_\alpha dI_s$ with treatment, where $f_\alpha$ represents the inhibitory effect of CAB-LA and is valued at the fraction of events unaffected by the drug. We discuss the model and evaluation of $f_\alpha$ in Sect. 4. Similarly, the viral-induced death for the DR-infected T-cells would also be reduced. The parameter $\phi$ represents the percentage of drug efficacy on the resistant strain. For example, the drug may be half as effective on the resistant strain in comparison to the sensitive one, thus $\phi = 0.50$. Then, $\phi (1-f_\alpha )$ represents the drug efficacy on the DR strain, and the viral-induced death changes from $dI_r$ to $(1-\phi (1-f_\alpha )) dI_r$. Each viral-induced death of the actively infected T-cells would lead to the generation of N virions. Hence, the DS virion concentration increases by $f_\alpha d I_s N$, and the DR virion concentration increases by $(1-\phi (1-f_\alpha )) dI_rN$. Both concentrations decrease due to natural clearance of the virus, c. It is well known that infected cells and virions are not cleared at a constant rate throughout infection because they are targeted and cleared by adaptive immune responses that expand in response to infection. However, our study focuses on the acute phase of HIV infection, and hence we assume a constant clearance rate in our implicitly modeled immune system.

We also assume that the DR virions arise from a naturally occurring mutation at the rate $m_{sr} = 10^{-5}$ [7]. We ignore backward mutations from DR to DS. Since extensive drug resistance to CAB requires multiple mutations [53], we assume that the chance to reverse multiple mutations is negligible. The forward mutations could occur in both latently infected cells and actively infected cells. Since we only consider the natural mutation rate per replication cycle and not the mutations that solely affect stage 3 of the HIV replication cycle, we have discounted the mutation rate by associating it only with the latently infected T-cells. However, in this model, we have a portion of latently DS-infected T-cells, $L_s$, mutating at rate $m_{sr}\chi$ to the actively DR-infected category, $I_r$, and the non-mutated portion $(1-m_{sr})$ moving at the rate $\chi$ to the actively DS-infected T-cells, $I_s$. The system of ODE representing this model is

$$\displaystyle \begin{aligned} \begin{array}{rcl} W& =&\displaystyle T+L_s +L_r+I_s +I_r, \\ \frac{dT}{dt} & =&\displaystyle \beta -k_s V_s T -k_r V_r T -\mu T +\gamma \Big(1-\frac{W}{K_T}\Big) T, \\ \frac{d L_s}{dt}& =&\displaystyle \sigma k_s V_s T -\chi L_s-\hat{\mu} L_s,\\ \frac{d I_s}{dt}& =&\displaystyle (1-\sigma) k_s V_s T + (1-m_{sr})\chi L_{s} -(\mu+f_\alpha d)I_s,{}\\ \frac{d L_r}{dt}& =&\displaystyle \sigma k_r V_r T -\chi L_r-\hat{\mu} L_r,\\ \frac{d I_r}{dt}& =&\displaystyle (1-\sigma) k_r V_r T +\chi L_r+ m_{sr}\chi L_s -(\mu+(1-\phi(1-f_\alpha) ) d ) I_r,\\ \frac{d V_s}{dt} & =&\displaystyle f_{\alpha}N dI_s-cV_s, \\ \frac{d V_r}{dt}& =&\displaystyle (1-\phi(1- f_\alpha)) N d I_r-c V_r. \end{array} \end{aligned} $$

(2)

The concentration of CD4${ }^+$ T-cells is large, and the number of virions per milliliter of blood is on a significantly different scale. To remove the numerical difficulties this will create, we preemptively scale the equations for our use in our simulations. We use the constant $T_0$ to scale the healthy, latently infected, and actively infected T-cells. The constant $V_0$ is used to scale the virions. From these constants, the dimensionless variables are defined: $\tilde {T} = T/T_0$, $\tilde {L}_j = L_j/T_0$, $\tilde {I}_j = I_j/T_0$, and $\tilde {V} = V/V_0$, where j is either s or r. Similarly, all other parameters are scaled: $\tilde {\beta }=\beta /T_0$, $\tilde {k}_s = k_s V_0$, $\tilde {k}_r = k_r V_0$, $\tilde {K_T}=K_T/T_0$, and $\tilde {N}=N T_0/ V_0$, where $T_0$ and $V_0$ are the initial conditions. The rescaled system looks identical to the system in Eq. (2).

4 Model for CAB-LA Drug Inhibitory Function

Cabotegravir (CAB) is an INSTI analog of dolutegravir (DTG) that is very potent ($50\%$ inhibitory concentration is about $0.22$ nM) and active against various subtypes of HIV [56]. CAB has the unique feature of a long half-life, which is about 40 days after oral administration, and can be formulated as a nanoparticle injection. Therefore, CAB has the potential to permit its formulation as a long-acting injection (LA) amenable to dosing every 2 months, making CAB-LA an attractive alternative to daily oral PrEP regimens [57]. Oral PrEP is an effective strategy to reduce the risk of HIV transmission in high-risk individuals. The key to the efficacy of any PrEP treatment is to maintain a high enough drug concentration in the body, consistently, to be effective. Drug concentration in the body varies over time due to various factors such as metabolism, excretion, and adherence to the prescribed regimen. Adherence to the regimen ensures consistent and sufficient drug levels in the blood, which help to suppress the replication of virus, and hence plays a critical role in the effectiveness of the drug. The efficacy of oral PrEP is highly dependent on user adherence, which some previous trials have struggled to optimize particularly in low- and middle-income settings [58]. By replacing the need for a daily pill with a bimonthly injection, CAB-LA removes one of the adherence obstacles.

4.1 General HIV Dose–Response

The standard form of dose–response function for antiviral drugs is the median effect model based on mass action, which plots the fraction of infection events unaffected by drug, $f_\alpha (t)$, against log of drug concentration $n (\log C)$, based on the Hill equation [59],

$$\displaystyle \begin{aligned} f_\alpha(t) = \frac{IC_{50}^n}{IC_{50}^n+C(t)^n}, {} \end{aligned} $$

(3)

where C is the drug concentration, $IC_{50}$ is the drug concentration that causes 50% of the maximum inhibitory effect, and n is a slope parameter. The slope parameter is mathematically analogous to the Hill coefficient, which is a measure of cooperativity in a binding process. A Hill coefficient of 1 indicates independent binding, while a value of greater than 1 shows positive cooperativity binding [60]. For antiviral drugs, the Hill slope values define intrinsic limitations on antiviral activity and are class specific. NRTIs and INSTIs have been shown experimentally to have Hill slopes of approximately 1, which is characteristic of noncooperative reactions. NNRTIs, PIs, and fusion inhibitors show positive cooperativity binding with slopes $>1$ [61]. Since CAB is an INSTI, Eq. (3) has a Hill slope of $n \approx 1$.

The potency of a drug is identified as $IC_{50}$, but with HIV, the 90% protein-adjusted maximal response is the number reported in experimental effectiveness reports. The formula for conversion is

$$\displaystyle \begin{aligned} IC_{X}= \left(\frac{X}{100-X} \right)^{\sqrt{n}} IC_{50}, \end{aligned}$$

where n is the same Hill coefficient as in Eq. (3) and $X = 90$. Thus, $IC_{50}= IC_{90}/9$. In the literature, the protein-adjusted PA-$IC_{90}$ is given instead of the $IC_{90}$; therefore we use the PA-$IC_{90}$ values throughout this chapter.

4.2 Human CAB-LA Data and Model

In Fig. 3, we fit an exponential curve to the human plasma drug concentration experimental data for CAB-LA versus time (in days) reported in [62]. We find the cabotegravir plasma concentration, denoted as $C(t)$, in $\mu \mathrm {g/mL}$ at time t. The concentration is given by $C(t) = C_{\max }e^{-k t}$ using experimental data values for $C_{\max }$ and $k=\ln (2)/\tau _{1/2}$, where $\tau _{1/2}=19.1$ days is the drug half-life measured by Shaik et al. [62]. Then, the drug–plasma concentration in humans is $C(t) = 5.04 e^{-0.0363 t}$, as illustrated in Fig. 3. For human (plasma) CAB levels, we have $IC_{90} = 0.166$ $\mu \mathrm {g/mL}$, and thus, $IC_{50} = 0.018$ $\mu \mathrm {g/mL}$. Therefore, the fraction of infection events unaffected by the drug, $f_\alpha ^h$ (t), for a single dose in human is described by the following equation:

$$\displaystyle \begin{aligned} f_\alpha^h(t) = \frac{0.018}{0.018+5.04 e^{-0.0363 t}}, {} \end{aligned} $$

(4)

Fig. 3

Cabotegravir plasma concentration in ($\mu \mathrm {g/mL})$ in humans after one PrEP injection. The diamonds represent the mean data reported by Shaik et al. [62]. The dotted and dashed horizontal lines represent $1\times$ PA-$IC_{90}$ and $4\times$ PA-$IC_{90}$, respectively. The solid line is the exponential fit to the mean data

where the superscript h represents human.

The WHO guideline announced in July 2022 recommends that the first two injections be administered 4 weeks apart, followed thereafter by an injection every 8 weeks [28]. In Fig. 4a, the simulated plasma CAB concentration solid curve always stays above $4\times IC_{90}$ shown by the dashed line. Figure 4b captures the fraction of infection events unaffected by the drug, $f_\alpha ^h$, during the 48 weeks. We note that the range of unaffected events varies from 0.5% to 2%.

Fig. 4

Human cases. (a) Pharmacokinetic profile of plasma CAB-LA concentration ($\mu \mathrm {g/mL}$) in humans represented by the solid black curve. The dotted and dashed horizontal lines represent $1\times$ PA-$IC_{90}$ and $4\times$ PA-$IC_{90}$, respectively. (b) Fraction of infection events unaffected by PrEP, $f^h_\alpha (t)$, versus time since first injection for humans

Since PrEP is given for HIV prevention for sexual exposures, it is important to note the relationship between the plasma concentration of CAB as compared to the drug concentration in rectal, cervical, and vaginal tissues and fluids. Shaik et al. [62] plotted CAB concentrations for each of these tissues and fluids versus plasma concentration. The slopes of these data plots ranged from a high slope of 1.173 (cervical tissue CAB concentration vs plasma) to 0.926 (vaginal fluids CAB concentration vs plasma). Rectal tissue and fluids plotted against plasma concentration had a slope of 1.012 and 0.929, respectively. Thus, the experimental data indicate that the relationship between the plasma–drug concentration and the drug concentration in rectal, cervical, and vaginal tissues and fluids is linear, as shown in Fig. 5.

Fig. 5

Human: concentration of CAB-LA over 48 weeks of treatment in plasma, tissues, and fluids. The CAB levels are represented by (top to bottom) in plasma with the solid black curve, in rectal fluid with the dashed black curve, in vaginal tissue with the dotted black curve, in cervical tissue with a solid gray curve, in rectal tissue with a dotted gray curve, and in cervicovaginal fluid with a dashed gray curve. The dotted and dashed horizontal lines represent $1\times$ PA-$IC_{90}$ and $4\times$ PA-$IC_{90}$, respectively

We note that to extend our model for the fraction of infection events unaffected by the drug, $f_\alpha ^h$, to rectal, cervical, and vaginal tissues and fluids, we would need to scale $IC_{50}$ and $C(t)$ by the slope-intercepts of this relationship. This may result in a value for $f_\alpha ^h$ in the tissues and fluids that is different from $f_\alpha ^h$ in plasma. The authors of this chapter are currently unable to find values for $IC_{50}$ or $IC_{90}$ in tissues or fluids for CAB-LA. So, we are unable to calculate $f_{\alpha }^h$ for tissues and fluids. However, our simulations in this chapter all model experiments with HIV infection via plasma, so we will be using $f_\alpha ^h$ as defined in Eq. (4) and in Fig. 4.

4.3 Macaque CAB-LA Data and Model

For macaques, $IC_{90}$ is equal to 0.166 $\mu \mathrm {g/mL}$ for CAB, thereby giving $IC_{50} = 0.018$ $\mu \mathrm {g/mL}$. The plasma drug concentration for macaques was calculated by using experimental data from [40]. The experiments measured $C_{\max }= 3.5$ and decay constant, $k=\ln (2)/\tau _{1/2}$, where $\tau _{1/2}=14.41$ days. The fraction of infection events unaffected by the drug for macaques, $f_\alpha ^m(t)$, is

$$\displaystyle \begin{aligned} f_\alpha^m(t) = \frac{0.018}{0.018+3.5 e^{-0.048 t}}. \end{aligned} $$

(5)

The timing of the CAB-LA injection series for macaques shown in Fig. 6 was chosen to match the experimental regime in [30]. Each CAB-LA injection for the macaques is given three times, each 4 weeks apart, following the experimental regime and data from [30]. In Fig. 6a, the simulated plasma CAB concentration solid black curve agrees with the individual macaque data shown by gray curves with circular markers. As shown in Fig. 6b, during the first 8 weeks, only a small portion of target cells will be unaffected by the drug. After the last injection, the drug concentration drops below $4\times IC_{90}$ 5 weeks after the last injection and below $1\times IC_{90}$ by 10 weeks after the last injection. With the drug–plasma concentration waning so quickly, a larger portion of the target cells will be unaffected. By the end of week 22, $30\%$ of target cells are unaffected by CAB.

Fig. 6

Macaque cases. (a) Pharmacokinetic profile of plasma CAB concentrations ($\mu \mathrm {g/mL}$) based on simulations, shown by solid black curve and in individual macaques data, shown in light gray curves with markers, from [30] for three injections. The dotted and dashed horizontal lines represent $1\times$ $IC_{90}$ and $4\times$ $IC_{90}$, respectively. (b) Fraction of infection events unaffected by PrEP, $f_\alpha$, versus time since the first injection for macaques. The dash-dotted line is a reference line of $f_\alpha ^m= 0.02$. Inset is an enlarged image for the first 13 weeks

5 Analysis

We are interested in the disease-free equilibrium (DFE) state where healthy T-cells are persistent and the actively and latently infected T-cells die out. Similarly, viral populations are totally cleared from the system. We begin by computing the DFE and the effective reproduction numbers for each of the two strains independently from the steady state with no virus present.

Let $\mathcal {E} = (T, L_s,I_s,V_s,L_r,I_r,V_r)$ denote an equilibrium of the system described by Eq. (2). The system always has the DFE, $\mathcal {E}_0=(T^*,0,0,0,0,0,0)$, where

$$\displaystyle \begin{aligned} T^* = \frac{(\gamma - \mu) +\sqrt{(\gamma - \mu)^{2} + 4 \gamma \beta / K_{T}} }{2 \gamma / K_{T}}. {} \end{aligned} $$

(6)

Following [63], we use the next generation matrix method to obtain the following expressions for the reproduction numbers for the DS and DR strains. The calculation of the reproduction numbers is given in the proof of Theorem 1. For the wild-type HIV(SHIV) infection, we find the reproduction number $\mathcal {R}_E^s$ as

$$\displaystyle \begin{aligned} \mathcal{R}_E^s=\frac{d \bar{f}_\alpha k_s (\chi (1 - \sigma m_{sr}) + (1 - \sigma) \hat{\mu}) N T^*}{c (\chi + \hat{\mu}) ( d \bar{f}_\alpha + \mu)} , {} \end{aligned} $$

(7)

where $\bar {f}_\alpha =$ average value of $f_s(\alpha )$. For the mutated strain, we also find the reproduction number for the $\mathcal {R}_E^r$ as

$$\displaystyle \begin{aligned} \mathcal{R}_E^r=\frac{(1-\phi (1-\bar{f_\alpha}))d k_r (\chi + (1 - \sigma) \hat{\mu}) N T^*}{c (\chi + \hat{\mu}) (\phi d \bar{f}_\alpha + \mu)}. {} \end{aligned} $$

(8)

With CAB-LA protocol levels, both $\mathcal {R}_E^s$ and $\mathcal {R}_E^r$ are less than 1 for humans and macaques.

Theorem 1

The disease-free equilibrium, $\mathcal {E}_0$, is locally asymptotically stable if both reproduction numbers, $\mathcal {R}_E^s$ and $\mathcal {R}_E^r$, are less than unity and is unstable if at least one of the reproduction numbers is greater than unity.

Proof

The proof follows the approach developed in [63]. Let $\mathcal {X} = (V_s, L_s,I_s,V_r, L_r,I_r,S)$ denote states. Define $\mathcal {F}_i(\mathcal {X})$ as the vector representing the rate of new infections and free virions into compartment i, $\mathcal {V}^+_i(\mathcal {X})$ ($\mathcal {V}^{-}_i(\mathcal {X})$) as the rate of transfer of cells into (out of) compartment i, and $\mathcal {V}=\mathcal {V}^{-} - \mathcal {V}^+$. Consider the systems $\dot {\mathcal {X}}=\mathcal {F}(\mathcal {X})-\mathcal {V}(\mathcal {X})$ where

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle \mathcal{F}(\mathcal{X})= \begin{bmatrix} \bar{f}_\alpha N d I_s \\[3pt] \sigma k_s V_s T \\[3pt] (1-\sigma) k_s V_s T \\[3pt] F_\phi N d I_r \\[3pt] \sigma k_r V_r T\\[3pt] (1-\sigma) k_r V_r T \\[3pt] 0 \end{bmatrix}, \mathcal{V}^+ = \begin{bmatrix} 0 \\[2pt] 0 \\[2pt] (1-m_{sr}) \chi L_s \\[2pt] 0 \\[2pt] 0 \\[2pt] \chi (L_r + m_{sr} L_s) \\[2pt] \beta + \gamma T \end{bmatrix}, \mbox{ and } \\ & &\displaystyle \quad \mathcal{V}^- = \begin{bmatrix} c V_s \\[2pt] (\chi + \hat{\mu})L_s \\[2pt] (\mu+ \bar{f}_\alpha d) I_s \\[2pt] c V_r \\[2pt] (\chi + \hat{\mu}) L_r \\[2pt] (\mu + \phi f_\alpha d) I_r \\[2pt] (\mu + \gamma W/K_T) T \end{bmatrix}, \end{array} \end{aligned} $$

where $W = T+L_s+L_r+I_s+I_r$ and $F_\phi = (1-\phi (1-\bar {f}_\alpha ))$. The following conditions need to be verified:

• (A1) If $\mathcal {X}\geq 0$, then $\mathcal {F}_i$, $\mathcal {V}^-_i$, $\mathcal {V}^+_i\geq 0$, for $i=1,\dots , 7$.

• (A2) If $\mathcal {X}=0$, then $\mathcal {V}^-_i(\mathcal {X})=0$.

• (A3) $\mathcal {F}_i=0$ if $i>6$.

• (A4) $\mathcal {F}_i(\mathcal {E}_0)=0$ and $\mathcal {V}^+_i(\mathcal {E}_0)=0$ for $i=1\dots 6$.

• (A5) If $\mathcal {F}(\mathcal {X})$ is set to zero, then all eigenvalues of $Df(\mathcal {E}_0)$ have negative real parts, where $Df(\mathcal {E}_0)$ represents the Jacobian matrix about $\mathcal {E}_0$.

Conditions (A1)–(A4) are easily verified. To verify Condition (A5), we need to calculate the Jacobian evaluated at the DFE.

$$\displaystyle \begin{aligned} \mathcal{D}f(\mathcal{E}_0) =\begin{bmatrix} J_1 & 0 \\ J_2 & J_3 \end{bmatrix}, \end{aligned}$$

where

$$\displaystyle \begin{aligned} \begin{array}{rcl} J_1 & =&\displaystyle \begin{bmatrix} -c &\displaystyle d \bar{f}_\alpha N &\displaystyle 0 &\displaystyle 0 \\[2pt] (1-\sigma) k_s T^* & -(d \bar{f}_\alpha+\mu) &\displaystyle (1-\sigma m_{sr}) \chi &\displaystyle 0 \\[2pt] \sigma k_s T^* & 0 &\displaystyle -(\chi+\hat{\mu}) &\displaystyle 0 \\[2pt] 0 & 0 &\displaystyle 0 &\displaystyle -c &\displaystyle d N F_\phi \\[2pt] 0 & 0 &\displaystyle \chi m_{sr} &\displaystyle (1-\sigma) k_r T^* \end{bmatrix}, \\ J_2 & =&\displaystyle \begin{bmatrix} 0 &\displaystyle 0 &\displaystyle 0 &\displaystyle \sigma k_r T^* \\[2pt] -k T^* & -\frac{\gamma T^*}{K_T} &\displaystyle -\frac{\gamma T^*}{K_T} &\displaystyle -k_r T^* \\ \end{bmatrix}, \mbox{ and } \\ & &\displaystyle J_3 = \begin{bmatrix} -d F_\phi-\mu &\displaystyle \chi &\displaystyle 0 \\[2pt] 0 & -\chi-\hat{\mu} &\displaystyle 0 \\[2pt] -\frac{\gamma T^*}{K_T} & -\frac{\gamma T^*}{K_T} &\displaystyle -\frac{2 \gamma T^*}{K_T}+\gamma-\mu \\ \end{bmatrix}. \end{array} \end{aligned} $$

The effective reproduction numbers, $\mathcal {R}_e^s$ and $\mathcal {R}_e^r$, are found by taking the maximum eigenvalues of the next generation matrix, given by $D\mathcal {F} (D\mathcal {V})^{-1}$.

The characteristic equation related to $\mathcal {D}f(\mathcal {E}_0)$ is

$$\displaystyle \begin{aligned} \biggl(\gamma \left(1 - 2 \frac{T^*}{K_T}\right) - \mu - \lambda\biggr) P_s(\lambda) P_r(\lambda)=0, \end{aligned}$$

where $P_s(\lambda )$ and $P_r(\lambda )$ are both third-order polynomials. Solving the first term for $\lambda$, we find the eigenvalue $\lambda =-\sqrt {\frac {4 \beta \gamma }{K_T}+(\gamma -\mu )^2}$, which has no positive real part. The polynomial, $P_s(\lambda )$, is

$$\displaystyle \begin{aligned} P_s(\lambda) = A_s + B_s \lambda + C_s \lambda^2 + \lambda^3, \end{aligned}$$

with the coefficients

$$\displaystyle \begin{aligned} \begin{array}{rcl} A_s & =&\displaystyle c (\chi+\hat{\mu}) (d \bar{f}_\alpha +\mu) (1-\mathcal{R}_e^s), \\ B_s & =&\displaystyle (\chi + \hat{\mu}) (d \bar{f}_\alpha +\mu) + c (\chi + d \bar{f}_\alpha +\mu + \hat{\mu}) - (1-\sigma) d \bar{f}_\alpha k_s N T^*, {}\\ C_s & =&\displaystyle c + \chi + d \bar{f}_\alpha + \mu + \hat{\mu}. \end{array} \end{aligned} $$

(9)

The coefficient $C_s$ is always positive, and the coefficient $A_s$ is positive when $\mathcal {R}_e^s<1$. The coefficient $B_s$ requires a little more effort to show it is always positive. We note that

$$\displaystyle \begin{aligned} d \bar{f}_\alpha k_s N T^* = \frac{(\chi+\hat{\mu})}{((1-\sigma) \chi + (1-\sigma m_{sr}) \hat{\mu})} (c (d \bar{f}_\alpha +\mu)\mathcal{R}_e^s) \geq c (d \bar{f}_\alpha +\mu)\mathcal{R}_e^s. \end{aligned}$$

Thus, we can substitute this inequality into the coefficient $B_s$ to find

$$\displaystyle \begin{aligned} B_s \geq (\chi + \hat{\mu}) (d \bar{f}_\alpha +\mu + c) + c (d \bar{f}_\alpha + \mu) (1-\mathcal{R}_e^s). \end{aligned}$$

Therefore $B_s$ is also always positive when $\mathcal {R}_e^s<1$. Then by Descartes’ law of signs, since the real polynomial $P_3(\lambda )$ has zero sign changes in the sequence of its nonzero coefficients, then it has zero roots with a positive real part. The polynomial $P_r(\lambda )$ is

$$\displaystyle \begin{aligned} P_r(\lambda) = A_r + B_r \lambda + C_r \lambda^2 + \lambda^3, \end{aligned}$$

with the coefficients

$$\displaystyle \begin{aligned} \begin{array}{rcl} A_r & =&\displaystyle c (\chi+\hat{\mu}) (d F_\phi +\mu) (1-\mathcal{R}_e^r), \\ B_r & =&\displaystyle (\chi + \hat{\mu})(d F_\phi +\mu) + c (\chi + d F_\phi +\mu + \hat{\mu}) - (1-\sigma) d F_\phi k_r N T^*, {}\\ C_r & =&\displaystyle c + \chi + d F_\phi + \mu + \hat{\mu}). \end{array} \end{aligned} $$

(10)

Similarly we can show that

$$\displaystyle \begin{aligned} B_r \geq (\chi + \hat{\mu}) (d F_\phi +\mu + c) + c (d F_\phi+ \mu) (1-\mathcal{R}_r^s). \end{aligned}$$

Therefore, the real roots of $P_r(\lambda )$ have only negative parts when $\mathcal {R}_e^r<1$. This shows condition (A5) holds. It follows from Theorem 2 in [63] that $\mathcal {E}_0$ is locally asymptotically stable when $\max \{{\mathcal R}_e^s, {\mathcal R}_e^r\}<1$ and unstable when $\max \{{\mathcal R}_e^s, {\mathcal R}_e^r\}>1$. □

6 Parameter Sensitivity

In modeling, equations usually depend on several unknown parameters. Finding the appropriate values and ranges is a critical step that is useful for screening outliers and as such requires careful consideration. In general, these values can be estimated through the least-squares fitting, Bayesian inference, or maximum likelihood. The ranges of the parameters can also be determined by setting upper and lower bounds on the values, either based on physical constraints or by exploring the parameter space through sensitivity analysis.

Previous literature has stated that understanding the peak of viral load data is vital to best predict T-cell latency and infection outcomes [64]. Thus, to best calibrate our HIV and SHIV models, we compared our model’s outputs to the viral load data extracted from Reeves et al. [38] and Vaidya et al. [39] for humans and macaques, respectively, using a simulated annealing technique in MATLAB. Simulated annealing is a local optimization technique that takes a user inputted initial guess for the global minimum of the system as defined by the user; for the purposes of this study, the smallest least squared error between the model predictions and the extracted viral load data [65]. Each parameter, in this case $c,$ the clearance rate of free virions and $N,$ the number of virions produced per infected T-cell, is given a lower and upper bound. These two parameters are chosen due to their direct, mathematical correlation with peak viral load. The simulated annealing algorithm then generates a random value within each boundary, calculates the user described error, and then generates a new random value to see if the error increases or decreases. This process is repeated until a local minimum is found and parameter values are produced [66]. We found when we performed this operation for the DS virion model, the resulting macaque values for N and c could not be matched to the experimental data. This indicates that the DR virions should not be discounted when fitting parameters.

6.1 Elasticity of the Effective Reproduction Number $\mathcal {R}_E$

In this section we test the sensitivity of the reproduction number, $\mathcal {R}_E$, to its parameters. We compute the elasticity (normalized forward sensitivity) index [67] to determine to what extent the value of $\mathcal {R}_E$ (for the macaque and human models) changes following changes to each parameter value. The sensitivity index with the reproduction number indicates the impact of the parameter on the disease-free equilibrium. Since $\mathcal {R}_E =\max \left (\mathcal {R}_E^s, \mathcal {R}_E^r \right )$, we compute the sensitivity with respect to both $\mathcal {R}_E^s$ and $\mathcal {R}_E^r$. The forward sensitivity indices for these parameters are represented by

$$\displaystyle \begin{aligned} {} \mathcal{F}_x =\left(\frac{\partial \mathcal{R}_E^i}{\partial x}\right) \left(\frac{x}{\mathcal{R}_E^i} \right), \end{aligned} $$

(11)

where x represents the parameter and i is s for sensitive and r for resistant. These forward sensitivity indices were evaluated using the baseline parameters given in Tables 2 and 3, except for $f_\alpha$ and $\phi$. We replace the time-varying value of $f_\alpha$ with a time-average of 0.01. $\phi$, which is between 0 and 1, and we set equal to +0.5. The elasticity results are shown in Table 4.

Table 2 Parameters (Pa.) of the model (Human)

Table 3 Parameters (Pa.) of the model (Macaque)

Table 4 Elasticity indices of $\mathcal {R}_E$ for HIV model evaluated at baseline human HIV parameters shown in Table 2 and baseline macaque SHIV parameters shown in Table 3, except for $f_\alpha$ and ${\phi }$

The positive sign of the elasticity index specifies that $\mathcal {R}_E^i$ increases with the parameter, and the negative sign specifies that $\mathcal {R}_E^i$ decreases. The magnitude of the elasticity determines the relative importance of the parameter. If $\mathcal {R}_E^i$ is given explicitly, then the elasticity index for each parameter can be explicitly computed and evaluated for a given set of parameters. The magnitudes of the elasticity indices depend on these parameter values.

For our model, we calculated elasticity indices for the 15 parameters, with the values for human set of parameters and macaque set given separately in Table 4. Although the parameter sets for humans (see Table 2) and macaques (see Table 3) are different, the response the basic reproduction number gives qualitatively and quantitatively is quite similar for both the resistant and sensitive strains, with $f_\alpha$ as the outlier. However, the behavior of $f_\alpha$ is to be expected; each index value can be thought of as a ratio of the effective change in the reproduction number with respect to the applied change in the given parameter. For example, for every $10\%$ increase in the infection rate of T-cells, $k_s$ or $k_r$, the reproduction number will increase by $5\%$. However, it will decrease by $5\%$ for every $10\%$ increase in the proliferation rate of T-cells, $\gamma$. When $f_\alpha$ increases, CAB-LA is less effective in blocking HIV infections. Hence, for the DS infections, the number of infections will rise. On the other hand, as the DS strain increases when $f_\alpha$ increases, the DR strain is outcompeted by the DS strain.

In terms of the effects of drug resistance, we note that the effective reproduction numbers, $\mathcal {R}_E^r$ for macaques and humans is twice as sensitive to the reduction in effectiveness of CAB-LA, represented by $\phi$ as it is to the fitness of the DR virions to infect a healthy T-cell, $k_r$.

6.2 Global Sensitivity

Sensitivity analysis is important for determining which parameters have the largest impact on the dynamics of the spread of HIV. Following [84], we employ partial rank correlation coefficient (PRCC) analysis to determine the sensitivity of the model, given by the system defined in Eq. (2), to each parameter for humans and macaques after 48 weeks of PrEP. In this instance, correlation provides a measure of the strength of a linear association between a parameter and the number of virions and infected T-cells. Rather than capturing the sensitivity of the total number of infected T-cells or virions to a single parameter at a time, partial correlation analysis reveals hidden true correlations and false correlations explained by the effect of other variables. The parameters, specified in Tables 2 and 3, are sampled using Latin hypercube sampling (LHS) [85]. LHS/PRCC sensitivity analysis is often employed in uncertainty analysis to explore the entire parameter space of a model.

The magnitude of the PRCC indicates the strength of the correlation between the parameter and the output, whereas the sign of the PRCC indicates whether there is a positive or negative correlation between the parameter and the output—the total number of infected T-cells or virions in this case.

In Fig. 7 the PRCC results are shown first for humans given PrEP (CAB-LA) before HIV exposure and second given PrEP 2 weeks after seroconversion. The black bars represent $L_s+I_s$, the gray bars $L_r+I_r$, the horizontal striped bars $V_s$, and the diagonal striped bars $V_r$. The parameter sensitivity for the human model does not change dramatically between the PrEP before HIV exposure and the PrEP after HIV exposure. The infected T-cells, both with the DS and DR strains, are sensitive to $k_s$ the infection rate of target cells, $\mu$ the natural death rate of T-cells, and $\chi$ the transition rate of target cells from latently infected to infected. The DR-infected T-cells and both types of virions are sensitive to the viral-induced death rate of infected cells, although the number of infected T-cells decreases as d increases and the number of virions increases as the infected cell bursts. Both classes of virions are more strongly sensitive to N, the number of virus produced per burst infected cell. When PrEP is given after HIV seroconversion, the infected T-cells become more sensitive to N, but less sensitive to c the clearance rate of virus, $f_\alpha$ the ability of CAB-LA to block HIV, and $k_r$ the infection rate of target cells by the resistant virions. The parameter sensitivity of the virions does not appear to change from the PrEP before HIV exposure to the PrEP after HIV infection scenarios.

Fig. 7

PRCC results for the Human HIV Model after 200 days of PrEP. The black bars represent the T-cells infected with the DS virus, $L_s+I_s$, the gray bars represent the T-cells infected with the DR virus, $L_r+I_r$, the horizontal striped bars represent the DS virions, $V_s$, and the diagonal striped bars represent the DR virions, $V_r$. The label DS indicates the drug-sensitive strain, and DR indicates the drug-resistant strain. (a) Human: PrEP before HIV exposure. (b) Human: PrEP after HIV infection

In Fig. 8 the PRCC results are shown first for macaques given PrEP before HIV exposure and second given PrEP 2 weeks after seroconversion in accordance with the experiment in [30]. The black bars represent $L_s+I_s$, the gray bars $L_r+I_r$, the horizontal striped bars $V_s$, and the diagonal striped bars $V_r$. For PrEP given as a preventative measure to the macaques, both the DS-infected and infectious T-cells and the DS virions are most sensitive to $k_s$ the infection rate of target cells, $\mu$ the natural death rate of T-cells, and c the clearance rate of the virus. The cells infected with the DR strain are most sensitive to $\chi$ the transition rate of target cells from latently infected to infected and $\sigma$ the fraction of “stronger” cells moved to latently infected. All the infected T-cell types and virions are equally sensitive to c, the clearance rate of the virus. This sensitivity does not change in the situation when PrEP is given before SHIV exposure or after SHIV seroconversion. Only the DS-infected T-cells are sensitive to CAB-LA, designated by $f_\alpha$.

Fig. 8

PRCC results for the Macaque SHIV Model after 200 days of PrEP. The black bars represent the T-cells infected with the DS virus, $L_s+I_s$, the gray bars represent the T-cells infected with the DR virus, $L_r+I_r$, the horizontal striped bars represent the DS virions, $V_s$, and the diagonal striped bars represent the DR virions, $V_r$. The label DS indicates the drug-sensitive strain, and DR indicates the drug-resistant strain. (a) Macaque: PrEP before SHIV exposure. (b) Macaque: PrEP after SHIV infection

In the situation when PrEP is given to the macaques 2 weeks after seroconversion, $\chi$ and $\sigma$ become sensitive for the infected T-cells and their accompanying virions. The infected T-cells become less sensitive to CAB-LA, but the virions become strongly more sensitive to changes in the ability of CAB-LA to block SHIV.

In terms of the effects of drug resistance, we note that the concentration of infected T-cells and virions are sensitive to the fitness of the DR virions to infect a healthy T-cell, $k_r$. However, the concentration of infected T-cells and virions are not sensitive to the reduction in effectiveness of CAB-LA, represented by $\phi$. The parameter $\phi$ does not appear on the PRCC bar charts since it has a p number greater than 0.5 and an insignificant PRCC.

The forward sensitivity and the global sensitivity appear to give contradictory results with regard to the reduction in effectiveness of CAB-LA toward the DR HIV strain, represented by $\phi$. However, these analyses are measuring the sensitivity of two different situations. The forward sensitivity measures how changing the parameter value will change the effective reproduction number, which indicates the sensitivity of the disease-free equilibrium to parameter changes. The forward sensitivity indicates that to achieve a disease-free state increasing the effectiveness of CAB-LA for the drug resistance will be twice as effective as reducing the fitness of the DR virions to infect a healthy T-cell, $k_r$. But in the global sensitivity after 200 days of PrEP, the reduction of effectiveness of CAB-LA against the DR strain from a baseline of $\phi =1/2$ is statistically insignificant to the concentration of infected T-cells and the number of virions. On the other hand, both sensitivity analyses indicate that the fitness of the DR virions to infect a healthy T-cell, $k_r$, is very important to achieving a disease-free equilibrium and after 200 days of PrEP. This indicates that the level of drug resistances and the fitness of the DR virions to infect are both very important to the effectiveness of CAB-LA as PrEP.

7 Treatment Simulations

In our simulations, we are modeling HIV or SHIV infection from the first onset and continuing just through the acute infection stage. We do not include antiretroviral treatment (ART) in our model. In the wild-type systems, we solely have the DS (CAB-LA sensitive) strain of HIV or SHIV, and we assume that there are no mutations occurring, i.e., $m_{sr}=0$. In the full model, we assume mutations can occur naturally, and this is how drug-resistant HIV or SHIV strains are introduced into the system. In other words, the SHIV or HIV exposures or initial strains upon seroconversion are assumed to be fully drug-sensitive.

In their paper describing their experimental results, Radzio-Basu et al. [30] emphasized that the experimental results for macaques administered PrEP during the acute SHIV infection indicated a strong emergence of drug resistance. In response, the FDA decided to require HIV tests prior to each CAB-LA dose for humans [31]. While caution to avoid the development and propagation of more drug-resistant strains of HIV is commendable, the question still remains: Does this development of drug resistance in macaques truly foretell the similar development of drug resistance in humans? This is a question that is difficult to answer with a dearth of clinical human data, but we are attempting to answer it with our mathematical model and numerical simulations. Hence the goal of this research is to establish a model that can capture the mechanistic behavior of SHIV and HIV virions and their interaction with healthy CD4${ }^+$ T-cells.

Toward this aim in this section, we validate our model by comparing our simulated results to experimental data for humans and macaques. We use the term validation in the sense of the National Academy of Sciences report [86] where validation is defined to be the process of determining the degree to which a model is an accurate representation of the real world from the perspective of the intended uses of the model. We note that we have not incorporated formal verification or validation methods in this chapter. Instead, our validation is restricted to illustrating graphically that the simulations match experimental results.

In Sect. 7.1 we present our numerical simulations and compare these to experimental data for humans and macaques without any HIV treatment or preventative measures, i.e., without ART or PrEP. In Sect. 7.2 we perform our numerical simulation for macaques administered PrEP before being exposed to SHIV and compare the results to the experiment in [40]. In Sect. 7.3 we perform our numerical simulation for macaques administered PrEP before being exposed to SHIV and compare the results to the experiment in [30].

Once behaviors of SHIV in macaques in pre-clinical settings were captured in silico in Sects. 7.1–7.3, human in silico clinical trials were run with standard of care in PrEP injection protocols in Sects. 7.4 and 7.5. To see the similarities and differences in the macaque and human simulations, we test several values for the effectiveness of CAB-LA on the DR strain and on the fitness of the DR virions to infect healthy T-cells.

7.1 Macaques and Humans: Numerical Validation Without ART or PrEP

We first plot our model simulations against data so that we may validate our model prior to running in silico pre-clinical and clinical trials. Our system represents wild-type HIV transmission without mutations, treatment for either post-infection ART, or pre-infection PrEP.

Figure 9a displays experimental viral loads from [38] in light gray with marker symbols. The experimental results for each individual human are differentiated by the different markers. The simulated viral loads are plotted as the dashed black curve. In the early stages, viral loads rapidly increase from $10^3$ RNA copies per mL to $10^6$ or higher, then decline, and reach a plateau around $10^4$ to $10^6$. The healthy T-cell concentration, represented by the solid black curve, rises slightly in the first 10 days, then drops to around $30\%$ shortly after maximum viral load, then raises to $40\%$, and stays stable. The actively infected T-cells, represented in black dash-dot curve, increase from 0 to about 400 cells per $\mu \mathrm {L}$ at the maximum viral load, then decline slowly and remain at 250 cells per $\mu \mathrm {L}$ after 40 days. The latently infected T-cells, represented by the black dotted curve remain low throughout the simulation. The combined T-cell concentration (healthy, latently infected, and actively infected) drops by about $35\%$ during this infection process. Overall, wild-type HIV model simulation viral load (DS) results match with the experimental data for humans without ART or PrEP.

Fig. 9

Simulation results for drug-sensitive only ($m_{sr} = 0$) infection model given by the system defined in Eq. (2) without ART or PrEP and experimental viral load data. In (a) the simulated human (HIV) viral load represented by black dash curve is compared to experimental viral load data from [38], shown in light gray with each individual recorded with a separate symbol. In (b) the simulated macaque (SHIV) viral load, represented by dash black curve, is compared to experimental viral load data from [41] given in light gray

Figure 9b displays the comparison between simulation results and experimental data, light gray with marker symbols, in Dobard et al. [41] for macaques. The experimental results for each individual macaque are differentiated by the different markers. The simulated viral loads are plotted as the dashed black curve. In the Dobard et al. [41] experiment the macaques are given a weekly SHIV challenge for the first 3 weeks of the experiment. This exposure, included in our simulation, is captured in Fig. 9b by the three vertical lines on the virus level. In the early stages, viral loads rapidly increase from $10^4$ RNA copies per mL to a plateau of $10^8$. This is on the higher end of the experimental viral loads in macaques. The healthy T-cell concentration, represented by the solid black curve, drops over 8 weeks, much slower than in the human simulation shown in Fig. 9a. The actively infected T-cells, represented by the black dash-dot curve, increase from 0 to about 200 cells per mL at the maximum viral load. The latently infected T-cells, represented by the black dotted curve, remain low throughout the simulation. Overall, wild-type SHIV model simulation viral load (DS) results match with the experimental data for macaques without ART or PrEP.

Once our model’s sensitive viral load output was calibrated, we varied $\phi$, the percentage of drug efficacy on the resistant strain. In addition, to capture a measure of the fitness of the mutated DR strain, we explore the relationship between $k_r$ the infection rate of healthy T-cells by DR virions and $k_s$ the infection rate of healthy T-cells by DS virions. These tests are conducted to better understand the dynamics of the DR viral load. The parameters being varied—$\phi$, the effectiveness of CAB-LA for the DR strain, and $k_r$, the fitness of DR virions to infect a T-cell—are not known in advance. The amount of drug resistance to CAB-LA of the mutated HIV strain is of utmost concern. Parikh et al. [53] note that multiple mutations are required for extensive CAB drug resistance. However, the macaque studies by Radzio-Basu et al. [30] indicate that these mutations are selected when CAB-LA is given to macaques previously infected with SHIV. The fitness of the DR virions to infect is included as parameters to vary. It is often the case that the mutation that makes the HIV strain at least partially drug-resistant may also trade off its ability to invade a T-cell.

In Fig. 10, we show simulation results of the full mutation model without ART and PrEP. For both human and macaques, we set the virus mutation rate from the wild-type DS strain to the mutated DR strain to be $m_{sr} = 10^{-5}$, based on [42]. In Fig. 10a, b, the healthy T-cells are represented by solid black curves. The latently infected and actively infected by the DS strain are combined and represented by black dash-dot curves and the DR by the black dashed curves. The DS virions are represented by the gray dotted curves and the DR virions by the black dotted curves.

Fig. 10

Simulation results for (a) human and (b) macaque model with mutations ($m_{sr} \neq 0$) given by the system defined in Eq. (2) without ART or PrEP. Each of the curves is labeled to designate the T-cells in various stages plus the virions. The label DS indicates the drug-sensitive strain, and DR indicates the drug-resistant strain. Solid curves: healthy T-cells. Dash-dot curve: actively infected T-cells. Dash curve: latently infected T-cells. Dark gray: DS, light gray: DR. Dark gray dotted curve: DS virus. Light gray dotted curve: DR virus. The concentration of DR infected T-cells and latently infected T-cells is close to zero in both humans and macaques

In humans, the concentrations of healthy, actively, and latently DS-infected T-cells and DS virus have almost identical behaviors in wild-type only ($m_{sr}=0$) and mutation ($m_{sr}\neq 0$) models during the first 10 weeks. In both Figs. 9a and 10a, the concentration of healthy T-cells changes from 1000 cells per $\mu \mathrm {L}$ to a lowest point, about 300, in 4 weeks and then increases slowly to reach the plateau in 3 weeks. The concentrations of actively DS-infected T-cells increase slowly during the 1st week and then quickly until it reaches its peak at the end of the 4th week. They slowly drop back to a steady state around the 6th week. The DS virus concentration reaches a small peak during the 4th week of infection, then drops slightly, and reaches a steady state. The concentrations of latently DS-infected T-cells and both types of DR-infected T-cells are negligible throughout the simulation.

In macaques, when comparing Figs. 9b and 10b, we notice concentrations of healthy T-cells stay steady around 1680 cells per $\mu \mathrm {L}$ for about 60 days, then drop to 1400 in next 20 days, and remain at that level. The concentration of actively DS-infected T-cells is steady and small and then increases during the same period when the concentration of healthy T-cells drops and then stays at a steady state. The DS virus concentration grows at an exponential rate during the first 60 days after infection and also stays flat afterward. The concentrations of latently DS-infected T-cells and both types of DR-infected T-cell concentrations stay low throughout the simulation.

7.2 PrEP Before SHIV Exposure: Numerical Validation for Macaque Experiment

In Andrews et al. [40], there were multiple experiments with three groups of macaques given three different PrEP regimes before being exposed to SHIV intravenously. The study was intended to evaluate the effectiveness of CAB-LA (as PrEP) against intravenous SHIV challenge. This macaque experimental model was intended to determine whether human studies for CAB-LA as PrEP in people who inject drugs are warranted.

We simulate the experiment where eight macaques are treated with PrEP twice, 4 weeks apart. There were also five control macaques who did not receive PrEP. On day 14, all the macaques are infected with SHIV. In the simulations we vary $\phi$ between 0.3 and 0.7 and allow $k_r=k_s$ or $k_r=k_s/2$. The parameter sensitivity studies indicated that the concentration of infected cells and virions are sensitive to the fitness of the DR virions to infect. The elasticity studies with the effective reproduction number for the DR strain indicated that there is sensitivity to the effectiveness of CAB-LA on the DR virions. We observe that the DS viral loads do not have noticeable changes, but there are obvious changes in the DR viral load. In Fig. 11, we present the experimental data of [40] with the control macaques (no PrEP) and the one macaque with a virion/mL count above 1. The other seven macaques given PrEP had virion concentrations of less than 1, but the exact virion concentration was not given. Also in Fig. 11 we display the numerical simulations for the DS viral load (solid black curve) and the two most extreme cases of the DR viral load (black dotted curve for $\phi = 0.3, k_r = k_s$, and black dotted-dashed curve for $\phi = 0.7, k_r = k_s/2$). Despite varying both the values of $\phi$ and $k_r$, the amount of DR virions is negligible when compared to DS virions. The combined viral load is almost identical to the DS viral load. As shown in Fig. 11, the DS viral load solid black curve is comparable to the one macaque received PrEP black dashed curve with triangles plus the seven macaques with even lower virion concentrations. The gray curves with crosses represent the viral load data in the control macaques who did not receive PrEP.

Fig. 11

PrEP injection on days 1 and 28, and SHIV infection on day 14. Macaque SHIV infection viral load data with PrEP treatment [40] compared with numerical simulations. The label DS indicates the drug-sensitive strain, and DR indicates the drug-resistant strain. Solid curve: simulation results for DS virus. Black dot and dash-dot curves: simulation results for DR virus with two most extreme levels of fitness and drug effectiveness. Light gray dash curves with crosses: experimental data for one macaque given PrEP. All other experimental data for macaques given PrEP had virion levels of less than 1 virion/mL and are not shown in this figure. Dark gray dash curves with triangles: experimental data for control macaques (no PrEP)

7.3 PrEP After SHIV Infection: Numerical Validation for Macaque Experiment

Radzio-Basu et al. [30] performed experiments infecting eight rhesus macaques intravenously with SHIV. Two of the macaques were used as controls. The other six macaques were given a treatment of CAB-LA 11 days after infection. The experiment was devised to test what would happen when PrEP was initiated during an acute HIV infection. In their experiments, Radzio-Basu et al. [30] found that DR mutations were frequently selected and maintained for several months.

In the Radzio-Basu et al. [30] experiment, the macaques were infected with SHIV on day 1 and then receive PrEP injections on days 11, 39, and 67. We replicated the experiment and then tested several levels of effectiveness of CAB-LA against the DR mutations and fitness for the DR virions to infect T-cells.

We run simulations by varying the effectiveness of CAB-LA, $\phi$, between 0.3 and 0.7. To test situations for fitness, we added two scenarios allowing $k_r=k_s$ for no loss of fitness or $k_r=k_s/2$, which corresponds to a 50% loss of fitness. Similar to Fig. 11, we observe that the DS viral loads do not have an obvious change, but there are noticeable changes in the DR viral load. In Fig. 12, we plot our numerical simulations for DS viral load (the solid black curve). We only plot the two most extreme cases of DR viral load with dotted curve for $\phi = 0.3, k_r = k_s$ and the black dash-dot curve for $\phi = 0.7, k_r = k_s/2$. The experimental data of [30] are included in Fig. 12. The dark gray dash curves with triangles represent viral load in macaques without PrEP, and the light gray curves with crosses illustrate the viral load in macaques given PrEP. It is clear to see that the sensitive viral load (solid black curve) agrees with the experimental observations. The amount of DR virions is negligible in comparison to the magnitude of DS virions. There is a noticeable difference between the simulated DR viral load when the DR drug effectiveness and virion fitness $\phi$, $k_r,$ and $k_s$ are varied.

Fig. 12

PrEP injections on days 11, 39, and 67 after SHIV infection. Macaque SHIV infection viral load data with PrEP treatment [30] compared with numerical simulations. The label DS indicates the drug-sensitive strain, and DR indicates the drug-resistant strain. Solid curve: simulation result for DS virus. Dot and dot-dash curves: DR virus with two levels of fitness and drug effectiveness. Light gray dash curves with triangles: experimental data for macaques given PrEP. Dark gray curves with crosses: experimental data for control macaques (no PrEP)

7.4 Macaque and Human Simulations: PrEP Before Exposure

Once we observed the DR viral load changes due to the efficacy of the PrEP drug, $\phi$, and due to the relationship between the infectivity of DR virions, $k_r$, and DS virions, $k_s$, we decided to run our own in silico pre-clinical trials in macaques to better understand the role of PrEP injection campaigns on the infectivity of DR and DS SHIV.

First, we designed a simulated trial for an extended version of the experiment in [40] (see Fig. 11) where there macaques were administered an injection of PrEP on day 0, exposed to SHIV on day 14, and then were given a second injection of PrEP on day 28 as shown in Fig. 13. We varied the value of the drug effectiveness against the mutated DR SHIV strain, $\phi$, between 30% (top row) and 70% (bottom row). We also set the fitness of the DR virions to infect healthy T-cells to equal the fitness of the DS virions, $k_r=k_s$ (first column), i.e., DR virions are equally as infective as DS. Next we set the DR virions to half of the fitness, $k_r= k_s/2$ (second column). We allow the in silico trial to last for 24 weeks to allow the system dynamics enough time to respond to the PrEP treatment and SHIV infection.

Fig. 13

Macaque in silico trial corresponding to the experiment in [40]. PrEP is administered on days 0 and 28. The macaque is exposed to SHIV on day 14. The label DS indicates the drug-sensitive strain, and DR indicates the drug-resistant strain. (a) PrEP is 30% effective against SHIV ($\phi =0.3$). DR virions are equally as infective as DS virions, thus $k_r=k_s$. (b) PrEP is 30% effective against SHIV ($\phi =0.3$). DR virions are half as infective as DS ($k_r=k_s/2$). (c) PrEP is 70% effective against SHIV ($\phi =0.7$). DR virions are equally as infective as DS ($k_r=k_s$). (d) PrEP is 70% effective against SHIV ($\phi =0.7$). DR virions are half as infective as DS ($k_r=k_s/2$). Solid curves: healthy T-cells. Dash-dot curve: actively infected T-cells. Dash curve: latently infected T-cells. Dotted curve: virus. Dark gray: DS, light gray: DR

We note that in all four of these simulations with both the DS and DR strains of SHIV, the healthy T-cell concentrations (black solid curves) dip significantly when the CAB-LA concentration is very low, 27 weeks after the second injection of PrEP. The drop in the healthy T-cell concentration follows the rise in DS virions in each of Fig. 13a–d. The significant difference between the subfigures is the level of DR virions. In Fig. 13a, where CAB-LA is only 30% effective against the DR strain and the DR virions are just as infective as the DS virions, the DR virion level rises above $10^2$ vRNA copies per mL. It does not outpace the DS virions, but it is at a level of concern. In Fig. 13c where the effectiveness of CAB-LA against the DR strain has risen to 70% with equal fitness between the DS and DR virions, there is a small increase in the DR virions to about 1 vRNA copy per mL. In Fig. 13b, d where the fitness of the DR virions is a half of the DS virions, the DR strain remains at a negligible level. This indicates that the virion fitness may be important as the drug efficacy in determining whether a mutated strain will grow to a significant level.

Once behaviors of SHIV in macaques in pre-clinical settings were captured in silico, human in silico clinical trials were run with standard of care PrEP injection protocols. Current standard of care for human injectable PrEP is two injections, 1 month apart, followed by an injection continually, every 2 months [28].

In Fig. 14a, where CAB-LA is only 30% effective for the DR strain and the fitness of the DR and DS virions is equal, the mutated DR strain overtakes the DS strain even before the second PrEP injection is administered. In Fig. 14b, where CAB-LA is 70% effective for the DR strain and the fitness of the DR and DS virions is equal, the DR strain takes longer, after the second administration of PrEP on week 18, to overtake the DS strain. However, in the scenario with $\phi =0.70$ and $k_r=k_s$ (Fig. 14c), the DR strain is already growing too much to be controlled by a third PrEP injection on week 19 (following the once a month PrEP protocol for macaques). The situation changes if the DR virion fitness is lowered to 50% of the DS virion fitness. In that case, the healthy T-cell count is maintained at a high level for both drug effectiveness levels as shown in Fig. 14b, d for the duration of the in silico trial. However, of these two cases, if a third PrEP injection were administered on week 19, the HIV seroconversion would be avoided if the drug were 70% effective against DS virions, but not if the drug were only 30% effective. This indicates that the virion fitness, along with the drug efficacy, is important in determining whether a mutated strain, and at what time, will overpower PrEP. About 50% fitness could have caused this seroconversion.

Fig. 14

Human in silico trial. PrEP is administered on days 0 and 28 and then every 2 months. The human is exposed to HIV on day 14. The label DS indicates the drug-sensitive strain, and DR indicates the drug-resistant strain. (a) PrEP is 30% effective against HIV ($\phi =0.3$). DR virions are equally as infective as DS virions ($k_r=k_s$). (b) $\phi = 0.3$. DR virions are half as infective as DS ($k_r=k_s/2$). (c) $\phi =0.7$, $k_r=k_s$. (d) $\phi =0.7$, $k_r=k_s/2$. Solid curves: healthy T-cells. Dash-dot curve: actively infected T-cells. Dash curve: latently infected T-cells. Dotted curve: virus. Dark gray: DS, light gray: DR

7.5 Macaque and Human Simulations: PrEP After Infection

Next, we implemented the PrEP injection protocol of [30], where SHIV exposure occurred on day 0, and then macaques were given PrEP on days 11, 39, and 67, as shown in Fig. 12. This in silico pre-clinical trial then lasted for 30 weeks to follow the protocol as described in [30]. We similarly varied $\phi$ between 30% efficacy (top row) and 70% efficacy (bottom row) and allowed $k_r=k_s$ (first column) and $k_r=0.5 \times k_s$ (second column).

We first observe in Fig. 15 that the healthy T-cell concentration, represented by the black solid curve, stays high through this whole experiment. The DS virions, represented by the black dotted curve, react to each of the PrEP injections, but not enough to reduce the overall DS virion level. In each of the in silico trials, the initial SHIV dose is $10^4$ vRNA copies per mL and grows to $10^5$ vRNA copies per mL by the end of 30 weeks. What does change, just as in Fig. 13 based on the Andrews et al. [40] experiment, is the concentration of the DR virions, represented by the gray dotted curve. In Fig. 15a where the CAB-LA is only 30% effective against the DR strain and the DR and DS virions have equal fitness, the DR virions grow to a concentration of $10^3$ and are still rising. Note that in macaques, experiments [30] have measured a mean of virion concentration level of $10^4$ RNA copies/mL at seroconversion. So this rise in DR virions is significant. In Fig. 15c where the CAB-LA is now 70% effective against the DR strain and the DR and DS virions still have equal fitness, the concentration of DR virions is about 10 vRNA copies per mL. In Fig. 15b, d where the DR virions are 50% as infective as the DS virions, the DR virions have a concentration of about $10^{-4}$ at the end of 30 weeks despite CAB-LA being 30% or 70% effective in the two subfigures.

Fig. 15

Macaque in silico trial corresponding to the experiment in [30]. The macaque patient is exposed to SHIV on day 0, and the PrEP injection is administered on days 11, 39, and 67. The label DS indicates the drug-sensitive strain, and DR indicates the drug-resistant strain. (a) PrEP is 30% effective against SHIV ($\phi =0.3$). DR virions are equally as infective as DS virions, thus $k_r=k_s$. (b) PrEP is 30% effective against SHIV ($\phi =0.3$). DR virions are half as infective as DS ($k_r=k_s/2$). (c) PrEP is 70% effective against SHIV ($\phi =0.7$). DR virions are equally as infective as DS ($k_r=k_s$). (d) PrEP is 70% effective against SHIV ($\phi =0.7$). DR virions are half as infective as DS ($k_r=k_s/2$). Solid curves: healthy T-cells. Dash-dot curve: actively infected T-cells. Dash curve: latently infected T-cells. Dotted curve: virus. Dark gray: DS, light gray: DR

Lastly, we observed an in silico clinical trial where humans were exposed to HIV on day and administered PrEP on days 14 and 42 and then every 2 months continually, as shown in Fig. 15. We notice initially, in Fig. 16a, that the healthy T-cell population depletes very quickly as the DR virion concentration increases to overtake the system. Since the drug efficacy, $\phi =0.3$, and the DR virions are just as infective as the DS virions, the DS virion concentration slowly depletes, but the DR virion concentration grows and saturates. We observe subtle differences between trials where the drug is 30% effective and $k_r=k_s/2$ (Fig. 16a) and where the drug is 70% effective and $k_r=k_s$ (Fig. 16b). In both of these cases, the DS virion concentration slowly depletes as a direct result of the PrEP injections, but the DR virion concentration increases linearly. The human patient would have become seropositive in both cases; however, we notice seroconversion happens much quicker in the case where the DR virions are just as infective as the DS virions. In the case where the drug is 70% effective and the DR virions are half as infective as the DS virions, both virion concentrations deplete over time, and the healthy T-cell population remains at a healthy steady state. It is interesting to notice that a combination of an effective, and consistent, drug schedule along with a decreased infectivity of DR virions is necessary to keep HIV seroconversion controlled.

Fig. 16

Human in silico trial. Human exposed to HIV on day 0. PrEP injection administered on days 14 and 42 and then every 2 months. The label DS indicates the drug-sensitive strain, and DR indicates the drug-resistant strain. (a) $\phi =0.3$, $k_r=k_s$. (b) $\phi = 0.3$, $k_r=k_s/2$. (c) $\phi =0.7$, $k_r=k_s$. (d) $\phi =0.7$, $k_r=k_s/2$. Solid curves: healthy T-cells. Dash-dot curve: actively infected T-cells. Dash curve: latently infected T-cells. Dotted curve: virus. Dark gray: DS, light gray: DR

8 Discussion

The FDA approved injectable pre-exposure prophylaxis (PrEP) has a long pharmacologic drug tail that is a plus and a minus. The long half-life is beneficial in reducing the burden of PrEP adherence from a daily pill to bimonthly injections. However, the long pharmacologic drug tail also means that there can be a long period when the CAB level is just high enough to inhibit the growth of the wild-type DS strain, but low enough that the DR strain can grow unchecked. This concern warranted experimentalists to conduct studies on macaque that showed that long-acting CAB may encourage the growth of the DR mutated strain [30], despite the fact that multiple INSTI mutations are required for extensive CAB drug resistance [53].

We developed a mathematical model of the within-host HIV infection with naturally occurring mutations that could result in PrEP resistant virions. Our model was validated against data for humans in the early stage of HIV before receiving antiretroviral therapy. We also parameterized our model using studies on macaques. This allowed validation of our model on experimental tests on macaques without antiretroviral therapy and several experiments involving PrEP and SHIV exposures. In this work we were not able to quantify the exact fitness of a mutation that is resistant to CAB-LA to infect a T-cell, $k_r$. This means we could not pinpoint how close our estimate is for the growth of drug resistance in humans when given CAB-LA and already infected with HIV. However, a study of parameter sensitivity showed that the infected T-cell concentration and the virion concentration were sensitive to the fitness, $k_s$ and $k_r$, of the DS and DR virions, respectively, to infect a healthy T-cell. Because we do not have a good estimate of $k_r$ and $\phi$ the reduction in the effectiveness of CAB-LA, we tested the forward sensitivity and global sensitivity of $k_r$ and $\phi$, along with several simulations at various $k_r$ and $\phi$ levels. The parameter studies showed that the effective reproduction numbers for the DR strain were more sensitive to the reduction in effectiveness of CAB-LA, $\phi$, than the infectivity fitness, $k_r$. This led to our in silico trials where we tested combinations of CAB-effectiveness and the fitness of the DR strain. We found that for the macaques, even in the case of 30% effectiveness in PrEP against the DR strain and equal fitness for infecting T-cells, the DR strain did not overtake the DS strain. It did show enough growth in the DR strain to warrant concern. For humans, this drug effectiveness and fitness combination gave more alarming results with the DR strain quickly outcompeting the DS strain.

This work is intended to be a starting point to connect experimental results from SHIV studies on macaques to HIV predictions for humans. Our next step will be to incorporate work estimating fitness costs associated with different types of mutations. We have recently learned of the work by Zanini et al. [87] that estimates the rates of mutation and the spectrum of different kinds of mutations along with fitness costs of HIV. Zanini et al. [87] make these estimates by using whole genome deep-sequencing data of nine untreated patients with HIV-1 [88]. The data contain 6–12 longitudinal samples per patient spanning 5–8 years of infection. In addition, the updated International Antiviral Society–USA (IAS–USA) drug resistance mutations list for HIV from 2019 includes the mutations for cabotegravir [89]. This work provides the mutations on the integrase gene associated with resistance to cabotegravir. While it will require a significant amount of work to combine these sources of information about fitness and specific mutations into a workable model, it should help us give a more accurate picture of the drug-resistant levels to expect when humans take CAB-LA while unknowingly infected with HIV. In addition, as a comparison, we can incorporate the drug mutation model used by Smith et al. [90] in their individual simulation studies.

There are many additional directions to which to continue to expand upon this work. We have lumped all mutations into a single DR HIV strain. One could filter the mutations into beneficial or deleterious mutations with regard to particular mutations taking advantage of tabulated information on key mutations for specific mutations giving rise to HIV drug resistance [89]. The experiments with SHIV positive macaques given 10 days after seroconversion track the specific mutations noted in the daily blood draws [30], so the simulation and experimental results could be aggregated and compared.

We have assumed that each T-cell is infected with only a single strain of the HIV virus. However, it has been shown in experimental studies that many cells in the body can be infected with multiple strains (see spleen studies [91], for one of many experiments). So a natural next step will be to allow co-infection of multiple strains. This can occur through cell-to-cell infection, i.e., viral synapsis, not presently included in our model. Multiple experiments have verified that HIV is spread through synaptic transmissions [92,93,94,95]. We could include synaptic transmission in our model using the model of [96] as a starting point.

Retroviruses in general, including HIV-1, are diploid, meaning that each virion contains two genomic RNA molecules. These viral RNA molecules then serve as the templates for proviral DNA synthesis by the means of virus-encoded enzyme reverse transcriptase RT. This creates another means to generate genetic changes and perhaps mutations through recombination. Since two RNA molecules are contained in each virion, the reverse transcription may switch from one template to another [91, 97].

While there are a number of ways to improve the precision of our current model, we believe that the current model results justify the FDA recommendation for extended-release cabotegravir users to take an HIV test before each PrEP injection.