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
References:
[1]

K. J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy, Econometrica, 22 (1954), 265-290. Google Scholar

[2]

J. P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag New York, 1984. doi: 10.1007/978-3-642-69512-4. Google Scholar

[3]

R. M. Axelrod, The complexity of cooperation: Agent-based models of competition and collaboration, Princeton University Press, 1997.Google Scholar

[4]

T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory, SIAM, 1995. Google Scholar

[5]

A. Bensoussan, Points de Nash dans le cas de fonctionnelles quadratiques et jeux differentiels lineaires a N personnes, SIAM J. Control, 12 (1974), 460-499. Google Scholar

[6]

Center for Disease Control and Prevention, Diagnoses of HIV Infection in the United States and Dependent Areas, HIV Surveillance Report, 2011.Google Scholar

[7]

B. Coburn and S. Blower, A major HIV risk factor for young men who have sex with men is sex with older partners, J. Acquir. Immune Defic. Syndr., 54 (2010), 113-114. Google Scholar

[8]

M. G. Cojocaru and L. B. Jonker, Existence of solutions to projected differential equations on Hilbert spaces, Proceedings of the American Mathematical Society, 132 (2004), 183-193. doi: 10.1090/S0002-9939-03-07015-1. Google Scholar

[9]

Cojocaru, M. -G., Wild, E., On Describing the Solution Sets of Generalized Nash Games with Shared Constraints, submitted to: Optimization and Engineering, 2016.Google Scholar

[10]

M. G. CojocaruC. T. Bauch and M. D. Johnston, Dynamics of vaccination strategies via projected dynamical systems, Bulletin of mathematical biology, 69 (2007), 1453-1476. doi: 10.1007/s11538-006-9173-x. Google Scholar

[11]

M. G. Cojocaru, Dynamic equilibria of group vaccination strategies in a heterogeneous population, Journal of Global Optimization, 40 (2008), 51-63. doi: 10.1007/s10898-007-9204-7. Google Scholar

[12]

J. P. DoddsA. NardoneD. E. Mercey and A. M. Johnson, Increase in high risk sexual behaviour among homosexual men, London 1996-8: Cross sectional, questionnaire study, BMI, 320 (2000), 1510-1511. Google Scholar

[13]

F. FacchineiA. Fischer and V. Piccialli, On generalized Nash games and variational inequalities, Operations Research Letters, 35 (2007), 159-164. doi: 10.1016/j.orl.2006.03.004. Google Scholar

[14]

F. Fachinei and C. Kanzow, Generalized Nash Equilibrium Problems, Ann Oper Res, 175 (2010), 177-211. doi: 10.1007/s10479-009-0653-x. Google Scholar

[15]

J. W. Friedman, Game theory with applications to economics, Oxford University Press New York, 1990.Google Scholar

[16]

D. Gabay and H. Moulin, On the uniqueness and stability of Nash-equilibria in noncooperative games, Applied Stochastic Control in Econometrics and Management Science, 130 (1980), 271-293. Google Scholar

[17]

R. H. GrayM. J. WawerR. BrookmeyerN. K. SewankamboD. SerwaddaF. Wabwire-MangenT. LutaloX. LiT. VanCott and T. C. Quinn, Probability of HIV-1 transmission per coital act in monogamous, heterosexual, HIV-1-discordant couples in Rakai, Uganda, The Lancet., 357 (2001), 1149-1153. Google Scholar

[18]

D. Greenhalgh and F. Lewis, Stochastic models for the spread of HIV amongst intravenous drug users, Stochastic Models, 17 (2001), 491-512. doi: 10.1081/STM-120001220. Google Scholar

[19]

T. B. HallettS. GregsonO. MugurungiE. Gonese and G. P. Garnett, Assessing evidence for behaviour change affecting the course of HIV epidemics: a new mathematical modelling approach and application to data from Zimbabwe, Epidemics, 1 (2009), 108-117. Google Scholar

[20]

Y. H. Hsieh and C. H. Chen, Modelling the social dynamics of a sex industry: Its implications for spread of HIV/AIDS, Bulletin of Mathematical Biology, 66 (2004), 143-166. doi: 10.1016/j.bulm.2003.08.004. Google Scholar

[21]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998. doi: 10.1017/CBO9781139173179. Google Scholar

[22]

D. M. Huebner and M. A. Gerend, The relation between beliefs about drug treatments for HIV and sexual risk behavior in gay and bisexual men, Annals of Behavioral Medicine, 4 (2001), 304-312. Google Scholar

[23]

T. Ichiishi, Game Theory for Economic Analysis, New York: Academic Press, New York, 1983. Google Scholar

[24] D. Kinderlehrer and D. Stampacchia, An Introduction to Variational Inequalities and their Application, Academic Press, New York, 1980. Google Scholar
[25]

K. Nabetani, P. Tseng and M. Fukushima, Parametrized Variational Inequality Approaches to Generalized Nash Equilibrium Problems with Shared Constraints, (Technical Report 2008), Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Japan.Google Scholar

[26]

J. F. Nash, Equilibrium points in n-person games, CMS, 2 (2005), 21-56. Google Scholar

[27]

J.-S. Pang and M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games, CMS, 2 (2005), 21-56. doi: 10.1007/s10287-004-0010-0. Google Scholar

[28]

R. H. RemienP. N. HalkitisA. O'LearyR. J. Wolitski and C. A. Gómez, Risk perception and sexual risk behaviors among HIV-positive men on antiretroviral therapy, AIDS and Behavior, 9 (2005), 167-176. Google Scholar

[29]

J. B. Rosen, Existence and uniqueness of equilibrium points for concave n-person games, Econometrica, 33 (1965), 520-534. Google Scholar

[30]

K. D. Schroeder and F. G. Rojas, A game theoretical analysis of sexually transmitted disease epidemics, Rationality and Society, 14 (2002), 353-383. Google Scholar

[31]

R. J. Smith and S. M. Blower, Could disease-modifying HIV vaccines cause population-level perversity?, The Lancet Infectious Diseases, 4 (2004), 636-639. Google Scholar

[32]

J. M. StephensonJ. ImrieM. M. D. DavisC. MercerS. BlackA. J. CopasG. J. HartO. R. Davidson and I. G. Williams, Is use of antiretroviral therapy among homosexual men associated with increased risk of transmission of HIV infection?, Sexually Transmitted Infections, 79 (2003), 7-10. Google Scholar

[33]

S. TullyM. G. Cojocaru and C. T. Bauch, Coevolution of risk perception, sexual behaviour, and HIV transmission in an agent-based model, Journal of theoretical biology, 337 (2013), 125-132. doi: 10.1016/j.jtbi.2013.08.014. Google Scholar

[34]

P. T. Harker, Generalized Nash games and quasi-variational inequalities, European Journal of Operational Research, 54 (1991), 81-94. Google Scholar

[35]

S. Tully, M. G. Cojocaru and C. T. Bauch, Sexual behaviour, risk perception, and HIV transmission can respond to HIV antiviral drugs and vaccines through multiple pathways, Scientific Reports, 5(2015).Google Scholar

[36]

J. X. Velasco-Hernandez and Y. H. Hsieh, Modelling the effect of treatment and behavioral change in HIV transmission dynamics, Journal of mathematical biology, 32 (1994), 233-249. Google Scholar

[37]

J. Von Neumann and O. Morgenstern, Theory of games and economic behavior, Bulletin of American Mathematical Society, 51 (1945), 498-504. Google Scholar

[38]

U. S. Census Bureau, Current Population Survey: Annual Social and Economic Supplement, 2012. Available at: https://www.census.gov/topics/population.html, (Accessed: Aug 2013).Google Scholar

[39]

University of Western News, HIV vaccine produces no adverse effects in trials, 2013, Available at: http://communications.uwo.ca/western_news/stories/2013/\September/hiv_vaccine_produces_no_adverse_effects_in_trials.html, (Accessed: Sept 2014).Google Scholar

[40]

Global Health Observatory, World Health Organization, 2014, Available at: http://www.who.int/gho/hiv/en/, (Accessed: Sept 2014).Google Scholar

[41]

Index Mundi, Zimbabwe Age structure, 2007, Available at: http://www.indexmundi.com/zimbabwe/age_structure.html, (Accessed: Aug 2014).Google Scholar

show all references

References:
[1]

K. J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy, Econometrica, 22 (1954), 265-290. Google Scholar

[2]

J. P. Aubin and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag New York, 1984. doi: 10.1007/978-3-642-69512-4. Google Scholar

[3]

R. M. Axelrod, The complexity of cooperation: Agent-based models of competition and collaboration, Princeton University Press, 1997.Google Scholar

[4]

T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory, SIAM, 1995. Google Scholar

[5]

A. Bensoussan, Points de Nash dans le cas de fonctionnelles quadratiques et jeux differentiels lineaires a N personnes, SIAM J. Control, 12 (1974), 460-499. Google Scholar

[6]

Center for Disease Control and Prevention, Diagnoses of HIV Infection in the United States and Dependent Areas, HIV Surveillance Report, 2011.Google Scholar

[7]

B. Coburn and S. Blower, A major HIV risk factor for young men who have sex with men is sex with older partners, J. Acquir. Immune Defic. Syndr., 54 (2010), 113-114. Google Scholar

[8]

M. G. Cojocaru and L. B. Jonker, Existence of solutions to projected differential equations on Hilbert spaces, Proceedings of the American Mathematical Society, 132 (2004), 183-193. doi: 10.1090/S0002-9939-03-07015-1. Google Scholar

[9]

Cojocaru, M. -G., Wild, E., On Describing the Solution Sets of Generalized Nash Games with Shared Constraints, submitted to: Optimization and Engineering, 2016.Google Scholar

[10]

M. G. CojocaruC. T. Bauch and M. D. Johnston, Dynamics of vaccination strategies via projected dynamical systems, Bulletin of mathematical biology, 69 (2007), 1453-1476. doi: 10.1007/s11538-006-9173-x. Google Scholar

[11]

M. G. Cojocaru, Dynamic equilibria of group vaccination strategies in a heterogeneous population, Journal of Global Optimization, 40 (2008), 51-63. doi: 10.1007/s10898-007-9204-7. Google Scholar

[12]

J. P. DoddsA. NardoneD. E. Mercey and A. M. Johnson, Increase in high risk sexual behaviour among homosexual men, London 1996-8: Cross sectional, questionnaire study, BMI, 320 (2000), 1510-1511. Google Scholar

[13]

F. FacchineiA. Fischer and V. Piccialli, On generalized Nash games and variational inequalities, Operations Research Letters, 35 (2007), 159-164. doi: 10.1016/j.orl.2006.03.004. Google Scholar

[14]

F. Fachinei and C. Kanzow, Generalized Nash Equilibrium Problems, Ann Oper Res, 175 (2010), 177-211. doi: 10.1007/s10479-009-0653-x. Google Scholar

[15]

J. W. Friedman, Game theory with applications to economics, Oxford University Press New York, 1990.Google Scholar

[16]

D. Gabay and H. Moulin, On the uniqueness and stability of Nash-equilibria in noncooperative games, Applied Stochastic Control in Econometrics and Management Science, 130 (1980), 271-293. Google Scholar

[17]

R. H. GrayM. J. WawerR. BrookmeyerN. K. SewankamboD. SerwaddaF. Wabwire-MangenT. LutaloX. LiT. VanCott and T. C. Quinn, Probability of HIV-1 transmission per coital act in monogamous, heterosexual, HIV-1-discordant couples in Rakai, Uganda, The Lancet., 357 (2001), 1149-1153. Google Scholar

[18]

D. Greenhalgh and F. Lewis, Stochastic models for the spread of HIV amongst intravenous drug users, Stochastic Models, 17 (2001), 491-512. doi: 10.1081/STM-120001220. Google Scholar

[19]

T. B. HallettS. GregsonO. MugurungiE. Gonese and G. P. Garnett, Assessing evidence for behaviour change affecting the course of HIV epidemics: a new mathematical modelling approach and application to data from Zimbabwe, Epidemics, 1 (2009), 108-117. Google Scholar

[20]

Y. H. Hsieh and C. H. Chen, Modelling the social dynamics of a sex industry: Its implications for spread of HIV/AIDS, Bulletin of Mathematical Biology, 66 (2004), 143-166. doi: 10.1016/j.bulm.2003.08.004. Google Scholar

[21]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998. doi: 10.1017/CBO9781139173179. Google Scholar

[22]

D. M. Huebner and M. A. Gerend, The relation between beliefs about drug treatments for HIV and sexual risk behavior in gay and bisexual men, Annals of Behavioral Medicine, 4 (2001), 304-312. Google Scholar

[23]

T. Ichiishi, Game Theory for Economic Analysis, New York: Academic Press, New York, 1983. Google Scholar

[24] D. Kinderlehrer and D. Stampacchia, An Introduction to Variational Inequalities and their Application, Academic Press, New York, 1980. Google Scholar
[25]

K. Nabetani, P. Tseng and M. Fukushima, Parametrized Variational Inequality Approaches to Generalized Nash Equilibrium Problems with Shared Constraints, (Technical Report 2008), Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Japan.Google Scholar

[26]

J. F. Nash, Equilibrium points in n-person games, CMS, 2 (2005), 21-56. Google Scholar

[27]

J.-S. Pang and M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games, CMS, 2 (2005), 21-56. doi: 10.1007/s10287-004-0010-0. Google Scholar

[28]

R. H. RemienP. N. HalkitisA. O'LearyR. J. Wolitski and C. A. Gómez, Risk perception and sexual risk behaviors among HIV-positive men on antiretroviral therapy, AIDS and Behavior, 9 (2005), 167-176. Google Scholar

[29]

J. B. Rosen, Existence and uniqueness of equilibrium points for concave n-person games, Econometrica, 33 (1965), 520-534. Google Scholar

[30]

K. D. Schroeder and F. G. Rojas, A game theoretical analysis of sexually transmitted disease epidemics, Rationality and Society, 14 (2002), 353-383. Google Scholar

[31]

R. J. Smith and S. M. Blower, Could disease-modifying HIV vaccines cause population-level perversity?, The Lancet Infectious Diseases, 4 (2004), 636-639. Google Scholar

[32]

J. M. StephensonJ. ImrieM. M. D. DavisC. MercerS. BlackA. J. CopasG. J. HartO. R. Davidson and I. G. Williams, Is use of antiretroviral therapy among homosexual men associated with increased risk of transmission of HIV infection?, Sexually Transmitted Infections, 79 (2003), 7-10. Google Scholar

[33]

S. TullyM. G. Cojocaru and C. T. Bauch, Coevolution of risk perception, sexual behaviour, and HIV transmission in an agent-based model, Journal of theoretical biology, 337 (2013), 125-132. doi: 10.1016/j.jtbi.2013.08.014. Google Scholar

[34]

P. T. Harker, Generalized Nash games and quasi-variational inequalities, European Journal of Operational Research, 54 (1991), 81-94. Google Scholar

[35]

S. Tully, M. G. Cojocaru and C. T. Bauch, Sexual behaviour, risk perception, and HIV transmission can respond to HIV antiviral drugs and vaccines through multiple pathways, Scientific Reports, 5(2015).Google Scholar

[36]

J. X. Velasco-Hernandez and Y. H. Hsieh, Modelling the effect of treatment and behavioral change in HIV transmission dynamics, Journal of mathematical biology, 32 (1994), 233-249. Google Scholar

[37]

J. Von Neumann and O. Morgenstern, Theory of games and economic behavior, Bulletin of American Mathematical Society, 51 (1945), 498-504. Google Scholar

[38]

U. S. Census Bureau, Current Population Survey: Annual Social and Economic Supplement, 2012. Available at: https://www.census.gov/topics/population.html, (Accessed: Aug 2013).Google Scholar

[39]

University of Western News, HIV vaccine produces no adverse effects in trials, 2013, Available at: http://communications.uwo.ca/western_news/stories/2013/\September/hiv_vaccine_produces_no_adverse_effects_in_trials.html, (Accessed: Sept 2014).Google Scholar

[40]

Global Health Observatory, World Health Organization, 2014, Available at: http://www.who.int/gho/hiv/en/, (Accessed: Sept 2014).Google Scholar

[41]

Index Mundi, Zimbabwe Age structure, 2007, Available at: http://www.indexmundi.com/zimbabwe/age_structure.html, (Accessed: Aug 2014).Google Scholar

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$$PSE(-, -)$=0.25$[0,1]$
$P_{1}$$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]$
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
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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