Sexual behavior, risk perception, and HIV transmission can respond to HIV antiviral drugs and vaccines through multiple pathways

There has been growing use of highly active antiretroviral treatment (HAART) for HIV and significant progress in developing prophylactic HIV vaccines. The simplest theories of counterproductive behavioral responses to such interventions tend to focus on single feedback mechanisms: for instance, HAART optimism makes infection less scary and thus promotes risky sexual behavior. Here, we develop an agent based, age-structured model of HIV transmission, risk perception, and partner selection in a core group to explore behavioral responses to interventions. We find that interventions can activate not one, but several feedback mechanisms that could potentially influence decision-making and HIV prevalence. In the model, HAART increases the attractiveness of unprotected sex, but it also increases perceived risk of infection and, on longer timescales, causes demographic impacts that partially counteract HAART optimism. Both HAART and vaccination usually lead to lower rates of unprotected sex on the whole, but intervention effectiveness depends strongly on whether individuals over- or under-estimate intervention coverage. Age-specific effects cause sexual behavior and HIV prevalence to change in opposite ways in old and young age groups. For complex infections like HIV—where interventions influence transmission, demography, sexual behavior and risk perception—we conclude that evaluations of behavioral responses should consider multiple feedback mechanisms.

The actor chooses between US and PS according to a Fermi-Dirac equation (Equation 6) for the probability f (∆E) of choosing US: where ∆E represents the difference between expected utilities. Actor 1 formulates ∆E as ∆E = E i (b t , s i , U S) -E i (b t , s i , P S), representing the difference in expected utilities calculated for US and PS. According to the Fermi-Dirac equation, when the value of ∆E is large and positive, or large and negative, then there is a high probability the individual will choose US, or PS, respectively. However, if ∆E is small, then the actor could choose US even though ∆E < 0, or PS even though ∆E > 0. This provides a more realistic model of decision-making whereby if the utility difference is small, factors other than the expressed utility (such as personal preferences or other sources of "randomness") enter into decisionmaking. We choose this functional form because it is relatively simple, and has been used in previous models of decision-making [47,48]. For every decision made, a random value is created between 0 and 1. If f (∆E) is greater than this value, the individual's choice becomes US, if it is less than this, PS is the action chosen. For Actor 2, they choose between the offer given to them by Actor 1 and NS, meaning their ∆E =E i (b t , s i , U S) -E i (b t , s i , N S) or ∆E =E i (b t , s i , P S) -E i (b t , s i , N S).

Understanding the evolution of b t -values
This section describes the evolution of b t -values through gameplay. Supplementary Table S4 visualizes how the game is played. Randomly selected Actor 1 will weigh their expected utilities to offer either US or PS. Actor 2 either increases or decreases their b t -value according to Equations 1 and 2, then chooses their option between what was offered to them or NS. After Actor 2 has made their offer, Actor 1 will either increase their b t -value if offered US or decrease it and accept the choice. The structure of the risky sex game was meant to portray interactions between highly active individuals with strangers of a similar non-monogamous life style. Having individuals choose solely between Actor 1s offer and NS was based on the assumption that these are quick one-off interactions with limited verbal interaction. As such, the Actor 2 is limiting their response as a result of not being able to engage in a lengthy negotiation. In support of this assumption, Elwood et al, reported a social norm for silence in bathhouse public areas. The norm facilitates efficient and anonymous sexual encounters, but precludes the ability to negotiate condom use verbally [49].

Choice threshold according to assigned utilities (No intervention)
An HIV+ person will always prefer US to all other options, and PS when deciding between PS and NS. The HIV-individuals' choice varies depending on what their b t -value is. According to baseline utilities, HIV-individuals with b t -value < 0.3 prefer US to PS. When choosing between US and NS, HIV-individuals with b t -value < 0.6 prefer US to NS. HIV-individuals will always prefer PS to NS under baseline utilities. This helps explain why the b t -values of HIV-individuals reach a ceiling in many simulations. When their values are < 0.3 They prefer US and will subsequently increase their b t -value when they are Actor 1, or Actor 2 when being offered US. However, when their b t -value surpasses 0.3 They will choose PS when they are Actor 1 and subsequently decrease their b t -value when Actor 2 inevitably accepts. Since every actor is randomly assigned to be Actor 1 or 2 at every instance of the game, their b t -value continues to oscillate between this value. The overall average b t -value may appear slightly higher as individuals who dip below < 0.3 and choose US may contract HIV and be removed from the average resulting in a slightly higher overall average. Under the current utilities assigned, when choosing between PS and NS, no individual has incentive to choose NS, as we would expect. HIV-individuals choosing between US and NS will choose US whenever their b t -value < 0.6, according to the baseline utilities. Because of the ceiling occurring at 0.3 the choice for NS would only occur by either adjusting the utilities or the parameters for the Fermi-Dirac equation. For HIV+ individuals, since US is always the favoured option, increased prevalence increases their overall average b t -value, see Figure S2.

Choice threshold according to assigned utilities (HAART)
Upon introducing HAART into the population, the U U S for an HIV-individual interacting with an HIV+ individual changes to Equation 3. Under the baseline settings, the threshold for an HIV-individual deciding between US and PS increases to 0.37, and the threshold between US and NS increases to 0.74. This explains why we see an increase in the overall average HIV-b t -value threshold when compared to the no intervention option. It also explains why increasing τ , the transmission probability, increases prevalence but does not increase the average overall b t -value beyond the established threshold as seen in Figure S9.

Choice threshold according to assigned utilities (Vaccine intervention)
The results for introducing HIV vaccine are more diverse since the duration of vaccine protection has a specified length, which may also differ from an individual's belief that they are protected by the vaccine. The utility assigned for an HIV-individual that believes they are protected changes from -50 to a value of 100, this means that they always prefer US as long as they believe they are protected, even if the protection of the vaccine has ended. A vaccinated individual checks their protection status at 5 and 10 years into the program, and stops assuming protection after 15 years. The protection is assigned through a random-normal distribution with a mean of 10 years, and standard deviation of 3 years. When offered US, the b t -value of an HIV-individual adjusts using Equation 4, where ρ is the vaccine coverage. For this model, we assume individuals have perfect knowledge of the age of the person they are interacting with. As the mean length of the vaccine is 10 years, individuals will formulate Equation 4 with ρ as the chosen coverage level when interacting with individuals less than 25 years of age. When interacting with individuals over 25 years of age, ρ will be set to 0 within this equation. This creates a number of situations.
1. For young individuals, the b t -value is slower to increase under US requests according to Equation 4 and our baseline settings.
2. In addition to reducing the incremental increase when offered US, the new equation creates a threshold of 0.25 for these young individuals when deciding between US and PS. An HIV-individual's b t -value will decrease when offered US according to our equation if the individuals b t -value is > 0.25. This fact, combined with 1) explains why the average b t -value is lower for the youngest age groups, (see Figure 5).
3. However, as an individual ages and begins to interact with individuals older than 25, the ρ value in Equation 4 drops to 0. Leaving HIV-individuals with a preference for US and a sharply increasing b t -value. What is interesting is that we also encounter individuals who still believe they are protected even though their vaccine protection has expired and they are between check-ups. This creates a group with a preference for US despite not having protection, even though they assume individuals over 25 to no longer be protected. This rapidly increases their b t -value as these individuals are always preferring US and continue to increase their b t -value when interacting with people over 25 years of age.
4. As HIV-individuals continue to age, their b t -value continues to rise which is why we see a spike in the average b t -value for 30-40 year-olds. All vaccinated individuals have a final check-up after 15 years from vaccination where an HIV-individual would then revert back to their original utility settings. Through interactions, their average b t -value eventually returns to the threshold we have seen in the no intervention option as seen in the older age groups (see Figure 5).

Interesting points
We make the following observations about the evolution of the b t -values.
1. The threshold found in the no intervention and HAART option is arguably a realistic portrayal of how an HIV-individual who randomly alternates between proposing and being propositioned would view their risk-assessment. An increase in threshold through HAART is also realistic since HAART aims to reduce the risk of spread, allowing HIVindividuals for an increased adjustment in their utility.
2. For the vaccination program, it is very interesting to see the effects of a population that understands the limitations of the length of the vaccine (assumes anyone older than 25 is no longer protected), but may overestimate their coverage between check-ups. The increase in average b t -value for HIV-30-40 year-olds reflects the consequences of individuals that prefers US, believing they are vaccinated while conservatively assuming the vaccination of others ended after 10 years. 30-40 year-olds having an increased b t -value and thus preferring PS is beneficial as this results in less chances to contract HIV or other sexually transmitted infections.
3. A booster program or catch-up vaccination scenario would likely shift the average increased b t -value average from 30-40 year-old's to a higher age group. Since these age groups are still sexually active, it would likely be beneficial to implement such a program.  ω Acu duration of time in acute stage. Once infected an individual will begin to age in the acute stage while exhibiting γ Acu relative influence of transmission. This will last for 5 months unless their age reaches 79.

months [34]
ω Lat duration of time in latent stage. After Acute stage an individual will begin to age in the latent stage while exhibiting γ Lat relative influence of transmission. This will last for 90 months unless total age reaches 79.

months [34]
ω P −AIDS duration of time in pre-AIDS stage. After Latent stage an individual will begin to age in the pre-aids stage while exhibiting γ P −AIDS relative influence of transmission. This will last for 24 months unless total age reaches 79.

months [34]
ω AIDS duration of time in AIDS stage. After pre-AIDS an individual will begin to age in the AIDS stage while exhibiting γ AIDS relative influence of transmission. This will last for 60 months unless total age reaches 79. If individual reaches end of this stage, it will die.

months [34]
ω P −AIDS−H duration of time in pre-AIDS stage, with HAART. After Latent stage, if individual is selected for treatment the individual will begin in the pre-AIDS with treatment stage while exhibiting γ P −AIDS−H relative influence of transmission. This will last for 2 months unless total age reaches 79.

months [34]
ω AIDS−H duration of time in AIDS stage, with HAART. After pre-AIDS with treatment stage an individual will begin in the AIDS with treatment stage while exhibiting γ AIDS−H level of infectiveness. This will last for 288 months unless total age reaches 79. If individual reaches end of this stage, it will die. γ Lat Relative change in transmission probability from an HIV+ actor to an HIV-actor through US during the latent stage.

[34]
γ P −AIDS Relative change in transmission probability from an HIV+ actor to an HIV-actor through US during the pre-AIDS stage.

[34]
γ AIDS Relative change in transmission probability from an HIV+ actor to an HIV-actor through US during the AIDS stage.

[34]
γ P −AIDS−H Relative change in transmission probability from an HIV+ actor to an HIV-actor through US during the pre-AIDS with HAART stage.

[51]
γ AIDS−H Relative change in transmission probability from an HIV+ actor to an HIV-actor through US during the AIDS with HAART stage.

[34]
Table S3: Relative reduction in transmission probability depending on stage of infection. The baseline HIV transmission probability τ (see Table 2) is modified by these values to produce the actual stage-specific HIV transmission rate.
Actor Action Actor 1 First offer Table S4: Layout of risky sex game between randomly selected actors. Actor 1 makes the first offer between US and PS. Actor 2 updates their b t -value using equation 1 or 2. Actor 2 then makes an offer consisting of either Actor 1's offer or NS. Actor 1 then updates their b t -value and the resulting action occurs.