Multiplayer games and HIV transmission via casual encounters

Stephen Tully; Monica-Gabriela Cojocaru; Chris T. Bauch

doi:10.3934/mbe.2017023

Article Contents

2017, Volume 14, Issue 2: 359-376. Doi: 10.3934/mbe.2017023

This issue Previous Article Next Article Optimal control analysis of malaria-schistosomiasis co-infection dynamics

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 26, 2015

Revised: June 26, 2016

Published: July 2016

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

Abstract Full Text(HTML) Figure(8) / Table(2) Related Papers Cited by

Abstract

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.

Keywords:

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

Citation:

FullText(HTML)

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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 $\mu=0.75$

Download: Full-size image PowerPoint slide

Table 1. 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	$[0,1]$

| Show Table

DownLoad: CSV

Table 2. 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

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	K. J. Arrow and G. Debreu, Existence of an equilibrium for a competitive economy, Econometrica, 22 (1954), 265-290.
[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.
[3]	R. M. Axelrod, The complexity of cooperation: Agent-based models of competition and collaboration, Princeton University Press, 1997.
[4]	T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory, SIAM, 1995.
[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.
[6]	Center for Disease Control and Prevention, Diagnoses of HIV Infection in the United States and Dependent Areas, HIV Surveillance Report, 2011.
[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.
[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.
[9]	Cojocaru, M. -G., Wild, E., On Describing the Solution Sets of Generalized Nash Games with Shared Constraints, submitted to: Optimization and Engineering, 2016.
[10]	M. G. Cojocaru, C. 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.
[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.
[12]	J. P. Dodds, A. Nardone, D. 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.
[13]	F. Facchinei, A. 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.
[14]	F. Fachinei and C. Kanzow, Generalized Nash Equilibrium Problems, Ann Oper Res, 175 (2010), 177-211. doi: 10.1007/s10479-009-0653-x.
[15]	J. W. Friedman, Game theory with applications to economics, Oxford University Press New York, 1990.
[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.
[17]	R. H. Gray, M. J. Wawer, R. Brookmeyer, N. K. Sewankambo, D. Serwadda, F. Wabwire-Mangen, T. Lutalo, X. Li, T. 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.
[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.
[19]	T. B. Hallett, S. Gregson, O. Mugurungi, E. 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.
[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.
[21]	J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998. doi: 10.1017/CBO9781139173179.
[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.
[23]	T. Ichiishi, Game Theory for Economic Analysis, New York: Academic Press, New York, 1983.
[24]	D. Kinderlehrer and D. Stampacchia, An Introduction to Variational Inequalities and their Application, Academic Press, New York, 1980.
[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.
[26]	J. F. Nash, Equilibrium points in n-person games, CMS, 2 (2005), 21-56.
[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.
[28]	R. H. Remien, P. N. Halkitis, A. O'Leary, R. 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.
[29]	J. B. Rosen, Existence and uniqueness of equilibrium points for concave n-person games, Econometrica, 33 (1965), 520-534.
[30]	K. D. Schroeder and F. G. Rojas, A game theoretical analysis of sexually transmitted disease epidemics, Rationality and Society, 14 (2002), 353-383.
[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.
[32]	J. M. Stephenson, J. Imrie, M. M. D. Davis, C. Mercer, S. Black, A. J. Copas, G. J. Hart, O. 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.
[33]	S. Tully, M. 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.
[34]	P. T. Harker, Generalized Nash games and quasi-variational inequalities, European Journal of Operational Research, 54 (1991), 81-94.
[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).
[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.
[37]	J. Von Neumann and O. Morgenstern, Theory of games and economic behavior, Bulletin of American Mathematical Society, 51 (1945), 498-504.
[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).
[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).
[40]	Global Health Observatory, World Health Organization, 2014, Available at: http://www.who.int/gho/hiv/en/, (Accessed: Sept 2014).
[41]	Index Mundi, Zimbabwe Age structure, 2007, Available at: http://www.indexmundi.com/zimbabwe/age_structure.html, (Accessed: Aug 2014).