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RESEARCH
A Bistable Switch in HIV Disease Outcome for Antibody Treatment Following the Onset of Recurrent Aphthous Oral Ulcers
ELISE PHAM, Harvard College '26
THURJ Volume 14 | Issue 2
Background
When challenged with foreign macromolecules, the humoral immune system produces an efficient response of antibody production (Skerra, 2003). However, the human immunodeficiency virus (HIV) evades CD4 T helper cells, the central component of generating neutralizing antibodies that orchestrate specific responses to a wide range of viruses (Wodarz & Nowak, 2002). When understanding the dynamic between HIV pathogenesis and immune system response, verbal reasoning proves insufficient because of the inherent complexity of the interactions involved. In particular, verbal reasoning may overlook subtle nuances, intricate feedback loops, and nonlinear relationships that are fundamental to comprehending the progression of HIV infection and the host immune response. Thus, mathematical models have proven useful in understanding how virus-antibody dynamics alter the course of HIV infection by capturing a set of assumptions and deducing logical conclusions through altering key parameters.
Many studies have utilized chimeric simian-human immunodeficiency virus (SHIV) as a comparable animal model for HIV pathogenesis (Ciupe et al., 2018). It has been found that broadly neutralizing monoclonal antibodies (bnMAbs) have exceptional in vitro potency against a high-dosage mucosal HIV challenge, but the strength of protection in both high-dose and low-dose challenges is dependent upon the ratio between concentration of bnMAbs and size of the viral challenge (Moldt et al., 2012; Hessell et al., 2009). In simpler terms, the effectiveness of vaccines relies heavily on the balance between the amount and strength of antibodies produced by the immune system and the size of the virus challenge; if there are not enough strong antibodies to fight off a large amount of virus, the vaccine may not be as effective. Therefore, understanding the concentration and avidity of antibodies in relation to the size of viral inoculum proves important in developing vaccines that offer robust protection against HIV.
Introduction
In Ciupe et al., 2018, a mathematical model incorporating the interaction between virus, recipient and donor antibody, and antibody-virus immune complexes was developed. Previous studies have demonstrated that 20 viral RNA (vRNA) copies from plasma obtained during the ramp-up stage of simian immunodeficiency virus (SIV) infection (7 days after inoculation) are required for establishing persistent infection in recipient rhesus macaques. Contrastingly, a substantially higher count of 1500 vRNA copies from plasma collected during the set-point stage (10-16 weeks after peak viremia) of infection is necessary for persistence (Ma et al., 2009). To test the hypothesis that the ramp-up plasma contains only free virus while the set-point plasma contains antibody-virus immune complexes that can still infect but at a lower infectivity rate, Ciupe et al., 2018 fits viral load data onto a developed mathematical model. This model estimates the long-run concentration of free virus, free donor and recipient antibodies, and donor and recipient antibody-viral immune complexes. The experimental protocol involved infecting rhesus macaques with SIV, collecting plasma at different infection stages, titrating the vRNA copies from each sample, and then using the plasma to infect SIV-naive rhesus macaques intravenously. Notably, while no donor antibodies were detected in ramp-up stage plasma, they were present in set-point stage plasma, suggesting a potential role of antibody-virus complexes in infection dynamics.
To predict the relationship between viral inoculum size and SIV disease outcome, Ciupe et al., 2018 develops a mathematical model consisting of seven differential equations describing the rate of change of uninfected CD4 T cells, infected CD4 T cells, free virus, recipient antibody concentration, donor antibody concentration, recipient antibody-virus complex concentration, and donor antibody-virus complex concentration. After fitting data to estimate model parameters, Ciupe et al., 2018 predicts the minimum virus inoculum to establish disease persistence in three different scenarios of infection:
1. In the first scenario, the recipient animal is infected with donor plasma containing SIV virions from the ramp-up stage of infection; a bistable switch between viral clearance and persistence was found at an initial free virus concentration of 17.5/300 vRNA copies per ml.
2. In the second scenario, the recipient animal is infected with donor plasma containing SIV virions from the set-point stage of infection; a bistable switch between viral clearance and persistence was found at an initial free virus concentration of 42/300 vRNA copies per ml and a ratio of 7.8 between initial donor antibody-virus complexes and free virus.
3. To address the effect of antibodies in changing disease outcome, in the third scenario, the recipient animal is infected with donor plasma containing SIV virions from the ramp-up stage of infection with heat-inactivated antibodies from the set-point stage of infection; a minimum of 7. 4 × 10 donor antibody molecules per ml are needed for viral clearance.
A key limitation to this study, however, is that while Ciupe et al., 2018 identifies the minimum number of antibody molecules needed to establish viral clearance at the time of infection, it is not realistic to assume that an animal would receive antibody treatment at the exact time of infection. Therefore, in this paper, I will not only attempt to replicate the mathematical model of recipient-virus infection in Ciupe et al., 2018, but I will also explore how changing the time at which antibodies are administered to the recipient animal will affect disease outcome and the concentration of donor antibodies needed to induce viral clearance.
Explanation of Differential Equations
Ciupe et al., 2018 uses the following variables and initial conditions stated in Chart 1.
Chart 1 (Ciupe et al., 2018) Parameter values and initial conditions used in the model of
recipient-virus interaction. Variables are constantly changing based on the given situation while parameters remain fixed.
When the recipient animal is challenged with donor plasma, Ciupe et al., 2018 assumes that the donor’s antibodies, 𝐴𝐷, and antibody-virus immune complexes, 𝑋𝐷, are transferred into the
recipient. It is also assumed that 𝑋𝐷 can still infect the recipient at a rate β1 < β, a de novo
antibody response that occurs in recipient animals, and that the recipient antibodies proceed independent of antigens at a maximum proliferation rate r and a maximum capacity of K. In simpler terms, the rate at which 𝑋𝐷 can cause infection in the recipient (represented by β1) is
assumed to be less than the initial infection rate denoted by β.
As a result, Ciupe et al., 2018 obtains the following model:
Model 1 (Ciupe et al., 2018) Model of Recipient-Virus Interaction; Listed as (4) in the article. This model explores the formation of virus-antibody immune complexes and their implications in transmission and defense. It demonstrates a bistable switch mechanism, indicating two distinct states: clearance and persistence.
We are most interested in the differential equations for 𝐴𝑅, 𝑋𝑅, 𝐴𝐷, and 𝑋𝐷, since the other equations build upon a basic Simian Immunodeficiency Virus (SIV) model. In analyzing the differential equations, notice that the assumption that the recipient antibodies reach a carrying capacity K result in the logistic term 𝑟𝐴 (1 − 𝐴𝑅) within 𝑑𝐴𝑅 , therefore indicating that the recipient antibodies cannot reproduce infinitely and thus disease persistence will possibly occur due to this carrying capacity. In addition, note that 𝑑𝑋𝑅 and 𝑑𝑋𝐷 both have terms that include 𝑑𝑡 𝑑𝑡 quotients, indicating that these are clearance terms. They describe the change in the rate of the clearance of antibody-virus immune complexes. The variable 𝑐𝐴𝑉 sets the maximum possible rate
of clearance of antibody-virus immune complexes, so the clearance terms vary between 0 and
𝑐 . When 𝑋 = M or 𝑋 = M, the clearance term is 𝑐𝐴𝑉 , which is half of the maximum clearance 𝐴𝑉𝐷𝑅 2 rate. Theoretically, large 𝑐𝐴𝑉 is therefore favorable, as it will allow 𝑋𝐷 and 𝑋𝑅 to decrease faster, thus allowing V to decrease faster.
Furthermore, it is important to further evaluate the relationship between 𝑑𝐴𝐷(𝑡) , 𝑑𝑋𝐷(𝑡) , 𝑑𝑡 𝑑𝑡
and 𝑑𝑉(𝑡) . Initially, when Ciupe et al., 2018 sets 𝐴 (0) = 7. 4 × 109, 𝑋 experiences an initial 𝑑𝑡 𝐷𝐷
surge. However, due to the conditions where 𝑐𝐴𝑉= 106 , 𝑘𝑚 = 100 >> 𝑘𝑝, and 𝐴𝐷 does not drastically change as t → ∞, 𝑋𝐷 gradually declines over time. Similarly, 𝑋𝑅 follows a trajectory
of initial increase followed by a decline over time, with 𝐴𝑅(0) = 3. 5 × 108 and 𝐴𝑅 remaining relatively stable over time. As both 𝑋𝐷 and 𝑋𝑅 decrease, V, consequently, diminishes over time.
A critical aspect lies in the dynamics of 𝐴𝐷, which decreases due to an association rate 𝑘𝑝
between virus and 𝐴𝐷 forming 𝑋𝐷, while simultaneously increasing due to a dissociation rate 𝑘𝑚
of 𝑋𝐷 into V and 𝐴𝐷. Notably, 𝑘𝑚 is much larger than 𝑘𝑝, leading to rapid dissociation of donor
antibody-virus immune complexes compared to their formation. Therefore, a drastic increase in 𝐴𝐷 would lead to a corresponding surge in 𝑋𝐷, subsequently elevating V significantly, owing to
the high dissociation rate. This relationship is important when I try to manipulate the model to include a term for a drastic increase in 𝐴𝐷 later in this paper.
Replicating the Model of Recipient-Virus Interaction
Aligning with the work of Ciupe et al., 2018 is necessary to validate my model. Through data fitting, Ciupe et al., 2018 estimates β = 1.34 × 10−7 and r = 1.95 for Animal 33815, a recipient animal (rhesus macaques) infected with donor plasma containing SIV virus from the ramp-up stage of infection.
The parameter β represents the rate at which susceptible individuals become infected upon contact with infectious individuals, reflecting the infectiousness of the virus. In the context of the model, a higher value of β indicates a greater likelihood of transmission from infected to susceptible individuals.
On the other hand, the parameter r represents the maximum proliferation rate of antibodies in the recipient animal's immune response. In the model, a higher value of r signifies a faster rate of antibody production by the immune system in response to the viral infection. This parameter is essential for understanding the strength and effectiveness of the immune response against the virus.
Thus, the following graphs were obtained (note that the function of total virus with respect to time 𝑉𝑇(𝑡) = 𝑉(𝑡) + 𝑋𝑅(𝑡) + 𝑋𝐷(𝑡)):
Figure 1 (Ciupe et al., 2018) The change in 𝑉𝑇 (total virus) over time given by Model 1 for
Animal 33815, which was infected by donor plasma containing ramp-up stage virions. At V(0) = 20/300, there is persistence of infection; at V(0) = 2/300, there is clearance of infection.
Figure 2 (Ciupe et al., 2018): The change in free antibody (solid black line), 𝑋𝐷(dotted line), and 𝑋𝑅(dashed line) over time given by Model 1 for Animal 33815, which was infected by donor plasma containing ramp-up stage virions. The gray lines represent the situation with condition V(0) = 2/300 vRNA copies per ml and the black lines represent the situation with condition V(0) = 20/300 vRNA copies per ml. For V(0) = 2/300 vRNA copies per ml, 𝑋𝑅 increases slightly above 0 and then proceeds back to 0; this is logical because the disease outcome is clearance, so while there is an initial response of recipient antibodies, the concentration of recipient antibody-virus complexes will eventually reach 0 as the virus is cleared.
Figure 1 and 2 show that at V(0) = 20/300, the disease and the presence of both donor and recipient antibody-virus complexes persist, while at V(0) = 2/300, the disease and the presence of both donor and recipient antibody-virus complexes are eliminated (Ciupe et al., 2018). The higher initial viral load (20/300) represents a more severe infection, while a lower initial viral load (2/300) represents a milder infection. Using the software Wolfram Mathematica and plugging in the model’s differential equations, initial conditions, and estimated parameters for this specific case, I obtained similar graphs:
Figure 3: Generated by Wolfram Mathematica. Analogous to the change in 𝑉𝑇 over time given by Model 1 for Animal 33815, which was infected by donor plasma containing ramp-up stage
virions. The vertical axis is on a logarithmic scale. (a) Duplication of Figure 1, the change in 𝑉𝑇
(total virus) over time given by Model 1 for Animal 33815, which was infected by donor plasma containing ramp-up stage virions. (b) V(0) = 20/300, comparable to the black line in Figure 3a. (c) V(0) = 2/300, comparable to the gray line in Figure 3a.
Verifying the Model of Recipient-Virus Interaction
To ensure that the generated model produces replicable outcomes, we reduce c, the virus clearance rate, from 23 to 2, while keeping V(0) = 2/300 and all other initial conditions the same. We get Figure 5, which indicates persistence of infection. This is logical, as dramatically decreasing the rate at which virus is cleared will result in persistence. It is important to note, however, that while Ciupe et al., 2018 uses c = 23, c can range anywhere between 9 and 36 (Ramratnam et al., 1999).
Figure 5: Generated by Wolfram Mathematica. The change in 𝑉𝑇 over time given by Model 1 for
Animal 33815, which was infected by donor plasma containing ramp-up stage virions, for V(0) = 2/300 and c = 2. This is a contrast to Figure 3b, where c = 23 resulted in virus clearance while keeping all other initial conditions the same. The vertical axis is on a logarithmic scale.
Let V(0) = 20/300 and c = 23, while keeping all other initial conditions the same. Then, change δ = 0.39 to δ = 3.9 to amplify the death rate of infected cells by ten-fold. We get Figure 6, which indicates that the concentration of recipient antibody-virus complexes will decrease to 0 over time, therefore indicating clearance of infection. This is logical, as dramatically increasing the rate at which infected cells die will result in clearance.
Figure 6: Generated by Wolfram Mathematica. The change in 𝑋𝑅 over time given by Model 1 for
Animal 33815, which was infected by donor plasma containing ramp-up stage virions, for V(0) = 20/300 and δ = 3.9. This is a contrast to the dashed black line in Figure 2, where δ = .39 resulted in the persistence of 𝑋𝑅 while keeping all other initial conditions the same. The vertical axis is on a logarithmic scale.
HIV Antibody Treatment from Symptom of Recurrent Aphthous Oral Ulcers
While Ciupe et al., 2018 finds that a minimum of 7. 4 × 109 donor antibody molecules per ml are needed for viral clearance at time t = 0, it is not reasonable to assume that individuals are treated with antibody treatment at the exact time that they become infected with HIV. In estimating the approximate amount of time between virus contraction and antibody treatment, it is important to determine the specific symptoms that indicate a potential HIV infection and the time elapsed between infection and embodiment of symptoms. Hecht et al., 2002 found that weight loss and oral ulcers were the most specific symptoms for HIV infection, and the best independent predictors for HIV infection were fever and skin rash.
Developing an injection term for 𝑑𝐴𝐷(𝑡)/𝑑(t)
Due to the relatively high prevalence of oral ulcers, affecting approximately 50% of patients with HIV at some point during the progression of their disease, recurrent aphthous ulcers remain a valuable diagnostic indicator. While the figure may seem modest, it's important to note that even though not all patients manifest this symptom, its occurrence spans various stages of HIV progression, from acute primary infection to advanced disease (Weinert, 1996). This persistence across disease stages underscores its significance in diagnosis despite its seemingly moderate prevalence. The most important oral indicator for HIV diagnosis is when the aphthous ulcer develops on nonkeratinized mucosa; however, it is important to note that other types of ulcers can also occur on nonkeratinized mucosa, so an aphthous ulcer must be distinguished by treatment response (MacPhail et al., 1992). Most HIV-infected patients acquire minor aphthous ulcers, which last up to 2 weeks (MacPhail et al., 1992). Therefore, since these aphthous ulcers develop during the acute stage of HIV infection (2-4 weeks after HIV exposure) and last for up to 2 weeks, and recurrent aphthous ulcers extend the experience of oral ulcers through the formation of new aphthous ulcers when old ones heal, we can predict that the earliest indication of potential HIV infection is when aphthous ulcers last for more than 4 weeks. Assuming a patient receives antibody treatment on the fourth week after HIV exposure, we can insert the same initial conditions that Ciupe et al., 2018 uses for recipient rhesus macaques infected with donor plasma containing SIV virions from the ramp-up stage of infection with heat-inactivated antibodies from the set-point stage of infection, except with 𝑔𝐴𝐷(0) = 0 and 𝐴𝐷(28) = I(t), where I(t) represents the infusion rate of antibodies, measured in units of antibodies per day, and is varied. Therefore, we can manipulate the differential equation for donor antibodies to include an I(t) term, where I(t) is an exponential function:
𝑑𝐴𝐷(𝑡) = −𝑑𝐴 −𝑘𝐴𝑉+𝑘 𝑋 +I(t) 𝑑𝑡 𝐴𝐷𝑃𝐷𝑀𝐷
I(t) = 10 𝑒−10(𝑡−28)2 π
I generated I(t) as a smooth rather than a discontinuous function to better complement the Wolfram Mathematica software, which has a difficult time interpreting a large discontinuous
𝑑𝐴𝐷(𝑡) 10 −10(𝑡−28)2 jump of 𝑑𝑡 . The reason why I used π 𝑒 is because its integral over the line is exactly 1. Thus, the injection I(t) parameter is the exact amount of increase that the infusion is delivering.
Modifying the Model of Recipient-Virus Interaction
In addition, I need to modify Model 1 developed by Ciupe et al., 2018 in order to account for a sudden drastic increase in 𝐴𝐷. Theoretically, in Model 1, 𝑋𝐷 will increase if 𝐴𝐷 increases. As a
result, because 𝑉𝑇 depends on 𝑋𝐷, 𝑉𝑇 also increases when 𝐴𝐷 increases. Since Ciupe et al., 2018 begins with an initial condition above 0 and does not add an injection term like the one I
propose, 𝑋𝐷 and 𝑉𝑇 do not experience a sudden drastic increase as t → ∞. However, if I keep
Model 1 as it is, I(t) will lead to an an extremely large and sudden increase in 𝑋𝐷, which will
ultimately lead to an increase in 𝑉𝑇 as t → ∞, since 𝑘𝑚 >> 𝑘𝑝; however, this is not biologically
reasonable as adding a large amount of donor antibodies should not lead to an even greater increase in total virus as t → ∞. Therefore, because Model 1 developed by Ciupe et al., 2018
may not be applicable to a situation in which there is an injection term – since 𝑑𝑉(𝑡) is designed 𝑑𝑡
to interpret a fairly small value of 𝑑𝐴𝐷(𝑡) and 𝑑𝑋𝐷(𝑡) (a small change in the rate of change of 𝑑𝑡 𝑑𝑡
𝐴𝐷(𝑡) and 𝑋𝐷(𝑡) in comparison to the large change in rate that 𝐴𝐷(𝑡) and 𝑋𝐷(𝑡) will experience from an injection term) – I will modify Model 1 to prevent I(t) from unrealistically changing the
virus outcome. The new model is as follows:
𝑑𝑉(𝑡) =𝑁δ𝐼−𝑐𝑉−𝑘(𝐴 +𝐴)𝑉+𝑘 𝑋 𝑑𝑡 𝑝𝑅𝐷 𝑚𝑅
𝑑𝑋𝑅(𝑡) = 𝑘 𝐴 𝑉 − 𝑘 𝑋 − 𝑐 𝑋𝑅 𝑑𝑡 𝑝𝑅 𝑚𝑅 𝐴𝑉𝑋𝑅+𝑀
Model 2: Modified version of the Recipient-Virus Interaction Model created by Ciupe et al., 2018. It incorporates an injection term, I(t), representing the infusion rate of antibodies over time, aiming to prevent unrealistic increases in the virus population. By integrating additional recipient antibodies into the system, the model ensures a more biologically plausible representation of HIV dynamics post-treatment.
I will not focus on 𝑑𝑇(𝑡) and 𝑑𝐼(𝑡) because they are not as relevant to this expansion as the 𝑑𝑡 𝑑𝑡
other differential equations since the emphasis lies on the impact of antibody infusion (I(t)) on the virus population and its interactions with recipient antibodies (𝐴𝑅) and donor antibodies (𝐴𝐷).
Notice that I completely removed 𝑑𝑋𝐷(𝑡) and 𝑋 from Model 1 because after a patient is treated 𝑑𝑡 𝐷
with an antibody injection, we can assume that these antibodies represent an input of extra recipient antibodies, 𝐴𝑅, so rather than forming their own antibody-virus immune complexes, 𝑋𝐷
, they become incorporated with the already-present 𝐴𝑅 to form 𝑋𝑅. This assumption will prevent the model from displaying a biologically unreasonable output from the sudden and large increase
in 𝐴𝐷 from the I(t) term.
Results
Using the baseline parameters from Ciupe et al., 2018, except with the adjustment of 𝐴𝐷 (0) = 0 and 𝐴𝐷(28) = I(t), a bistable switch between persistent and cleared virus populations
occurs within the range of 1. 04 × 1012 to 1. 05 × 1012 for I(t). Figure 7 illustrates the corresponding variations in 𝑉𝑇 for different values of I(t).
Therefore, Figure 7a and 7b show that there are two different stable equilibrium states that the system can achieve when donor antibodies are injected at t=28 days. When I(t)=1.04 × 1012, the stable equilibria that the system approaches is 𝑉𝑇=8.5 × 106, or viral persistence. When
I(t)≥ 1.05 × 1012, the system approaches 𝑉𝑇=0, or viral clearance, at approximately t=70 days.
While viral persistence occurs when V(0)=20/300 and I(t)=1.04 × 1012, manipulating the value of V(0) will allow investigation of whether changing the initial concentration of free virus will also confer viral clearance. As seen in Figure 8, viral clearance occurred for I(t) =1.04 × 1012 when V(0)≤ 18.6/300:
Figure 8 Generated by Wolfram Mathematica. The change in 𝑉𝑇 over time for Model 2, with 𝐴𝐷 (0) = 0 and I(t) = 1. 04 × 1012. The vertical axis is on a logarithmic scale. (a) V(0) = 18.6/300(b)V(0)= 18.7/300
In addition, I collected the value of t when 𝑉𝑇 = 0 for 1/300 ≤ V(0) ≤ 18/300 in order to investigate how changing the initial concentration of free virus can affect when the patient will experience full recovery. I obtain Table 1 and Figure 9.
Table 1: The values of t collected when 𝑉𝑇 = 0 for 1/300 ≤ V(0) ≤ 18/300 using the “Get Coordinates” feature on Wolfram Mathematica. As V(0) increases, the Time t days when Total Virus 𝑉𝑇 = 0 also increases.
Figure 9 The correlation between the time at which 𝑉𝑇 = 0 and an integer value of V(0) when virus
clearance occurs for I(t) = 1. 04 × 1012. For 1/300 ≤ V(0) ≤ 12/300, there is a positive correlation between V(0) and the time at which 𝑉𝑇 = 0, and this positive correlation seems to occur at a decreasing, positive exponential weight.
Gender and Court Decisions
Ciupeetal.,2018foundthataconcentrationof7.4 × 109donorantibodiesareneededat time t = 0 in order to achieve viral clearance for an animal with V(0) = 20/300, β =
1. 46 × 10−7 𝑚𝑙 𝑑𝑎𝑦 − 𝑣𝑖𝑟𝑖𝑜𝑛−1, and r = 2.14 𝑑𝑎𝑦−1. However, rather than simulating a situation in which the animal is treated with the antibody treatment at the exact time of infection, I propose a more realistic situation in which the recipient animal is infected by the HIV virus with 𝐴𝐷(0) = 0, but then the animal receives antibody treatment on the 28th day after infection, a
reasonable estimate for the earliest indication of potential HIV infection when tracking the
symptomofrecurrentaphthousoralulcers.Ifindthataninjectionof1.05 × 1012donor antibodies at 28 days confers viral clearance. More specifically, as seen in Figure 7, I predict a bistableswitchbetweenviralpersistenceandviralclearanceexistsbetweenI(t)=1.04 × 1012 andI(t)=1.05 × 1012.
Thebistableswitchseparatesverydifferentoutcomesforsimilar initial conditions. For example, if I(t) = 1. 04 × 1012, the system approaches a stable equilibriumof𝑉𝑇=8.5 × 106,orviralpersistence.Ontheotherhand,ifI(t)=1.05 × 1012, the system approaches a stable equilibrium of 𝑉𝑇 = 0, or viral clearance. If antibody treatment is given at t = 28 rather than t = 0, we can predict that more antibodies are needed to confer viral clearance; this prediction proves true, as 1. 05 × 1012 > 7. 4 × 109, the value of 𝐴𝐷(0) in Ciupe et al., 2018 .
It is worth investigating why the patient experiences an overall decrease and then increase in
total virus before experiencing viral persistence in Figure 7. When the patient receives treatment on Day 28, we observe a sharper decrease in 𝑉𝑇, which is biologically reasonable because more antibodies means more defense against the virus. However, at approximately Day 50, the patient experiences an increase in 𝑉𝑇 before 𝑉𝑇 approaches an equilibrium of 8. 5 × 106. This increase is due to the increase in 𝑋𝑅 at approximately t = 50 days (Figure 10), since V can be generated through the dissociation of 𝑋𝑅 and 𝑉𝑇 = V + 𝑋𝑅, as seen in Model 2.
Figure 10: Generated by Wolfram Mathematica. The change in 𝑋𝑅 over time for Model 2 with 𝐴𝐷 (0) = 0 and I(t) = 1. 04 × 1012. The vertical axis is on a logarithmic scale.
While the biological cause for the increase in 𝑋𝑅 at approximately t = 50 days is not precisely
known, it can be inferred that due to insufficient levels of I(t), the patient initially underwent damped oscillatory behavior in total virus levels after 28 days before stabilizing into a more persistent equilibrium.
Furthermore,whileviralpersistenceoccursforI(t)=1.04 × 1012whenV(0)=20/300,I investigate if changing the initial concentration of free virus while maintaining I(t) =
1.04 × 1012 can result in viral clearance. As seen in Figure 8, I find that a bistable switch between viral clearance and persistence exists between V(0) = 18. 6/300 and V(0)= 18.7/300; I observed viral clearance for I(t)=1.04 × 1012 when V(0)≤ 18.6/300. Consequently, as seen in Table 1 and Figure 9, I find that when viral clearance is observed, changing the value of V(0) changes the time at which the patient experiences full viral recovery. For 1/300 ≤ V(0) ≤ 12/300, there is a positive correlation between V(0) and the time at which 𝑉𝑇 = 0, and this positive correlation seems to occur at a decreasing, positive exponential rate
(Figure 9); therefore, changing the concentration of V(0) at lower concentrations has more impact on the time at which the patient achieves full recovery than changing the concentration of V(0) at higher concentrations. However, between V(0) = 12/200 and V(0) = 13/300, there is a much larger increase in the time at which the patient achieves full recovery; as seen in Table 1, at V(0) = 12/300, the patient achieves full recovery at 37.30 days, but at V(0) = 13/300, the patient achieves full recovery at 63.22 days. Between V(0) = 13/300 and V(0) = 20/300, the correlation between V(0) and the time at which 𝑉𝑇 = 0 increases gradually as before.
Overall, administering antibodies at 28 days post-infection can still lead to viral clearance, albeit requiring a higher concentration of antibodies compared to immediate treatment. A bistable switch phenomenon, indicating a threshold effect between viral persistence
andclearance,existsbetweenI(t)=1.04 × 1012andI(t)=1.05 × 1012.Additionally, altering the initial virus concentration affects the time to achieve viral clearance, with lower concentrations exhibiting a more pronounced impact on recovery time. These findings highlight the complexity of antibody treatment dynamics in HIV infection and underscore the importance of timing and initial viral load in treatment outcomes.
Future Expansion and Failures
Future Expansion
In this paper, I(t) = π 𝑒 , where 𝑎 = 10 and 𝑏 = 10. Here, 𝑎 represents the amplitude
of the antibody infusion, while 𝑏 controls the rate at which the infusion peaks and decays. A key remark about this I(t) term, however, is that the values for 𝑎 and 𝑏 may not realistically represent the time length of one antibody treatment session.
Therefore, by increasing the value of 𝑏 and adjusting 𝑎 accordingly to maintain the integral of I(t) as 1, we can effectively shorten the duration of the antibody infusion administered to a patient. This adjustment is crucial as it allows for a more realistic representation of treatment duration, potentially improving the model's accuracy in simulating real-world scenarios. Maintaining the integral of I(t) as 1 ensures that the total amount of antibodies delivered remains consistent, which is important for accurately assessing the efficacy and impact of the treatment regimen. More research regarding the average time length of antibody treatment for patients with HIV is needed. In addition, it is unclear whether my model can be applied to mucosal virus infections or non-neutralizing antibody responses.
Difficulties
It is important to note that due to the mediocre occurrence rate of oral ulcers in patients with HIV, using oral ulcers as a proxy may not be entirely accurate.
I initially developed I(t) as a discontinuous function in order to better replicate an antibody infusion in which all of the antibodies are given at the same time, over a time frame of t = 0.1 days:
𝑑𝐴𝐷(𝑡) = −𝑑𝐴 −𝑘𝐴𝑉+𝑘 𝑋 +I(t) 𝑑𝑡 𝐴𝐷𝑃𝐷𝑀𝐷
I(t) = some varied dosage, DSG; if 28 ≤ t ≤ 28.1 0;
otherwise
However, the Wolfram Mathematica software was only able to sustain low values of I(t) before breaking and producing unreasonable outputs, such as 𝑉𝑇 → ∞ at t = 28 days. Thus, I
modified the I(t) term to a continuous function in order to comply with the limits of the Wolfram Mathematica software.
In other words, the main failure that I encountered was that I was unable to find a bistable switch without the Wolfram Mathematica software breaking when I used a discontinuous function of I(t). I attempted to manipulate the other parameters while maintaining a discontinuous function of I(t) in order to see if I could produce a reasonable result. When the I(t) was within a range that Wolfram Mathematica could handle, I did experience clearance when I significantly decreased 𝑘𝑚, or significantly increased 𝑐𝐴𝑉 or c, but of course, dramatically decreasing the unbinding rate, or dramatically increasing the immune complexes clearance rate or virus clearance rate, is not biologically reasonable. In addition, I noticed that Ciupe et al., 2018 assumes that antibody-virus immune complexes can still infect but at a rate of β1 < β.
Because the Wolfram Mathematica software broke from the drastic increase of 𝑋𝐷 since 𝑘𝑚 >> 𝑘𝑝, I realized that if I assume that the antibody-virus immune complexes cannot infect or dissociate (thus allowing the deletion of 𝑘𝑚 and any variables it attaches to in Model 1), then the program can produce a reasonable output without breaking. Again, however, this assumption would not be biologically correct, as these immune complexes are still infectious since antibodies are incapable of fully rupturing covalent bonds in viruses (Fox & Cottler-Fox, 1992).
Conclusion
Recurrent aphthous oral ulcers, prevalent across all stages of HIV infection, stand out as a particularly specific symptom of the condition (Weinert, 1996; Hecht et al., 2002). These ulcers, persisting for at least four weeks or more, serve as a crucial early indicator of HIV infection. This emphasis on oral ulcers is justified by their distinctiveness and reliability in signaling potential HIV infection compared to other symptoms. I modify the recipient-virus interaction model developed by Ciupe et al., 2018, using the same initial conditions that Ciupe et al., 2018 uses for recipient rhesus macaques infected with donor plasma containing SIV virions from the ramp-up stage of infection with heat-inactivated antibodies from the set-point stage of infection,
except 𝐴 (0) = 0 and 𝐴 (28) = I(t), where I(t) is varied. In my modification, I remove 𝑑𝑋𝐷(𝑡) and 𝐷𝐷 𝑑𝑡
𝑋𝐷 by assuming that the infusion of 𝐴𝐷 is interpreted as an extra input of 𝐴𝑅. I find that a bistable switchbetweenviralpersistenceandviralclearanceexistsbetweenI(t)=1.04 × 1012andI(t)
=1.05 × 1012.
When I(t)=1.04 × 1012, viral clearance can be observed when V(0) ≤ 18.6/300, and a positive correlation exists for V(0) and the time at which the patient experiences full recovery when 1/300≤V(0)≤ 18/300. Although this paper highlights a bistable switch between viral persistence and clearance, shedding light on the dynamics of antibody treatment, further investigation into the optimal duration of antibody treatment for HIV patients is warranted to enhance the accuracy of treatment simulations. Future research in this realm could explore the interplay between treatment timing and efficacy, as well as the incorporation of additional factors influencing HIV dynamics, such as immune response variations and viral mutations. Such inquiries would contribute to refining predictive models and improving treatment strategies in combating HIV infection.
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