# American Institute of Mathematical Sciences

April  2017, 14(2): 359-376. doi: 10.3934/mbe.2017023

## Multiplayer games and HIV transmission via casual encounters

 1 Department of Mathematics & Statistics, University of Guelph, Guelph ON Canada N1G 2W1, Canada 2 Department of Applied Mathematics & Statistics, University of Waterloo, Waterloo ON Canada, Canada

Received  January 27, 2015 Revised  June 27, 2016 Published  August 2016

Fund Project: The second author is supported by NSERC DG grant: 400684; the third author is supported by NSERC RGPIN-04210-2014.

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.

Citation: Stephen Tully, Monica-Gabriela Cojocaru, Chris T. Bauch. Multiplayer games and HIV transmission via casual encounters. Mathematical Biosciences & Engineering, 2017, 14 (2) : 359-376. doi: 10.3934/mbe.2017023
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##### References:
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
The 2-player game showing $x^{1*}_{-}$, $x^{2*}_{+}$ and $\epsilon_{+}$ varying $USE(-, +)$ and $USE(-, +)$
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
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
Heat map for 4-player game showing ($x^{1}$, $x^{2}$, $x^{3}$, $x^{4}$) equilibrium values for the respective initial conditions
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
Compounded $\epsilon_{+}$ using U.S. census data (left) over 5 games vs. the same using Zimbabwe data (right)
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 $\mu=0.75$
This table outlines the base case preferences for different sexual acts given a players' status
 $P_{1}$ $P_{2}$ Utility for USE Utility for PSE Range 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 $[0,1]$
 $P_{1}$ $P_{2}$ Utility for USE Utility for PSE Range 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 $[0,1]$
Parameter definitions and parameter values for baseline scenario. Here $\tau$ is a fixed probability of transmission per contact
 Term Definition Baseline value Range $\tau$ 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. 0.05 5% of population $\epsilon_{-}(0)$ Initial proportion of $HIV-$ individuals in the population. 0.95 95% of population
 Term Definition Baseline value Range $\tau$ 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. 0.05 5% of population $\epsilon_{-}(0)$ Initial proportion of $HIV-$ individuals in the population. 0.95 95% of population
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