By Lakshmi Chandrasekaran
As the world began to cautiously emerge from the ravages of COVID-19 in early 2022, a new global outbreak arose: monkeypox/mpox, a viral infectious zoonotic disease. According to the World Health Organization (WHO), the virus has spread to 110 countries thus far. WHO also reported that smallpox vaccines are roughly 85 percent effective in preventing mpox. Nevertheless, public health experts still debate a central question: Should public policy mandate vaccinations to prevent the spread of mpox, or should the decision to vaccinate be left to the individual? A group of undergraduate researchers at Virginia Commonwealth University—led by Jan Rychtář and Dewey Taylor—explored this question with mathematical modeling and game theory [1].
Endemic to Central and West Africa, mpox is classified as one of the neglected tropical diseases that primarily impact impoverished communities. The virus, which can have especially serious consequences for expectant mothers, may cause physical impairments and limit productivity. “While these diseases don’t have high death tolls, they are neglected in the true meaning of the word,” Rychtář said. He contrasted mpox with the fact that scientists developed the first mathematical model of malaria more than 100 years ago; nowadays, hundreds of models for malaria exist. “Our 2020 paper on monkeypox prevention [3] was one of the first mathematical studies of that disease,” Rychtář continued. “Only due to the current threat of spread in Europe and the U.S. is there now an uptick in monkeypox modeling efforts.”
Rychtář and his colleagues approached this problem with an unconventional tactic; they utilized a game theoretical perspective to understand vaccination strategies in the context of human behavior. This method begins with the susceptible-vaccinated-exposed-infected-recovered (SVEIR) compartmental model of mpox [6], given by
\[\frac{dS}{dt}=\Lambda-\left(\psi+\mu+\beta\frac{I}{N}\right)S,\tag1\]
\[\frac{dV}{dt}=\psi S-\left(\mu+(1-e)\beta\frac{I}{N}\right)V,\tag2\]
\[\frac{dE}{dt}=\left(\beta\frac{S}{N}+(1-e)\beta\frac{V}{N}\right)I-(\mu+\sigma)E,\tag3\]
\[\frac{dI}{dt}=\sigma E -(\mu+\gamma)I,\tag4\]
\[\frac{dR}{dt}=\gamma I-\mu R.\tag5\]
Here, \(\Lambda\) is the natural birth rate and \(\mu\) is the rate of death due to natural causes. Without vaccination, a susceptible individual becomes exposed after contact with an infectious person at rate \(\beta\frac{I}{N}\), where \(\beta\) is the transmission rate and \(N=\Lambda/\mu\) is the population size. Susceptible individuals are vaccinated at rate \(\psi\) with a vaccine efficacy of \(e\). Equations \((3)\) and \((4)\) capture the dynamics of exposed and infected states, and the infectious and incubation periods in equation \((5)\) are respectively given by \(\frac{1}{\gamma}\) and \(\frac{1}{\sigma}\).
The disease-free equilibrium is represented by \(\varepsilon^0=(S^0,V^0,E^0,I^0,R^0)\), where
\[S^0=N\frac{\mu}{\mu+\psi} \quad \textrm{and} \quad V^0=N\frac{\psi}{\mu+\psi}.\]
The basic reproduction number \((R_0)\) is the number of secondary infections that stem from a single infected individual in an otherwise healthy, unvaccinated population:
\[R_0=\frac{\sigma\beta}{(\sigma+\mu)(\gamma+\mu)}.\tag6\]
The researchers extended this SVEIR model by applying a game theoretical framework for a vaccination game, wherein individuals decide whether or not to vaccinate themselves. “Game theory models are useful for studying complex epidemiological scenarios in which self-interested individuals take actions based on the decisions of the rest of the population,” Rychtář said. He explained that vaccination games are akin to the tragedy of the commons — a well-studied model that pits the interests of a group against those of the individuals that comprise it [7]. For instance, one set of individuals may decide to pay taxes and contribute to the common good, while another set opts not to pay taxes but still reaps the benefits from those who do. Similarly, vaccination could be prone to “free riders” who avoid the costs of getting vaccinated themselves while benefiting from the immunity that vaccines confer on others. People tend to balance the perceived costs of vaccination against a vaccine’s effectiveness, and individuals’ actions generally maximize their own self-interest rather than the interests of the group.
Previous studies have utilized the extended model of disease transmission with game theory in the context of COVID-19 [5], as well as measles, smallpox, and other childhood illnesses [4]. The present model assumes that the vaccination game consists of susceptible and presumably rational individuals who act in their own self-interest.
Figure 1. Nash equilibria (NEs) as \(e\) and \(C^*_{MPX}\) vary. In the blue region, \(0\) is the only NE and it is a convergent stable NE (CSNE). In the brown region, positive \(\psi_{NE}\) is the only NE and is also a CSNE. Figure courtesy of [1].
The expected cost of mpox infection relative to the cost of vaccination is \(C^*_{MPX}=\frac{C_{MPX}}{C_V}\). Rychtář, Taylor, and their collaborators noted that there are many types of vaccination costs, including the actual price of the vaccine as well as indirect factors such as time away from work, travel expenses, and potential side effects. These costs may negatively affect the probability of achieving complete vaccination. On the other hand, mpox infection generally does not seem to impose a significant cost to individuals, apart from mild symptoms and time off of work (although some mpox cases can be severe). As a result, the researchers’ model assumes that the value of \(C^*_{MPX}\) will range from \(1\) to \(10\). In a real-world scenario, this value will likely differ for each individual.
With a population vaccination rate of \(\psi\), the incentive function \(h(\psi)\) for an individual to receive a vaccination is given by the difference between the expected costs of vaccination versus non-vaccination (including the potential cost of contracting mpox):
\[h(\psi)=C^*_{MPX}\left(\frac{\sigma}{\sigma+\mu}\right)(\pi_{NV}-\pi_V)-1.\tag7\]
Here, \(\left(\frac{\sigma}{\sigma+\mu}\right)\) is the probability that an exposed individual becomes infected. The probability that an unvaccinated individual is exposed to mpox is then
\[\pi_{NV}=\frac{\beta\frac{I^*}{N}}{\beta\frac{I^*}{N}+\mu},\tag8\]
where \(\beta\frac{I^*}{N}+\mu\) signifies the rate at which individuals who have no intention to vaccinate leave the “susceptible” compartment. The model assumes that the epidemic has reached a steady state with \(I^*\) infected individuals [8].
The probability that a vaccinated individual becomes exposed to mpox is
\[\pi_V=\frac{(1-e)\beta\frac{I^*}{N}}{(1-e)\beta\frac{I^*}{N}+\mu}.\tag9\]
The Nash equilibrium (NE)—the solution to the vaccination game—is the population vaccination rate \(\psi_{NE}\), which ensures that individuals do not have an incentive to deviate from the population strategy. “The existence of multiple optimal strategies was truly unique but not unexpected,” Rychtář said. “This was the consequence of an imperfect vaccine against mpox.” A perfect vaccine would imply that the efficacy \(e=1\), which is usually the assumption in vaccination games. But in the more general imperfect vaccine scenario of the current work, \(e<1\); consequently, there can be multiple NEs and roots of \(h\).
Figure 1 depicts the efficacy \(e\) versus the expected cost of infection \(C^*_{MPX}\), with \(R_0 \approx 2.11\) and \(\beta=0.09\). The blue region indicates that when \(C^*_{MPX}\) is low, people have minimal incentive to vaccinate. The only NE is \(\psi=0\), which is a stable solution that holds true even when \(C^*_{MPX}\) is high (and the disease is more dangerous) but \(e\) is low (between \(0.2\) and \(0.4\)). Under these conditions, not vaccinating is an optimal strategy from an individual’s perspective. Yet as \(C^*_{MPX}\) and \(e\) increase, the vaccination rate becomes positive \(\psi_{NE}\) — the only NE and stable. The brown region in Figure 1 indicates this optimal strategy, wherein individuals choose to vaccinate. For a low value of \(\beta\), the model predicts these scenarios as the only two optimal strategies.
But how do higher transmission rates of mpox impact the vaccination game? In Figure 2, \(\beta\) has increased to \(0.18\) and \(R_0 \approx 4.23\). There are now four patterns of NEs, which are marked by four different colored regions. In the bottommost blue region at low values of \(C^*_{MPX}\) (between \(0\) and \(2\)) or \(e\) (between \(0\) and \(0.4\)), the cost of mpox is too low for individuals to get vaccinated. Here, \(0\) is the only NE and it is stable.
“The brown region on the top is where we want to be or should be,” Rychtář said. In this area, \(C^*_{MPX}\) is high (the disease is more dangerous) but \(e\) is also high. The brown region signifies where people tend to vaccinate the most, and a positive \(\psi_{NE}\) is the only NE and is stable. The red region to the right is an unfortunate case, as the high \(C^*_{MPX}\) and low \(e\) indicate that mpox would not be eradicated even if people vaccinated as much as possible.
Figure 2. Nash equilibria (NEs) as \(e\) and \(C^*_{MPX}\) vary for a higher transmission rate, \(\beta=0.18\). In the blue region, \(0\) is the only NE and it is a convergent stable NE (CSNE). In the brown region, positive \(\psi_{NE}<\psi_{\textrm{max}}\) is the only NE and it is a CSNE. In the light blue region, there are three NEs; \(0\) and the larger NE are CSNEs. In the red region, the maximal feasible vaccination rate is the only CSNE. Figure courtesy of [1].
Finally, the light blue region in the middle of Figure 2 is particularly interesting because it has multiple NEs. Individuals in this region have two potentially optimal strategies that are stable NEs: to vaccinate at a higher rate or refrain from vaccination entirely. There is also a third, unstable NE wherein some people do vaccinate, but the voluntary vaccination rate may not be sufficient enough to substantially decrease the number of mpox cases. Vaccinating a larger group of individuals is therefore necessary to avoid the unstable region. Based on these predictions, Rychtář, Taylor, and their collaborators recommended a minimal mandated vaccination rate from a public health standpoint; otherwise, the overall population vaccination rate could decline to \(0\).
“Without game theory, our model would be largely like the model from previous work in [6],” Rychtář said. “It would calculate \(R_0\) and specify what vaccination coverage is needed to achieve herd immunity, i.e., what kind of vaccination effort the government must implement to protect its citizens. [A game theory perspective has] highlighted human behavior and the actions of individuals, as well as what they need to do to protect themselves from infection.”
The researchers mentioned that their current work includes many simplifying assumptions, such as a homogenous and well-mixed population — i.e., that everyone has the same amount of contact with infected individuals and that the risk of infection is homogenous. Additionally, if the perceived cost of mpox is smaller than the cost of vaccination in the current model, people will decide not to vaccinate. But in a real-world heterogenous population, some nonnegligible proportion of the group may still choose to receive the vaccination.
Nevertheless, scientists frequently incorporate human behavior into disease modeling to enhance a model’s realism. “The significance of this paper is that it shows the existence of multiple equilibria, a phenomenon that has not been previously studied in many details,” Hyunju Oh, an associate professor at the University of Guam who is not affiliated with the study, said. She also noted similar results for other infectious diseases [2], including COVID-19 [5].
“A game theory approach can even be applied to novel infectious diseases, especially because people may be hesitant to get new vaccines due to concerns about side effects,” Eunha Shim, a professor at Soongsil University who was also not associated with the study, said. Future research could better guide public health officials to make strategic vaccination decisions and help limit the spread of fatal infectious diseases.
References
[1] Augsburger, I.B., Galanthay, G.K., Tarosky, J.H., Rychtář, J., & Taylor, D. (2022). Voluntary vaccination may not stop monkeypox outbreak: A game-theoretic model. PLOS Negl. Trop. Dis., 16(12), e0010970.
[2] Augsburger, I.B., Galanthay, G.K., Tarosky, J.H., Rychtář, J., & Taylor, D. (2023). Imperfect vaccine can yield multiple Nash equilibria in vaccination games. Math. Biosci., 356, 108967.
[3] Bankuru, S.V., Kossol, S., Hou, W., Mahmoudi, P., Rychtář, J., & Taylor, D. (2020). A game-theoretic model of monkeypox to assess vaccination strategies. PeerJ, 8, e9272.
[4] Bauch, C.T., & Earn, D.J.D. (2004). Vaccination and the theory of games. Proc. Natl. Acad. Sci., 101(36), 13391-13394.
[5] Choi, W., & Shim, E. (2021). Optimal strategies for social distancing and testing to control COVID-19. J. Theor. Biol., 512, 110568.
[6] Grant, R., Nguyen, L.-B.L., & Breban, R. (2020). Modelling human-to-human transmission of monkeypox. Bull. World Health Organ., 98(9), 638-640.
[7] Hintze, A., Staudacher, J., Gelhar, K., Pothmann, A., Rasch, J., & Wildegger, D. (2020). Inclusive groups can avoid the tragedy of commons. Sci. Rep., 10, 22392.
[8] Usman, S., & Adamu, I.I. (2017). Modeling the transmission dynamics of the monkeypox virus infection with treatment and vaccination interventions. J. Appl. Math. Phys., 5(12), 2335-2353.
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Lakshmi Chandrasekaran holds a Ph.D. in mathematical sciences from the New Jersey Institute of Technology and a master’s degree in science journalism from Northwestern University. She is a freelance science writer whose work has appeared in MIT Technology Review, Quanta, Science News, and other outlets. |