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Nash-MFG equilibrium in a SIR model with time dependent newborn vaccination

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Abstract

We study the newborn, non compulsory, vaccination in a SIR model with vital dynamics. The evolution of each individual is modeled as a Markov chain. His/Her vaccination decision optimizes a criterion depending on the time-dependent aggregate (societal) vaccination rate and the future epidemic dynamics. We prove the existence of a Nash-Mean Field Games equilibrium among all individuals in the population. Then we propose a novel numerical approach to find the equilibrium and test it numerically.

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References

  1. Abakuks, A.: Optimal immunisation policies for epidemics. Adv. Appl. Probab. 6, 494–511 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anand, S., Hanson, K.: Disability-adjusted life years: a critical review. J. Health Econ. 16(6), 685–702 (1997)

    Article  Google Scholar 

  3. Anderson, R.M., May, R.M.: Infectious Diseases of Humans Dynamics and Control. Oxford University Press, Oxford (1992)

    Google Scholar 

  4. Bacaër, N.: A Short History of Mathematical Population Dynamics. Springer, London (2011)

    Book  MATH  Google Scholar 

  5. Bai, F.: Uniqueness of Nash equilibrium in vaccination games. J. Biol. Dyn. 10(1), 395–415 (2016). PMID: 27465224

    Article  MathSciNet  Google Scholar 

  6. Bauch, C.: Imitation dynamics predict vaccinating behaviour. Proc. Biol. Sci. 272(1573), 1669–1675 (2005)

    Article  Google Scholar 

  7. Bauch, C.T., Earn, D.J.D.: Vaccination and the theory of games. Proc. Natl. Acad. Sci. USA 101(36), 13391–13394 (2004). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bauch, C.T., Galvani, A.P., Earn, D.J.D.: Group interest versus self-interest in smallpox vaccination policy. Proc. Nat. Acad. Sci. 100(18), 10564–10567 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Benoussan, A., Frehse, J., Yam, P.: Mean Field Games and Mean Field Type Control Theory. Springer, New York (2013)

    Book  MATH  Google Scholar 

  10. Breban, R., Vardavas, R., Blower, S.: Mean-field analysis of an inductive reasoning game: application to influenza vaccination. Phys. Rev. E 76, 031127 (2007)

    Article  Google Scholar 

  11. Bressan, A.J.R., Rampazzo, F.: Impulsive control systems with commutative vector fields. J. Optim. Theory Appl. 71(1), 67–83 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brito, D.L., Sheshinski, E., Intriligator, M.D.: Externalities and compulsary vaccinations. J. Public Econ. 45(1), 69–90 (1991)

    Article  Google Scholar 

  13. Buonomo, B., d’Onofrio, A., Lacitignola, D.: Global stability of an SIR epidemic model with information dependent vaccination. Math. Biosci. 216(1), 9–16 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, F.H.: A susceptible-infected epidemic model with voluntary vaccinations. J. Math. Biol. 53(2), 253–272 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen, F.H.: Modeling the effect of information quality on risk behavior change and the transmission of infectious diseases. Math. Biosci. 217(2), 125–133 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Clarke, F.: Functional Analysis, Calculus of Variations and Optimal Control. Graduate Texts in Mathematics, vol. 264. Springer, London (2013)

    Book  Google Scholar 

  17. Clarke, F.H.: Generalized gradients and applications. Trans. Am. Math. Soc. 205, 247–262 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  18. Codeço, C.T., Luz, P.M., Coelho, F., Galvani, A.P., Struchiner, C.: Vaccinating in disease-free regions: a vaccine model with application to yellow fever. J. R. Soc. Interface 4(17), 1119–1125 (2007)

    Article  Google Scholar 

  19. Coelho, F.C., Codeço, C.T.: Dynamic modeling of vaccinating behavior as a function of individual beliefs. PLoS Comput. Biol. 5(7), e1000425 (2009)

    Article  MathSciNet  Google Scholar 

  20. Cojocaru, M.-G.: Dynamic equilibria of group vaccination strategies in a heterogeneous population. J. Global Optim. 40(1–3), 51–63 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cojocaru, M.-G., Bauch, C.T., Johnston, M.D.: Dynamics of vaccination strategies via projected dynamical systems. Bull. Math. Biol. 69(5), 1453–1476 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Diekmann, O., Heesterbeek, J.: Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation. Wiley Series in Mathematical and Computational Biology. Wiley, Chichester (1999)

    MATH  Google Scholar 

  23. d’Onofrio, A., Manfredi, P., Salinelli, E.: Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases. Theor. Popul. Biol. 71(3), 301–317 (2007)

    Article  MATH  Google Scholar 

  24. d’Onofrio, A., Manfredi, P., Salinelli, E.: Fatal SIR diseases and rational exemption to vaccination. Math. Med. Biol. 25(4), 337–357 (2008)

    Article  MATH  Google Scholar 

  25. Fan, K.: Fixed-point and minimax theorems in locally convex topological linear spaces. Proc. Natl. Acad. Sci. USA 38, 121–126 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fine, P.E.M., Clarkson, J.A.: Individual versus public priorities in the determination of optimal vaccination policies. Am. J. Epidemiol. 124(6), 1012–1020 (1986)

    Article  Google Scholar 

  27. Francis, P.J.: Optimal tax/subsidy combinations for the flu season. J. Econ. Dyn. Control 28(10), 2037–2054 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Fudenberg, D., Tirole, J.: Game theory. MIT Press, Cambridge (1991)

    MATH  Google Scholar 

  29. Fukuda, E., Kokubo, S., Tanimoto, J., Wang, Z., Hagishima, A., Ikegaya, N.: Risk assessment for infectious disease and its impact on voluntary vaccination behavior in social networks. Chaos Solitons Fractals 68, 1–9 (2014)

    Article  MATH  Google Scholar 

  30. Funk, S., Salathé, M., Jansen, V.A.A.: Modelling the influence of human behaviour on the spread of infectious diseases: a review. J. R. Soc. Interface 7(50), 1247–1256 (2010)

    Article  Google Scholar 

  31. Galvani, A.P., Reluga, T.C., Chapman, G.B.: Long-standing influenza vaccination policy is in accord with individual self-interest but not with the utilitarian optimum. Proc. Nat. Acad. Sci. 104(13), 5692–5697 (2007)

    Article  Google Scholar 

  32. Geoffard, P.-Y., Philipson, T.: Disease eradication: private versus public vaccination. Am. Econ. Rev. 87(1), 222–230 (1997)

    Google Scholar 

  33. Glicksberg, I.: A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points. Proc. Am. Math. Soc. 3, 170–174 (1952)

    MathSciNet  MATH  Google Scholar 

  34. Gomes, D.A., Mohr, J., Souza, R.R.: Continuous time finite state mean field games. Appl. Math. Optim. 68(1), 99–143 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Guéant, O., Lasry, J.-M., Lions, P.-L.: Mean field games and applications. In: Paris-Princeton Lectures on Mathematical Finance, vol. 2003 of Lecture Notes in Math. Springer, Berlin, vol. 2011, pp. 205–266 (2010)

  36. Hethcote, H.W., Waltman, P.: Optimal vaccination schedules in a deterministic epidemic model. Math. Biosci. 18(3–4), 365–381 (1973)

    Article  MATH  Google Scholar 

  37. Huang, M., Malhamé, R.P., Caines, P.E.: Nash equilibria for large-population linear stochastic systems of weakly coupled agents. In: Boukas, E., Malhamé, R.P. (eds.) Analysis, Control and Optimization of Complex Dynamic Systems, pp. 215–252. Springer, New York (2005)

    Chapter  Google Scholar 

  38. Huang, M., Malhamé, R.P., Caines, P.E.: Large population stochastic dynamic games: closed-loop mckean-vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–252 (2006)

    MathSciNet  MATH  Google Scholar 

  39. Kelley, J.L.: General Topology, 2nd edn. Springer, New York (1975)

    MATH  Google Scholar 

  40. Lachapelle, A., Salomon, J., Turinici, G.: Computation of mean field equilibria in economics. Math. Models Methods Appl. Sci. 20(4), 567–588 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Laguzet, L., Turinici, G.: Global optimal vaccination in the SIR model: Properties of the value function and application to cost-effectiveness analysis. Math. Biosci. 263, 180–197 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. Laguzet, L., Turinici, G.: Individual vaccination as Nash equilibrium in a SIR model with application to the 2009–2010 influenza A (H1N1) epidemic in France. Bull. Math. Biol. 77(10), 1955–1984 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. Laguzet, L., Turinici, G., Yahiaoui, G.: Equilibrium in an individual—societal sir vaccination model in presence of discounting and finite vaccination capacity. In: New Trends in Differential Equations, Control Theory and Optimization, pp. 201–214

  44. Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. I: Le cas stationnaire. C.R. Math. Acad. Sci. Paris 343(9), 619–625 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. II: Horizon fini et contrôle optimal. C.R. Math. Acad. Sci. Paris 343(10), 679–684 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  46. Manfredi, P., d’Onofrio, A.: Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases. Springer, New York (2013)

    Book  MATH  Google Scholar 

  47. Morton, R., Wickwire, K.H.: On the optimal control of a deterministic epidemic. Adv. Appl. Probab. 6, 622–635 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  48. Müller, J.: Optimal vaccination strategies—for whom? Math. Biosci. 139(2), 133–154 (1997)

    Article  MATH  Google Scholar 

  49. Reluga, T.C., Bauch, C.T., Galvani, A.P.: Evolving public perceptions and stability in vaccine uptake. Math. Biosci. 204(2), 185–198 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  50. Reluga, T.C., Galvani, A.P.: A general approach for population games with application to vaccination. Math. Biosci. 230(2), 67–78 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  51. Salvarani, F., Turinici, G.: Optimal individual strategies for influenza vaccines with imperfect efficacy and limited persistence. Math. Biosci. Eng., in print (2017)

  52. Sassi, F.: Calculating QALYs, comparing QALY and DALY calculations. Health Policy Plan. 21(5), 402–408 (2006)

    Article  Google Scholar 

  53. Sethi, S.P., Staats, P.W.: Optimal control of some simple deterministic epidemic models. J. Oper. Res. Soc. 29(2), 129–136 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  54. Shim, E., Chapman, G.B., Townsend, J.P., Galvani, A.P.: The influence of altruism on influenza vaccination decisions. J. R. Soc. Interface 9(74), 2234–2243 (2012)

    Article  Google Scholar 

  55. Turinici, G.: Metric gradient flows with state dependent functionals: the Nash-MFG equilibrium flows and their numerical schemes. Nonlinear Anal., in print (2017)

  56. Vardavas, R., Breban, R., Blower, S.: Can influenza epidemics be prevented by voluntary vaccination? PLoS. Comput. Biol. 3(5), e85 (2007)

    Article  Google Scholar 

  57. Wang, Z., Bauch, C.T., Bhattacharyya, S., d’Onofrio, A., Manfredi, P., Perc, M., Perra, N., Salathé, M., Zhao, D.: Statistical physics of vaccination. Phys. Rep. 664, 1–113 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  58. Zeckhauser, R., Shepard, D.: Where now for saving lives? Law Contemp. Probl. 40, 5–45 (1976)

    Article  Google Scholar 

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Acknowledgements

G.T. acknowledges support from the Agence Nationale de la Recherche (ANR), projects EMAQS (ANR-2011-BS01-017-01), CINE-PARA and MFG (ANR-16-CE40-0015-01).

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Correspondence to Gabriel Turinici.

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This article belongs to the Special Issue: Demographic and temporal heterogeneity in infectious disease epidemiology.

Appendices

Clarke generalized gradients

To recall the definition of the Clarke generalized gradients we follow the presentation in [16, section 10.1 page 194] and [17]. Let X be a Banach space, \(X^*\) its dual and \(x \in X\); also take \(f: X \rightarrow {\mathbb R}\) to be a functional which is Lipschitz with constant \(L>0\) in a neighborhood of x, that is, for some \(\epsilon > 0\), we have \(\Vert f(y)-f(z) \Vert \le L \Vert y-z \Vert \) for all yz in the ball of center x and radius \(\epsilon \). The generalized directional derivative of f at x in the direction v, denoted \(f^o(x ; v)\), is defined as

$$\begin{aligned} f^o(x,v) = \limsup _{y \rightarrow x, t \downarrow 0 } \frac{f(y+tv)-f(y)}{t}. \end{aligned}$$
(46)

Note that \(\Vert f^o(x,v) \Vert \le L \Vert v \Vert \) for any \(v \in X\); moreover, as function of v, the directional derivative \(f^o(x,v)\) is subadditive i.e. \( f^o(x,v+w) \le f^o(x,v)+ f^o(x,w)\), \(\forall v,w \in X\). In particular it can be lower bounded by a linear functional in \(X^*\). The (Clarke) generalized gradient of f at x denoted \(\overline{\partial }{f}(x)\) or \(\dot{f}(x)\) is the set of all such linear functionals; the formal definition is the following:

$$\begin{aligned} \overline{\partial }{f}(x) = \{ \xi \in X^* | f^o(x,v) \ge \langle v , \xi \rangle , \forall v \in X \}. \end{aligned}$$
(47)

It can be shown that the Clarke generalized gradient is a non empty, convex, (weakly-\(*\)) compact subset of \(X^*\). In particular when \(X={\mathbb R}^k\) for some \(k \in {\mathbb N}^*\), \(\overline{\partial }{f}(x)\) is the convex hull of the set \(\{ \lim _{\ell \rightarrow \infty } \nabla f(x_\ell ) \}\) for any sequence \(x_\ell \) converging to x such that:

  • \(\nabla f(x_\ell )\) exists \(\forall \ell \) (recall that since f is Lipschitz it is differentiable a.e.) and

  • the limit \(\lim _{\ell \rightarrow \infty } \nabla f(x_\ell )\) exists.

Technical details concerning the probability of infection

Recall that \( \phi _I^{u} \left( . \right) \) is a function from \( \mathbb {R}_{+} \) to \( \left[ 0, 1 \right] \) such that, for any \(t \in \mathbb {R}_{+} \), \( \phi _I^{u} \left( t \right) \) is the probability of infection during the life of an individual, born in t and not vaccinated, when the population follows the vaccination strategy u. In mathematical terms, for any individual born in \(t \ge 0\),

$$\begin{aligned} \phi _I^{u} \left( t \right) = \mathbb {P} \left( \exists \tau \ge t \text { such that } M_{ \tau }^t = I \; \vert \; M_{ t }^t = S \right) . \end{aligned}$$
(48)

In order to compute \(\phi _I^{u} \left( t \right) \) we introduce the probability \(\varphi _I^{u,t} \left( . \right) \) of infection before \(\tau \):

$$\begin{aligned} \varphi _I^{u,t} \left( \tau \right) = \mathbb {P} \left( \exists s \in \left[ t, \tau \right] \text { such that } M^{t}_{ s } = I \; \vert \; M^{t}_{ t } = S \right) . \end{aligned}$$
(49)

Hence, we have:

$$\begin{aligned} \varphi _I^{u,t} \left( \tau + \varDelta \tau \right)&= \mathbb {P} \left( M^{t}_{ \tau + \varDelta \tau } \in I \cup R \cup D_2 \; \vert \; M^{t}_{ t } = S \right) \\&= \mathbb {P} \left( M^{t}_{ \tau + \varDelta \tau } \in I \cup R \cup D_2 \; \vert \; M^{t}_{ \tau } = S \right) \times \mathbb {P} \left( M^{t}_{ \tau } = S \; \vert \; M^{t}_{ t } = S \right) \\&\quad + \mathbb {P} \left( M^{t}_{ \tau + \varDelta \tau } \in I \cup R \cup D_2 \; \vert \; M^{t}_{ \tau } = I \right) \times \mathbb {P} \left( M^{t}_{ \tau } = I \; \vert \; M^{t}_{ t } = S \right) \\&\quad + \mathbb {P} \left( M^{t}_{ \tau + \varDelta \tau } \in I \cup R \cup D_2 \; \vert \; M^{t}_{ \tau } = R \right) \times \mathbb {P} \left( M^{t}_{ \tau } = R \; \vert \; M^{t}_{ t } = S \right) \\&\quad + \mathbb {P} \left( M^{t}_{ \tau + \varDelta \tau } \in I \cup R \cup D_2 \; \vert \; M^{t}_{ \tau } = D_1 \right) \times \mathbb {P} \left( M^{t}_{ \tau } = D_1 \; \vert \; M^{t}_{ t } = S \right) \\&\quad + \mathbb {P} \left( M^{t}_{ \tau + \varDelta \tau } \in I \cup R \cup D_2 \; \vert \; M^{t}_{ \tau } = D_2 \right) \times \mathbb {P} \left( M^{t}_{ \tau } = D_2 \; \vert \; M^{t}_{ t } = S \right) , \end{aligned}$$

with

$$\begin{aligned} \mathbb {P} \left( M^{t}_{ \tau + \varDelta \tau } \in I \cup R \cup D_2 \; \vert \; M^{t}_{ \tau } = D_1 \right)&= 0 \\ \mathbb {P} \left( M^{t}_{ \tau + \varDelta \tau } \in I \cup R \cup D_2 \; \vert \; M^{t}_{ \tau } = I \right)&= 1 \\ \mathbb {P} \left( M^{t}_{ \tau + \varDelta \tau } \in I \cup R \cup D_2 \; \vert \; M^{t}_{ \tau } = R \right)&= 1 \\ \mathbb {P} \left( M^{t}_{ \tau + \varDelta \tau } \in I \cup R \cup D_2 \; \vert \; M^{t}_{ \tau } = D_2 \right)&= 1 \\ \mathbb {P} \left( M^{t}_{ \tau + \varDelta \tau } \in I \cup R \cup D_2 \; \vert \; M^{t}_{ \tau } = S \right)&= \beta I^{u} \left( \tau \right) \varDelta \tau + o(\varDelta \tau ). \end{aligned}$$

Hence,

$$\begin{aligned} \varphi _I^{u,t} \left( \tau + \varDelta \tau \right)&= \beta I^{u} \left( \tau \right) \varDelta \tau \times \mathbb {P} \left( M^{t}_{ \tau } = S \; \vert \; M^{t}_{ t } = S \right) \\&\quad + \mathbb {P} \left( M^{t}_{ \tau } \in I \cup R \cup D_2 \; \vert \; M^{t}_{ t } = S \right) + o(\varDelta \tau ) \\&= \beta I^{u} \left( \tau \right) \varDelta \tau \times \mathbb {P} \left( M^{t}_{ \tau } = S \; \vert \; M^{t}_{ t } = S \right) + \varphi _I^{u,t} \left( \tau \right) + o(\varDelta \tau ). \end{aligned}$$

We denote \( r^{u,t} \left( \tau \right) = \mathbb {P} \left( M^{t}_{ \tau } = S \; \vert \; M^{t}_{ t } = S \right) \) the probability of staying susceptible between t and \( \tau \). We compute this probability:

$$\begin{aligned} r^{u,t} \left( \tau + \varDelta \tau \right)&= \mathbb {P} \left( M^{t}_{ \tau + \varDelta \tau } = S \; \vert \; M^{t}_{ \tau } = S \right) \times \mathbb {P} \left( M^{t}_{ \tau } = S \; \vert \; M^{t}_{ t } = S \right) \\&= \left( 1 - \beta I^{u} \left( \tau \right) \varDelta \tau - \mu \varDelta \tau \right) r^{u,t} \left( \tau \right) + o(\varDelta \tau ). \end{aligned}$$

Hence, the probability of staying susceptible between t and \( \tau \) is:

$$\begin{aligned} r^{u,t} \left( \tau \right) = e^{ - \mu \left( \tau - t \right) } \exp \left( - \int _t^{ \tau } \beta I^{u} \left( s \right) ds \right) . \end{aligned}$$

Hence,

$$\begin{aligned} \varphi _I^{u,t} \left( \tau + \varDelta \tau \right) = \beta I^{u} \left( \tau \right) \varDelta \tau e^{ - \mu \left( \tau - t \right) } \exp \left( - \int _t^{ \tau } \beta I^{u} \left( s \right) ds \right) + \varphi _I^{u,t} \left( \tau \right) + o(\varDelta \tau ), \end{aligned}$$

which leads to

$$\begin{aligned} \frac{d \varphi _I^{u,t} \left( \tau \right) }{d \tau }&= \beta I^{u} \left( \tau \right) e^{ - \mu \left( \tau - t \right) } \exp \left( - \int _t^{ \tau } \beta I^{u} \left( s \right) ds \right) \\&= e^{ - \mu \left( \tau - t \right) } \exp \left( \int _0^t \beta I^{u} \left( s \right) ds \right) \times \beta I^{u} \left( \tau \right) \exp \left( - \int _0^{ \tau } \beta I^{u} \left( s \right) ds \right) \\&= - e^{- \mu \left( \tau - t \right) } \exp \left( \int _0^t \beta I^{u} \left( s \right) ds \right) \dfrac{d \left[ \exp \left( - \int _0^{ \tau } \beta I^{u} \left( s \right) ds \right) \right] }{d \tau } \\&= - e^{- \mu \left( \tau - t \right) } \exp \left( \int _0^t \beta I^{u} \left( s \right) ds \right) \dfrac{d \psi _I^{u} \left( \tau \right) }{d \tau }, \\ \end{aligned}$$

by setting \( \psi _I^{u} \left( \tau \right) = \exp \left( - \int _0^{ \tau } \beta I^{u} \left( s \right) ds \right) \).

Finally, we just have to compute the probability of infection during the life of an individual born in t who is not vaccinated, which is:

$$\begin{aligned} \phi _I^{u} \left( t \right)&= \int _t^{ + \infty } d \varphi _I^{u,t} \left( \tau \right) \\&= - \int _t^{ + \infty } \exp \left( - \mu \left( \tau - t \right) + \int _0^t \beta I^{u} \left( s \right) ds \right) \left[ \psi _I^{u} \left( \tau \right) \right] ' \; d \tau \\&= - \exp \left( \mu t + \int _0^t \beta I^{u} \left( s \right) ds \right) \int _t^{ + \infty } e^{ - \mu \tau } \left[ \psi _I^{u} \left( \tau \right) \right] ' \; d \tau . \end{aligned}$$

By an integration by parts, we obtain:

$$\begin{aligned} \phi _I^{u} \left( t \right)&= 1 - \int _t^{ + \infty } \mu \exp \left( - \int _t^{ \tau } \left( \mu + \beta I^{u} \left( s \right) \right) ds \right) \; d \tau . \end{aligned}$$

In fact, this probability is solution of the differential equation:

$$\begin{aligned} \dfrac{d \phi _I^{u} \left( t \right) }{dt}&= \mu \exp \left( - \int _t^t \left( \mu + \beta I^{u} \left( s \right) \right) ds \right) \\&\quad - \int _t^{ + \infty } \mu \dfrac{\partial }{\partial t} \left[ \exp \left( - \int _t^{ \tau } \left( \mu + \beta I^{u} \left( s \right) \right) ds \right) \right] \; d \tau \\&= \mu - \mu \int _t^{ + \infty } \left( \mu + \beta I^{u} \left( t \right) \right) \exp \left( - \int _t^{ \tau } \left( \mu + \beta I^{u} \left( s \right) \right) ds \right) \; d \tau \\&= \mu - \left( \mu + \beta I^{u} \left( t \right) \right) \left( 1 - \phi _I^{u} \left( t \right) \right) = \left( \mu + \beta I^{u} \left( t \right) \right) \phi _I^{u} \left( t \right) - \beta I^{u} \left( t \right) . \end{aligned}$$

In order to get an explicit form for \(\phi _I^{u} \left( 0 \right) \), we define, for all \(t \ge 0\):

$$\begin{aligned} f_I^{u} \left( t \right)&= \exp \left[ - \int _0^t \left( \mu + \beta I^{u} \left( \tau \right) \right) d \tau \right] \\ F_I^{u} \left( t \right)&= \int _0^t \mu f_I^{u} \left( \tau \right) d\tau \\ \mathcal {F}_I^{u} \left( t \right)&= 1 - \phi _I^{u} \left( t \right) . \end{aligned}$$

The last function satisfies the following differential equation:

$$\begin{aligned} \left[ \mathcal {F}_I^{u} \left( t \right) \right] '&= - \mu + \left( \mu + \beta I^{u} \left( t \right) \right) \mathcal {F}_I^{u} \left( t \right) \\ \mathcal {F}_I^{u} \left( 0 \right)&= \int _0^{ + \infty } \mu \exp \left( - \int _0^{ \tau } \left( \mu + \beta I^{u} \left( s \right) \right) ds \right) \; d \tau \\&= \int _0^{ + \infty } \mu f_I^{u} \left( \tau \right) d\tau = F_I^{u} \left( \infty \right) . \end{aligned}$$

Note that:

$$\begin{aligned} \left[ \mathcal {F}_I^{u} \left( t \right) \times f_I^{u} \left( t \right) \right] '&= \left\{ \left[ \mathcal {F}_I^{u} \left( t \right) \right] ' - \mathcal {F}_I^{u} \left( t \right) \left( \mu + \beta I^{u} \left( t \right) \right) \right\} \times f_I^{u} \left( t \right) = - \mu f_I^{u} \left( t \right) \end{aligned}$$

Thus, \( \mathcal {F}_I^{u} \left( t \right) \times f_I^{u} \left( t \right) = \mathcal {F}_I^{u} \left( 0 \right) - \int _0^t \mu f_I^{u} \left( \tau \right) \) and therefore

$$\begin{aligned} \mathcal {F}_I^{u} \left( t \right) = \dfrac{1}{ f_I^{u} \left( t \right) } \left[ F_I^{u} \left( \infty \right) - F_I^{u} \left( t \right) \right] . \end{aligned}$$

Hence, for all \(t \ge 0\), we obtain:

$$\begin{aligned} \phi _I^{u} \left( t \right) = 1 - \dfrac{ F_I^{u} \left( \infty \right) - F_I^{u} \left( t \right) }{ f_I^{u} \left( t \right) }, \ \phi _I^{u} \left( 0 \right) = 1 - F_I^{u} \left( \infty \right) . \end{aligned}$$

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Hubert, E., Turinici, G. Nash-MFG equilibrium in a SIR model with time dependent newborn vaccination. Ricerche mat 67, 227–246 (2018). https://doi.org/10.1007/s11587-018-0365-0

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