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Multiplayer games and HIV transmission via casual encounters

The second author is supported by NSERC DG grant: 400684; the third author is supported by NSERC RGPIN-04210-2014.
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  • Population transmission models have been helpful in studying the spread of HIV. They assess changes made at the population level for different intervention strategies.To further understand how individual changes affect the population as a whole, game-theoretical models are used to quantify the decision-making process.Investigating multiplayer nonlinear games that model HIV transmission represents a unique approach in epidemiological research. We present here 2-player and multiplayer noncooperative games where players are defined by HIV status and age and may engage in casual (sexual) encounters. The games are modelled as generalized Nash games with shared constraints, which is completely novel in the context of our applied problem. Each player's HIV status is known to potential partners, and players have personal preferences ranked via utility values of unprotected and protected sex outcomes. We model a player's strategy as their probability of being engaged in a casual unprotected sex encounter ($ USE $), which may lead to HIV transmission; however, we do not incorporate a transmission model here. We study the sensitivity of Nash strategies with respect to varying preference rankings, and the impact of a prophylactic vaccine introduced in players of youngest age groups. We also study the effect of these changes on the overall increase in infection level, as well as the effects that a potential prophylactic treatment may have on age-stratified groups of players. We conclude that the biggest impacts on increasing the infection levels in the overall population are given by the variation in the utilities assigned to individuals for unprotected sex with others of opposite $ HIV $ status, while the introduction of a prophylactic vaccine in youngest age group (15-20 yr olds) slows down the increase in $ HIV $ infection.

    Mathematics Subject Classification: Primary: 91A80; Secondary: 91C99.

    Citation:

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  • Figure 1.  Heat map for 2-player game showing ($x^1_{-}, x^{1}_{+}, x^{2}_{-}, x^{2}_{+})=(0, 1, 1, 0)$ equilibrium values for the respective initial conditions

    Figure 2.  The 2-player game showing $x^{1*}_{-}$, $x^{2*}_{+}$ and $\epsilon_{+}$ varying $USE(-, +)$ and $USE(-, +)$

    Figure 3.  Heat map for 3-player game showing $(\underline{x}^{1*}, \underline{x}^{2*}, \underline{x}^{3*})$ equilibrium values for a uniform spread of initial conditions

    Figure 4.  Results for 3-player game. The three upper panels show the $(x^{1*}_{1-}\, x^{1*}_{1+}, \, x^{1*}_{2-})$ choices of $P_1$, whereas the lower left panels show the $x^{2*}_{1+}$ choice of $P_{2}$, $x^{3*}_{1+}$ for $P_3$ dependent on $USE(-, +)$ and $USE(+, -)$. Lower right panel shows the $\epsilon_{+}(game_1)$ variation

    Figure 5.  Heat map for 4-player game showing ($x^{1}$, $x^{2}$, $x^{3}$, $x^{4}$) equilibrium values for the respective initial conditions

    Figure 6.  3-dimensional results for 4-player game2 showing choices for varying $USE(+, -)$ and $USE(-, +)$ utilities. The upper panels show the change in equilibrium strategies of $P_1$: the upper left panel shows the strategies $x^{1*}_{2+}=0$, $x^{1*}_{1-}=x^{1*}_{2-}=x^{1*}_{3-}\approx 0.334$. The likelihoods of $P_2$ to engage in USE with $P_1$ are shown in lower left panel, while $x^{3*}_{2+}=x^{4*}_{2+}=0$ are not shown. The effect on the infected fraction due to this game is shown in lower right panel

    Figure 7.  Compounded $\epsilon_{+} $ using U.S. census data (left) over 5 games vs. the same using Zimbabwe data (right)

    Figure 8.  Compounded $\epsilon_{+} $ using U.S. (left panel) and Zimbabwe (right panel) census data comparing $USE(+, -)\in[0,1]$ and $USE(-, +)\in[0.25, 1]$ values, while with $U(USE, -, +, vacc ) = 0.5$ and $\mu=0.75$

    Table 1.  This table outlines the base case preferences for different sexual acts given a players' status

    $P_{1}$$P_{2}$Utility for USEUtility for PSERange
    HIV+HIV+ $USE(+, +)=1$$PSE(+, +)$=0.25 $[0,1]$
    HIV+HIV- $USE(+, -)=0$$PSE(+, -)$=0 $[0,1]$
    HIV-HIV+ $USE(-, +)=0$$PSE(-, +)$=0$[0,1]$
    HIV-HIV- $USE(-, -)=1$$PSE(-, -)$=0.25$[0,1]$
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    Table 2.  Parameter definitions and parameter values for baseline scenario. Here $\tau$ is a fixed probability of transmission per contact

    TermDefinitionBaseline valueRange
    $\tau$Probability of HIV spread from an $HIV+$ player to an $HIV-$ player through $USE$0.02-
    $\epsilon_{+}(0)$Initial proportion of $HIV+$ individuals in the population.0.055% of population
    $\epsilon_{-}(0)$Initial proportion of $HIV-$ individuals in the population.0.9595% of population
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