## Abstract

We consider a vaccination game that results with the introduction of premature and possibly scarce vaccines introduced in a desperate bid to combat the otherwise ravaging deadly pandemic. The response of unsure agents amid many uncertainties makes this game completely different from the previous studies. We construct a framework that combines SIS epidemic model with a variety of dynamic behavioral vaccination responses and demographic aspects. The response of each agent is influenced by the vaccination hesitancy and urgency, which arise due to their personal belief about efficacy and side-effects of the vaccine, disease characteristics, and relevant reported information (e.g., side-effects, disease statistics etc.). Based on such aspects, we identify the responses that are stable against static mutations. By analysing the attractors of the resulting ODEs, we observe interesting patterns in the limiting state of the system under evolutionary stable (ES) strategies, as a function of various defining parameters. There are responses for which the disease is eradicated completely (at limiting state), but none are stable against mutations. Also, vaccination abundance results in higher infected fractions at ES limiting state, irrespective of the disease death rate.

### Keywords

- Vaccination games
- ESS
- Epidemic
- Stochastic approximation
- ODEs

The work of first and second author is partially supported by Prime Minister’s Research Fellowship (PMRF), India.

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- 1.
Agents use dynamic policies, while mutants use static variants (refer Sect. 2).

- 2.
We assume mutants are more rational, estimate various rates using reported data.

- 3.
This is because the chances of infection before the next vaccination epoch decrease with increase in the availability rate \(\nu \).

- 4.
Observe that \(\hat{A}(\theta ) A = (\hat{A}(\hat{\theta }) + \widetilde{\theta }) A\), and \(\hat{A}(\hat{\theta }) = 0\).

- 5.
When \(\tilde{q}(\varUpsilon ) > 1\), the zeros are , otherwise they are the zeros of a quadratic equation with varying parameters, we have real zeros in this regime.

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## Appendices

### Appendix A: Stochastic approximation related proofs

### Lemma 2

Let \(\delta = 2/(N(0)-1)\). Then for any *k*, \(\eta _k \ge {\bar{\delta }} := \frac{N(0)-3}{(N(0)-1)^2}\) a.s. And thus,

Proof is provided in [12]. \(\blacksquare \)

### Lemma 3

The term \(\alpha ^m_k \rightarrow 0\) a.s., and, \(\sum _k \epsilon _k |\alpha ^m_k|<\infty \) a.s. for \(m=\theta , \psi \).

### Proof:

We will provide the proof for \(\alpha ^{\psi }_k\) and proof goes through in exactly similar line for \(\alpha ^\theta _k\). From equation (9), as in (10),

By Lemma 2 and because \(| G_{V,k+1} - (N_{k+1}- N_k) \psi _k | \le 2\) a.s., we have:

Thus we have:

\(\blacksquare \)

### Lemma 4

\(\sup _k E|L^m_k|^2<\infty \) for \(m=\theta , \psi ,\eta \).

### Proof:

The result follows by Lemma 2 and (9) (for an appropriate *C*):

\(\blacksquare \)

**Proof of Theorem** 1**:** As in [10], we will show that the following sequence of piece-wise constant functions that start with \(\varUpsilon _k\) are equicontinuous in extended sense. Then the result follows from [10, Chapter 5, Theorem 2.2]. Define \((\varUpsilon ^k(t))_k:=( \theta ^k(t),\psi ^k(t), \eta ^k(t))_k\) where,

where \(m(t):=\max \{k:t_k\le t\}\). This proof is exactly similar to that provided in the proof of [10, Chapter 5, Theorem 2.1] for the case with continuous *g*, except for the fact that \(g (\cdot )\) in our case is not continuous. We will only provide differences in the proof steps towards \((\theta ^k(t))_k\) sequence, and it can be proved analogously for others. Towards this we define \(M_k^{\theta }=\sum _{i=0}^{k-1}\epsilon _i\delta M^{\theta }_i \) with \(\delta M^{\theta }_k := L^{\theta }_{k+1}-g^{\theta }(\varUpsilon _k)-\alpha ^\theta _k\) as in [10] and show the required uniform continuity properties in view of Lemmas 3–4. Observe that \(\eta _k \le 1+ N(0)/k\), \(\theta _k \le 1\) for any *k*. Now the uniform continuity of integral terms like the following is achieved because our \(g (\cdot ) \) are bounded:

Such arguments lead to the required equicontinuity (details are in [12]). \(\blacksquare \)

### Appendix B: ODE Attractors Related Proofs

**Proof of Theorem** 2**:** Let \(\varUpsilon := (\theta , \psi , \eta )\). Let \(\hat{\varUpsilon }\) represent the corresponding attractors from Table 1. Here \(q(\theta , \psi ) = \min \{\tilde{q}(\theta , \psi ), 1\} \) with \(\tilde{q}(\theta , \psi ) = \beta \psi \).

We first consider the case where \(\tilde{q}(\hat{\theta }, \hat{\psi }) < 1\). Further, note that one can re-write ODEs, \(\dot{\varUpsilon } = g(\varUpsilon )\), as below:

\(A = A(\varUpsilon ) := (1- \theta - \psi )\lambda - r - b\), \(B = B(\varUpsilon ) := (1- \theta - \psi )\beta \nu - b\) and . To this end, we define the following Lyapunov function based on the regimes of parameters:

where \(\hat{A}(\theta ) := A(\theta , \hat{\psi }, \hat{\eta })\), \(\hat{B}(\psi ) := B(\hat{\theta }, \psi , \hat{\eta })\). We complete this proof using the above functions and the details are in [12]. \(\blacksquare \)

### Lemma 5

Let \(\hat{\theta }, \hat{\psi }> 0\). If \(\tilde{q}(\hat{\theta }, \hat{\psi }) \ne 1\), there exists a Lyapunov function such that \((\hat{\theta }, \hat{\psi }, \hat{\eta })\) is locally asymptotically stable attractor for ODE (11) in the sense of Lyapunov.

### Proof:

We use similar notations as in previous proof. Let us first consider the case where \(\tilde{q}(\hat{\theta }, \hat{\psi }) < 1\), i.e., \(q(\hat{\theta }, \hat{\psi }) = \tilde{q}(\hat{\theta }, \hat{\psi })\). Then, one can choose a neighborhood (further smaller, if required) such that \(\hat{q} - \delta< q(\theta , \psi ) < \hat{q} + \delta \), and \(q(\theta , \psi ) = \tilde{q}(\theta , \psi )\) for some \(\delta > 0\). Define the following Lyapunov function (for some \(w_1, w_2 > 0\), which would be chosen appropriately later):

\(\hat{A}(\theta ) := 1-\theta -\hat{\psi }- \frac{1}{\rho }\), and \(\hat{B}(\psi ) := \hat{q}(1-\hat{\theta }-\psi ) - \mu \psi \) (recall \(\hat{q} := q(\hat{\theta },\hat{\psi })\)). Call \(\widetilde{\theta }:= \hat{\theta }- \theta \) and \(\widetilde{\psi }:= \hat{\psi }- \psi \). The derivative of \(V(\varUpsilon (t))\) with respect to time is:

One can prove that the last component, i.e., \(- (C + \hat{\eta }- \eta ) C\) is strictly negative in an appropriate neighborhood of \(\hat{\varUpsilon }\) as in proof of Theorem 2. Now, we proceed to prove that other terms in \(\dot{V}\) (see (15)) are also strictly negative in a neighborhood of \(\hat{\varUpsilon }\).

Consider the term^{Footnote 4} \(\left( \hat{A}(\theta ) + \widetilde{\theta }\right) A\), call it \(A_1\):

where \(c_1\) will be chosen appropriately in later part of proof. Similarly the term corresponding to *B* is (details in [12]),

Thus, we get: \(\dot{V} < -A_1 \frac{ \theta \lambda w_1}{\eta \varrho } - B_1 \frac{\nu w_2}{\eta \varrho } \). Now, for \(\dot{V}\) to be negative, we need (using terms, in (16) and (17), corresponding to \(\widetilde{\theta }^2, \widetilde{\psi }^2\)):

By appropriately choosing the constants (for various agents), we complete the proof (details in [12]). \(\blacksquare \)

### Appendix C: ESS Related Proofs

### Lemma 6

Let \(\rho > 1\). Assume \(\tilde{q}(\varUpsilon )\ne 1\) where \(q(\varUpsilon ) = \min \{\tilde{q}(\varUpsilon ), 1\}\). Consider a policy \(\pi (\hat{\beta })\) where \(\pi \in \varPi \) and \(\hat{\varUpsilon }\) is the attractor of the corresponding ODE (11). Let \(\hat{\varUpsilon }_\epsilon \) be attractor corresponding to \(\epsilon \)-mutant of this policy, \(\pi _\epsilon (\hat{\beta }, p)\) for some \(p\in [0,1]\). Then, i) there exists an \({\bar{\epsilon }} (p) > 0\) such that the attractor is unique and is a continuous function of \(\epsilon \) for all \(\epsilon \le {\bar{\epsilon }}\) with \(\hat{\varUpsilon }_0 = {\hat{\varUpsilon }}\).

ii) Further \(\bar{\epsilon }\) could be chosen such that the sign of \(h(\hat{\varUpsilon }_\epsilon )\) remains the same as that of \(h(\hat{\varUpsilon })\) for all \(\epsilon \le {\bar{\epsilon }}\), when the latter is not zero. \(\blacksquare \)

### Proof:

We begin with an interior attractor. Such an attractor is a zero of a function like the following (e.g., for VFC1 it equals, see (11)):

Under mutation policy, \(\pi _\epsilon (\hat{\beta }, p)\), the function modifies to the following:

By directly computing the zero of this function, it is clear that we again have unique zero and these are continuous^{Footnote 5} in \(\epsilon \) (in some \({\bar{\epsilon }}\)-neighbourhood) and that they coincide with \(\hat{\varUpsilon }\) at \(\epsilon =0\). Further using Lyapunov function as defined in the corresponding proofs (with obvious modifications) one can show that these zeros are also attractors in the neighborhood. The remaining part of the proof is completed in [12]. The last result follows by continuity of *h* function (2). \(\blacksquare \)

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Singh, V., Agarwal, K., Shubham, Kavitha, V. (2021). Evolutionary Vaccination Games with Premature Vaccines to Combat Ongoing Deadly Pandemic. In: Zhao, Q., Xia, L. (eds) Performance Evaluation Methodologies and Tools. VALUETOOLS 2021. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 404. Springer, Cham. https://doi.org/10.1007/978-3-030-92511-6_12

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