Fig 1.
Scheme of the SVEIR model for MPX.
Table 1.
Times are in days, rates are per capita per day (and Λ is individuals per day). Details on model calibration are shown in Section 4.
Table 2.
Estimated effective reproduction number for the 2022 MPX outbreak using method from [83] on MPX reported cases data from [85].
Fig 2.
Parameter values and ranges as specified in Table 1. The average value of R0 is about 2.53.
Fig 3.
Actual (black dots) and predicted incidence.
Full red line is for β = 0.09, dashed is for β = 0.19 and dotted is for β = 0.29. All other parameters are as specified in Table 1.
Fig 4.
Uncertainty analysis of μ/ψHI.
Parameter values and ranges as specified in Table 1. For those values, μ/ψHI ≈ 0.61 meaning that to achieve herd immunity, the whole population at risk has to be vaccinated in about 61% of the average lifespan. The average value of μ/ψHI is about 0.53.
Fig 5.
Sensitivity analysis of μ/ψHI.
Parameter values and ranges as specified in Table 1. For those values, μ/ψHI ≈ 0.61 meaning that to achieve herd immunity, the whole population at risk has to be vaccinated in about 61% of the average lifespan.
Fig 6.
The optimal voluntary vaccination rates.
The lines are color coded corresponding to the regions shown in Fig 9 which shows a diagram for Nash equilibria as e and vary. Blue: 0 is the only NE and it is CSNE. Brown: positive ψNE is the only NE and it is CSNE. Red dashed line shows the value of ψHI. Unless varied or otherwise specified, the parameters are as in Table 1.
Fig 7.
MPX prevalence in a population that uses optimal voluntary vaccination rates.
The lines are color coded corresponding to the regions shown in Fig 9 which shows a diagram for Nash equilibria as e and vary. Blue: 0 is the only NE and it is CSNE. Brown: positive ψNE is the only NE and it is CSNE. Unless varied or otherwise specified, the parameters are as in Table 1.
Fig 8.
Annual MPX incidence in a population that uses optimal voluntary vaccination rates.
The lines are color coded corresponding to the regions shown in Fig 9 which shows a diagram for Nash equilibria as e and vary. Blue: 0 is the only NE and it is CSNE. Brown: positive ψNE is the only NE and it is CSNE. Unless varied or otherwise specified, the parameters are as in Table 1.
Fig 9.
Nash equilibria as e and vary.
Blue: 0 is the only NE and it is CSNE. Brown: positive ψNE is the only NE and it is CSNE. Unless varied or otherwise specified, the parameters are as in Table 1.
Fig 10.
Nash equilibria as e and vary for higher transmission rate, β = 0.18.
Other parameters are as in Table 1 unless they vary or are otherwise specified. Blue: 0 is the only NE and it is CSNE. Brown: positive ψNE < ψmax is the only NE and it is CSNE. Light blue: three NEs, 0 and the larger NE are CSNE. Red: maximal feasible vaccination rate is the only CSNE.
Fig 11.
Nash equilibria as R0 and vary; β is estimated by (21) as β ≈ R0γ.
Other parameters are as in Table 1. Blue: 0 is the only NE and it is CSNE. Brown: positive ψNE < ψmax is the only NE and it is CSNE. Light blue: three NEs, 0 and the larger NE are CSNE. Red: maximal feasible vaccination rate is the only CSNE.
Fig 12.
The incentive function for the parameters in the blue region of Fig 9 where 0 is the only NE and CSNE.
Full black circle is the CSNE, the red circle corresponds to ψHI. Unless varied or otherwise specified, the parameters are as in Table 1.
Fig 13.
The incentive function for the parameters in the brown region of Fig 9 where ψNE > 0 is the only NE and CSNE.
Full black circle is the CSNE, the red circle corresponds to ψHI. Unless varied or otherwise specified, the parameters are as in Table 1.
Fig 14.
The incentive function for the parameters in the light blue region of Fig 10.
There are three NE at the same time. Full black circles are the CSNE. The empty circle is NE that is not CSNE. The red circle corresponds to ψHI. Unless varied or otherwise specified, the parameters are as in Table 1.
Fig 15.
The incentive function for the parameters in the red region of Fig 10.
Full black circle is the CSNE. There is no ψHI. Unless varied or otherwise specified, the parameters are as in Table 1.
Fig 16.
Optimal voluntary vaccination rates when β = 0.18. Other parameters are as in Table 1 unless they vary or are otherwise specified.
The lines are color coded corresponding to the regions shown in Fig 10 which shows a diagram for Nash equilibria as e and vary. Blue: 0 is the only NE and it is CSNE. Brown: positive ψNE is the only NE and it is CSNE. Red dashed line shows the value of ψHI. Light blue shows a backward bifurcation when there are three NE at the same time. The full lines are CSNE, the dotted line is not.
Fig 17.
MPX prevalence in a population that uses optimal voluntary vaccination rates; β = 0.18 and other parameters as specified in Table 1.
The lines are color coded corresponding to the regions shown in Fig 10 which shows a diagram for Nash equilibria as e and vary. Blue: 0 is the only NE and it is CSNE. Brown: positive ψNE is the only NE and it is CSNE. Light blue: three NEs, 0 and the larger NE are CSNE.
Fig 18.
MPX incidence in a population that uses optimal voluntary vaccination rates; β = 0.18 and other parameters as specified in Table 1.
The lines are color coded corresponding to the regions shown in Fig 10 which shows a diagram for Nash equilibria as e and vary. Blue: 0 is the only NE and it is CSNE. Brown: positive ψNE is the only NE and it is CSNE. Light blue: three NEs, 0 and the larger NE are CSNE.
Fig 19.
Uncertainty analysis for the MPX prevalence in 104 unvaccinated population (the average is approximately 6.5).
Fig 20.
Uncertainty analysis for the MPX annual incidence per 104 unvaccinated population (the average is approximately 110).