1 Introduction

Vaccination is the best response available in the control of most infectious diseases. A huge effort is put on the development of new and better vaccines. When humans are directly involved, the role of direct experimentation is naturally limited and therefore mathematical models have been used to evaluate the effect of control measures, such as vaccination, to assist in policy decisions. One central result of classical mathematical models for the spread of infectious diseases is that persistence of an infectious disease within a population requires the density of susceptible individuals to exceed a strictly positive critical value such that, on average, each primary case of infection generates more than one secondary case. It is therefore not necessary to vaccinate everyone within a community to eliminate infection. This phenomenon is known as herd immunity and is one of the key epidemiological questions in defining a vaccination strategy.

In this work, we consider a SIRS model with periodic transmission. The model consists of a non-autonomous system of ordinary differential equations in which we introduce periodic vaccination of adults. For simplicity, we considered that vaccination confers the same protection as natural infection. We study the consequences of two extreme types of vaccination strategies: mandatory vaccination, where the population is vaccinated at a predefined rate; and voluntary vaccination, where individuals can choose freely to be vaccinated or not, according to their risk perception.

Mathematical models have been widely used to help health authorities in the definition of vaccination strategies for very different contexts. Typically, the objective is to define an optimal vaccination strategy by minimizing combinations of the vaccination effort/cost and of the effective reproduction number \(\mathcal {R}_0\) (i.e., the number of secondary infections generated by a primary case) (Müller and Hadeler 1996; Castillo-Chavez and Feng 1998; Laguzet and Turinici 2015a). This can give rise to particularly interesting problems when we consider non-homogeneous models for which vaccination strategy depends on age (Müller and Hadeler 1996; Castillo-Chavez and Feng 1998; Tartof et al. 2013), risk-groups (Scott et al. 2015; Long and Owens 2011) or when it is time-dependent (Onyango and Müller 2014; d’Onofrio 2002; Browne et al. 2015; Houy 2016). The current work is concerned with time-dependent epidemic models with vaccination when both the transmission rate and the vaccination are assumed to be periodic. Note that periodic vaccinations are used by public health services; e.g., influenza vaccine is only available in a specific season of the year.

Whenever the goal is long term disease elimination, optimal vaccination will consist on reducing \(\mathcal {R}_0\) below one, in the sense that it implies the attractiveness and the asymptotic stability of the disease-free state. Here, we choose to work with an alternative definition of optimal vaccination, where the goal is, not only to eliminate disease, but also to prevent outbreaks. Hence, we define a class of preventive vaccination profiles such that, for any sufficiently small perturbation of the disease free state, the number of infectious individuals is monotonically decreasing. We construct the optimal vaccination strategy as the limit of preventive strategies for which vaccination effort is minimized. We start by revisiting basic concepts in mathematical epidemiology to recall that for constant transmission, the condition \(\mathcal {R}_0\le 1\), and subsequent stability of the disease-free state, is equivalent to the condition that infectious population I is monotonically decreasing in time, when the initial number of susceptible individuals is below the number of susceptible individuals in the disease free state. However, as we move towards more general situations this equivalence may not hold. Note that the former condition refers to the long term behaviour of the system, while the latter considers also the short time behaviour which, in principle, is more restrictive (Hastings 2004, 2010). In particular, the former condition restricts the average transmission over a period and the latter is defined pointwise in time. Our approach is particularly suitable for diseases with high mortality or morbidity rates, for which it is imperative to prevent outbreaks. Despite that, so far our model and examples will not consider disease related death.

On the opposite end of vaccination policies is voluntary vaccination, which is increasingly common in industrialized countries. Even when vaccines are offered by the public health system without costs, vaccination is, at least in part, voluntary. Opposition to vaccines can be philosophical, religious and depend on social contacts and information available. It puts important challenges to disease control by decreasing vaccine uptake. The case which is best known is the measles, since the unproven hypotheses that measles-mumps-rubella (MMR) vaccine was linked to autism led to a decrease in vaccination followed by measles epidemics in UK (Fitzpatrick 2004; Jansen et al. 2003). Voluntary vaccination can also give rise to free-rider phenomenon, where individuals or families choose not to be vaccinated, or to not have their children vaccinated, taking advantage of herd immunity created in the population by others, avoiding the possible negative effects of vaccination. In this work, we consider a population of rational individuals that compares the risk of the vaccination (more precisely, its perception of the risk of the vaccination) and the risk of the disease and make options that minimizes the joint risk. Despite the fact that some countries are implementing fines for parents that prefer not to vaccinate their childrenFootnote 1, we do not introduce in the model a risk of non-vaccination, other than the one associated to the disease.

In this work, we model human behaviour using game theory. In a seminal paper by Bauch and Earn (2004), it has been shown that voluntary vaccination cannot lead to disease eradication. The authors coupled a SIR model for disease spread in a partially vaccinated population with a theoretical game framework describing a population of rational individuals. Many subsequent developments were made in order to include the human behaviour in epidemiological models (cf. Chen 2006; d’Onofrio et al. 2007; Manfredi et al. 2009; Coelho and Codeço 2009; Mbah et al. 2012; Manfredi and D’Onofrio 2013; Morin et al. 2013; Bhattacharyya et al. 2015; Laguzet and Turinici 2015b). See also (Funk et al. 2010; Wang et al. 2015) for a review, and (Funk et al. 2015) for further discussion on the subject.

In this paper, we generalize the framework of Bauch and Earn (2004) to the SIRS model with periodic transmission function. From the modelling point of view, we consider that all choices in the population influence the dynamics, and the resulting dynamics also has effect in the rational behaviour of the population. Due to the richness of the non-autonomous system that describes our model, several technical problems arise. For instance, the risk of disease no longer depends simply on the constant steady state as before. Considering a rational individual, we assume that he/she is going to choose to be vaccinated only when the risk of disease times the probability of being infected is higher than the risk of the vaccine, as perceived by the taker. As we analyse only stationary states of our periodic system, the risk to be minimized is the joint risk of vaccination and disease during one season. Hence, we define the set of herd immunity provider vaccination strategies, for which the rational strategy for a given focal (rational) individual is not to be vaccinated, taking advantage of the herd immunity provided by the choices of the rest of the population. Moreover, we define a Nash vaccination strategy as the strategy that minimizes the joint risk for every individual taking into account the strategy of all other individuals, i.e., the natural strategy to be expected in a population of rational individuals with full knowledge of all epidemiological data.

Existence of optimal and Nash vaccination strategies are proved in this work in a very general setting; however, these strategies may not be functions but Radon measures, even for transmission rates given by a real function. This is a consequence of the fact that the set of continuous functions in a given compact interval is not closed under any reasonable metric. Many results used in the existence proofs presented in the appendices require compactness and after introducing a convenient topology in the set of continuous functions, we are naturally led to the introduction, in this framework, of Radon measures. For more information on the topic of Radon measures we refer to (Schwartz 1973; Athreya and Lahiri 2006).

The paper is organized as follows. In Sect. 2, we introduce the mathematical model and derive some preliminary results. Section 3 is dedicated to the vaccination strategies. First, we give rigorous definitions of preventive vaccination strategies, and of vaccination effort and we define the optimal strategy as one strategy that can be arbitrarily approximated by a preventive strategy and such that its associated effort is never superior to the effort of any given preventive strategy. In the context of voluntary vaccination, we define the set of herd immunity provider strategies and the concept of Nash strategy, in which all individuals minimizes the joint risk of vaccination and disease. In the end of the Sect. 3.2, we state the main theorem, which guarantees the existence of an optimal and a Nash vaccination strategies in the set of Radon measures. Explicit formulas for the optimal and Nash strategies are provided in Sect. 3.3, for specific forms of the transmission rate. In Sect. 4, we present some examples such as the constant transmission case, the sinusoidal case and also a critical case to illustrate the results from previous sections. We finish with two appendices, the first one guaranteeing the existence of periodic solutions in the model and the second proving the existence of optimal and Nash strategies.

2 The model

Consider a SIRS model. Let S(t), I(t), R(t) be the fraction of susceptible, infectious and recovered individuals at time \(t\ge 0\). We assume non negative normalized initial conditions, i.e, \(S(0),I(0),R(0)\ge 0\), \(S(0)+I(0)+R(0)=1\). We also assume the transitions \(S+I\mathop {\longrightarrow }\limits ^{\beta }2I\), \(I\mathop {\longrightarrow }\limits ^{\gamma }R\), \(R\mathop {\longrightarrow }\limits ^{\alpha }S\), \(S\mathop {\longrightarrow }\limits ^{p}R\). Constants \(\mu \) (mortality/birth rate), \(\alpha \) (temporary immunity) and \(\gamma \) (recovery rate) are strictly positive. The disease is assumed to be non-fatal, i.e., the death rate \(\mu >0\) does not depend on the disease class. By normalization, we also consider the birth rate as \(\mu \). These are common assumptions of compartmental epidemiological models.

We consider functions \(\beta ,p:\mathbb {R}_+\rightarrow \mathbb {R}_+\), representing the transmission and vaccination rates at time t, respectively. More precise assumptions on these functions will be introduced latter on.

From now on, we call SIRS model to the following system of differential equations:

$$\begin{aligned} S'&=\mu +\alpha R-\beta (t)IS-p(t)S-\mu S \nonumber \\&=\mu +\alpha -\alpha I-\beta (t)IS-p(t)S-(\mu +\alpha )S \end{aligned}$$
(1a)
$$\begin{aligned} I'&=\beta (t)IS-\gamma I - \mu I \end{aligned}$$
(1b)
$$\begin{aligned} R'&=\gamma I + p(t) S-\mu R-\alpha R \end{aligned}$$
(1c)

A schematic representation of the SIRS model with vaccination is represented in Fig. 1. Due to the normalization \(S(t)+I(t)+R(t)=S(0)+I(0)+R(0)=1\), the equation for R is always redundant and will be ignored from now on. We define \({\varDelta }^2{:}{=}\{(x,y)\in \mathbb {R}^2|\,x\ge 0,y\ge 0,x+y\le 1\}\).

Fig. 1
figure 1

Schematic diagram of the SIRS model with death and vaccinations

Full size image

We begin by analysing the solutions and stability of system (1).

Lemma 1

Let us consider that functions \(\beta \) and p are continuous functions with commensurable periods, i.e., there exists \(T>0\) such that \(p(t+T)=p(t)\) and \(\beta (t+T)=\beta (t)\) for all t. Equivalently, we assume that \(\beta ,p \in C([0,T])\) with \((p(0),\beta (0))=(p(T),\beta (T))\).

Therefore, there exists only one periodic solution of system (1) in the subspace \(\{I(t)=0,\ \forall t\}\), given by \((S_0(t),0)\in {\varDelta }^2\). We call this solution the disease-free solution. This solution attracts all initial conditions of the form (S(0), 0). We define \(I_0(t)=0\).

Depending on the choices of the parameters \(\gamma ,\mu \) and the functions \(p,\beta \), we may have one of two possibilities:

  1. 1.

    The disease-free solution is globally stable in \({\varDelta }^2\);

  2. 2.

    There are other periodic solutions (with period multiple of T), called endemic solutions \((S_i(t),I_i(t))\in {\varDelta }^2\), with \(I_i(t)>0\), for all \(t\in \mathbb {R}_+\) and \(i\in \mathbb {N}\). In this case, there is \(\eta >0\) such that for any initial condition (S(0), I(0)) with \(I(0)>0\), we have \(\liminf _{t\rightarrow \infty } I(t)> \eta \). In this case, we say that the solution of the SIRS model is persistent.

Furthermore, for any initial condition, the solution (S(t), I(t)) depends continuously on \(\beta \) and p; namely, if \(p_n\rightarrow p\) and \(\beta _n\rightarrow \beta \) weakly as measures and both sequences are uniformly integrable, then \(S_i[p_n,\beta _m]\rightarrow S_i[p,\beta ]\) and \(I_i[p_n,\beta _m]\rightarrow I_i[p,\beta ]\) uniformly in [0, T], when \(n,m\rightarrow \infty \), \(i\in \mathbb {N}\).

Proof

For the first part, see Rebelo et al. (2012); see “Appendix 1” for further details. The second part follows from Heunis (1984, Theorem 2.1).

Remark 1

The assumptions on \(\beta ,p \) in the previous lemma are extremely restrictive and used only for the first part of the result (existence of disease free solution and periodicity of the endemic solution). If we relax our assumptions to require only that \(\beta \) is of bounded variation and p is a measurable function, then existence of solutions (not necessarily periodic) and convergence of solutions (as in the second part of Lemma 1) is guaranteed by (Heunis 1984). This will be explored in the examples. Note that if \(\beta _n,p_n\) are continuous then, \(\beta {:}{=}\lim \beta _n\) and \(p{:}{=}\lim p_n\) necessarily satisfy these more relaxed assumptions.

For constant transmission and vaccination, we establish the following result, that is going to motivate our definition of optimal vaccination.

Lemma 2

Consider that \(\beta (t)=\beta _0>0\) and \(p(t)=p_0\ge 0\). The only stationary disease free solution of system (1) is given by \(\hat{S}_0=\hat{S}_0[p]=S_0[p](t)=\frac{\mu +\alpha }{p_0+\mu +\alpha }\le 1\). Furthermore, the three conditions below are equivalent:

  1. 1.

    \(\frac{\beta _0\hat{S}_0}{\gamma +\mu } \le 1\).

  2. 2.

    The disease free solution \((\hat{S}_0,0)\) is globally asymptotically stable.

  3. 3.

    \(I'<0\) for all \(I>0\) and all \(S<\hat{S}_0\).

Proof

1\(\Leftrightarrow \)3. Note that \(I'=I(\gamma +\mu )\left( \frac{\beta _0 S}{\gamma +\mu }-1\right) \), and therefore, assuming \(I>0\), \(I'<0\) if and only if \(S<\frac{\gamma +\mu }{\beta _0}\).

      1 \(\Rightarrow \) 3. \(S<\hat{S}_0\le \frac{\gamma +\mu }{\beta _0}\) and then \(I'<0\).

      3 \(\Rightarrow \) 1. \(I'<0\) if and only if \(S<\frac{\gamma +\mu }{\beta _0}\) and therefore \(\hat{S}_0\le \frac{\gamma +\mu }{\beta _0}\).

1 \(\Leftrightarrow \) 2.

      1 \(\Rightarrow \) 2. We follow ideas from (Cruz 2009); see also (Capasso 1993) for other examples of use of Lyapunov functions in Mathematical Epidemiology.

Let us define

$$\begin{aligned} V(S,I)=\frac{1}{2}\left( S-\hat{S}_0+I\right) ^2+ a I, \quad a=\frac{2(\mu +\alpha )+\gamma +p_0}{\beta _0}. \end{aligned}$$

We differentiate V with respect to t and obtain

$$\begin{aligned} \begin{aligned} V'(S,I)=\frac{dV}{dt}=&\left( S-\hat{S}_0+I\right) (S'+I')+a I'\\ =&\left( S-\hat{S}_0+I\right) ( (\hat{S}_0-S)(p_0+\mu +\alpha ) -(\gamma +\alpha + \mu ) I)+a I'\\ =&-\left( S-\hat{S}_0\right) ^2(p_0+\mu +\alpha )-I^2(\gamma +\alpha + \mu )\\ {}&- (S-\hat{S}_0)I(2(\mu +\alpha )+\gamma +p_0) +a( \beta _0 IS-(\gamma + \mu )) I.\\ =&-\left( S-\hat{S}_0\right) ^2(p_0+\mu +\alpha )-I^2(\gamma +\alpha + \mu )\\ {}&-a \beta _0 I \left( \frac{\gamma + \mu }{\beta _0}-\hat{S}_0\right) .\\ \end{aligned} \end{aligned}$$

Let \(G=\{(S,I)\in [0,1]\times (0,1]|S+I\le 1\}\). Note that \(V'\) is a continuous function in G and, by Condition 1, it is negative. It is immediate to verify that V is a Lyapunov function in G. Let \(\bar{G}\) be the closure of G. As \(\{(S,I) \in \bar{G}| V'(S,I)=0\}\) is the singleton with the equilibrium point \(\{(\hat{S}_0,0)\}\) we conclude from Hale (2009, Corollary 1.2 in Chapter X) that \((\hat{S}_0,0)\) is globally asymptotically stable.

      2 \(\Rightarrow \) 1. After system linearisation around the disease free solution \((\hat{S}_0,0)\), we find the Jacobian matrix

$$\begin{aligned} \left( \begin{matrix} -p_0-\mu -\alpha &{}-\alpha -\beta _0\hat{S}_0\\ 0&{}(\gamma +\mu )\left( \frac{\beta _0\hat{S}_0}{\gamma +\mu }-1\right) \end{matrix}\right) . \end{aligned}$$

If \(\frac{\beta _0\hat{S}_0}{\gamma +\mu } > 1\), the equilibrium would not be stable, which leads to a contradiction.

From the above lemma, we recover the effective reproductive number for the constant parameter case, \(\mathcal {R}_0{:}{=}\mathcal {R}_{0}[p]{:}{=}\frac{\beta _0\hat{S}_0[p]}{\gamma +\mu }\). Condition \(\mathcal {R}_{0}\le 1\) guarantees at the same time that all epidemics will be eventually extinct and that I(t) decreases monotonically in time, from \(I(0)>0\).

However, in the time dependent case (in particular in the periodic case), these two phenomena are not equivalent. In general, even for linear systems, it is possible that before being attracted to an asymptotic equilibrium, the trajectory of (S(t), I(t)) drifts away from this equilibrium (Hastings 2004, 2010).

For the periodic case, we can compute the effective reproduction number, following (Thieme 2000) (see also Wang and Zhao 2008), as

$$\begin{aligned} \mathcal {R}_0\,{:}{=}\,\frac{1}{\gamma +\mu }\langle \beta S_0\rangle =\frac{1}{T(\gamma +\mu )}\int _0^T\beta (t)S_0(t)\mathrm {d}t. \end{aligned}$$
(2)

Note that, for the periodic case, condition \(\mathcal {R}_0<1\) still guarantees asymptotic stability of the disease free case (Wang and Zhao 2008), but does not necessarily prevent the existence of outbreaks; see, e.g., (Zhao 2008).

In this work, we will look for conditions that generalize, for time-dependent parameters, Condition 3 in Lemma 2, i.e., that guarantees that the number of infectious is monotonically decreasing for small perturbations of the disease free solution. From the modelling point of view, no particular definition can be considered better than the other; in fact, for certain particular diseases (e.g., polio, tuberculosis) vaccination policy aims to eradicate/eliminate the disease in the long run, while for other diseases, governments act to prevent the existence of large outbreaks (e.g., influenza, cholera) (WHO 2015). Our approach describes better this second setting.

From now on, we assume that, for a given vaccination strategy p(t), system (S(t), I(t)) is in its stationary (periodic) state, and we will consider two different cases:

  1. (C1)

    The disease free state \((S_0[p](t),0)\);

  2. (C2)

    A certain endemic state \((S_1[p](t),I_1[p](t))\). (There is no uniqueness for the endemic state; for the sake of simplicity, we will consider from now on only one endemic solution. There is no essential change if we consider more than one.)

Both solutions are assumed to be periodic, possibly with period multiple of T; however, without loss of generality, we will consider the period given by T. Note that for a different set of parameters more complicated behaviour (possible chaotic) can be found, cf. (Kuznetsov and Piccardi 1994).

3 Vaccination strategies

In this section we will consider two types of vaccination: mandatory and voluntary vaccination. For each one, we will define one special case: for the former, an optimal vaccination is defined as one vaccination strategy that is able to prevent outbreaks while having the minimum number of vaccinations possible and for the latter, a Nash vaccination strategy is defined as a strategy in which all individuals in the population minimize the joint risk of both disease and vaccine.

3.1 Optimal vaccination

For the optimal vaccination, we choose to work with a generalization of Condition 3 in Lemma 2. More specifically, we say that a certain vaccination strategy p is a preventive strategy when the fraction of individuals in the class I decreases monotonically in time for any small enough perturbation of the disease free state. We then construct the optimal vaccination strategy as the limit of the preventive strategies for which the vaccination effort is minimized. In Onyango and Müller (2014), optimality for time dependent vaccination profiles is defined based on the effective reproductive number. Note that in our model, only susceptibles are vaccinated, which implies a full knowledge of the current status of an individual.

Definition 1

We define the vaccination effort associated to a given strategy p, as the average number of vaccinations in one period, i.e. \(\mathbb {E}[p]{:}{=}\langle p S[p]\rangle {:}{=}\frac{1}{T}\int _0^T p(t)S[p](t)\mathrm {d}t\), where \(\langle \cdot \rangle \) denotes the average in one period, and S is the relevant solution, given by (C1) or (C2), defined above.

We denote a cumulative distribution function, associated with p, by \(\mathsf {P}(t)=\int _0^t\mathrm {d}p\), or in a more relaxed notation \(\mathrm {d}\mathsf {P}(t)=p(t)\mathrm {d}t\). To simplify the notation, we will use indistinctly \(\mathrm {d}\mathsf {P}\) and \(p\mathrm {d}t\), whenever there is no risk of confusion. Therefore, we now write \(\mathbb {E}[p]=\frac{1}{T}\int _0^TS(\tau )\mathrm {d}\mathsf {P}(\tau )\). For technical reasons, we need to consider bounds in the set of vaccination profiles. More precisely:

Definition 2

We say that a certain vaccination function p is admissible if its cumulative distribution is such that

$$\begin{aligned} \mathsf {P}([0,T])=\int _{0}^T\mathrm {d}\mathsf {P}\le (\mu +\alpha \ )T\frac{\bar{\beta }}{\gamma +\mu }, \end{aligned}$$
(3)

where \(\bar{\beta }=\sup _{t\in [0,T]}\beta (t)\). Furthermore, we use \(\mathsf {RM}_+\) to denote the set of non-negative Radon measures \(\mathsf {P}\) in [0, T] such that \(\mathsf {P}([0,T])\le (\mu +\alpha \ )T\frac{\bar{\beta }}{\gamma +\mu }\) and \(C_+([0,T])\) to denote the set of continuous functions in [0, T], with \(p(0)=p(T)\) such that \(\int _0^Tp(t)\mathrm {d}t\le (\mu +\alpha )T\frac{\bar{\beta }}{\gamma +\alpha }\). We also consider the natural immersion \(C_+([0,T])\subset \mathsf {RM}_+\).

Now we show that the definition of vaccination effort can be extend continuously for the case of Radon measures.

Lemma 3

Let \(p_n\in C_+([0,T])\) be such that \(p_n\rightarrow p\in \mathsf {RM}_+\), in the weak topology, cf. (Koralov and Sinai 2007), and let \(\mathbb {E}[p]{:}{=}\lim \mathbb {E}[p_n]\). Then, \(\mathbb {E}[p]\) is independent of the choice of the sequence \(p_n\).

Proof

Let \(p_n,q_n\in C_+([0,T])\) such that \(p_n,q_n\rightarrow p\in \mathsf {RM}_+\). Let \(\mathsf {P}_n\) and \(\mathsf {Q}_n\) be the associated cumulative distribution functions, respectively. Note that

$$\begin{aligned} \mathbb {E}[p_n]-\mathbb {E}[q_n]= & {} \int _0^T \left( S[p_n]\mathrm {d}\mathsf {P}_n-S[q_n]\mathrm {d}\mathsf {Q}_n\right) \\= & {} \int _0^T S[p_n]\mathrm {d}\left( \mathsf {P}_n-\mathsf {Q}_n\right) +\int _0^T\left( S[p_n]-S[q_n]\right) \mathrm {d}\mathsf {Q}_n. \end{aligned}$$

From the fact that \(S[p_n]\) is bounded and \(\mathsf {P}_n-\mathsf {Q}_n\rightarrow 0\), we conclude that the first integral converges to 0. For the second integral, the convergence to 0 follows from the continuity of \(p\mapsto S[p]\) in the appropriate topology. See (Heunis 1984, Theorem 2.1) for further details.

Definition 3

Let \(\beta \in C([0,T])\), \(\beta (0)=\beta (T)\) be given. For a given vaccination strategy p, let \((S_0(t),0)=(S_0[p](t),0)\) be the disease-free solution of System (1). We say that p is a preventive strategy if \(\beta (t)S_0[p](t)< \gamma +\mu \) for all t. We call \(\chi _{\mathrm {p}}=\chi _{\mathrm {p}}[\beta ]\) the set of admissible strategies that are preventive, i.e, \( \chi _{\mathrm {p}}=\{p \in C_+([0,T]),\ \text {with}\ p(0)=p(T),\ \text {and}\ \beta (t)S_0[p](t)< \gamma +\mu ,\) \( \text { for all } t \in [0,T]\}. \)

Now, we analyse the preventive strategies. First, we explicitly characterize the disease free state and then we show the existence of at least one preventive strategy. Afterwards, we define the concept of optimal strategy. Here, we reproduce the result from (Thieme 2003, Theorem 3.7).

Lemma 4

Let \(S_0[p](t)\) be the time dependent periodic number of susceptibles in the unique disease free state of system (1). Then

$$\begin{aligned} S_0[p](0)=\frac{(\mu +\alpha )\int _0^T\mathrm {e}^{-\int _s^T(p+\mu +\alpha )(\tau )\mathrm {d}\tau }\mathrm {d}s}{1-\mathrm {e}^{-\int _0^T(p+\mu +\alpha )(\tau )\mathrm {d}\tau }}. \end{aligned}$$

Before looking for optimal strategies, we prove that the set of preventive strategies is not empty.

Lemma 5

For any choice of parameters, there exists at least one preventive strategy, i.e., \(\chi _{\mathrm {p}}[\beta ]\ne \emptyset \) for all \(\beta \in C([0,T])\), \(\beta (0)=\beta (T)\).

Proof

Given \(\beta \), assume \(p(t)=p_0>(\mu +\alpha )\left( \frac{\bar{\beta }}{\gamma +\mu }-1\right) \), constant, corresponding to a preventive strategy in the case of the maximum transmission rate. Note that \(S_0[p](t)=\frac{\mu +\alpha }{p_0+\mu +\alpha }\) is the only stationary solution of the Eq. (1a) with \(I(t)=0\) and is, additionally, the solution obtained from the initial condition given by Lemma 4. From the definition of \(p_0\), we conclude that \(\beta (t)S_0[p](t)<\gamma +\mu \) for all t, and then \(\{p(t)=p_0\}\in \chi _{\mathrm {p}}\).

Finally, we construct the optimal vaccination strategy as one strategy that can be arbitrarily approximated by a preventive strategy and such that its associated effort is never superior to the effort of any preventive strategy. More rigorously, we define an optimal vaccination strategy by

Definition 4

Let \(\beta \in C([0,T])\), \(\beta (0)=\beta (T)\) be given. We say that a given strategy \(p_{\mathrm {opt}}=p_{\mathrm {opt}}[\beta ]\) is optimal if the following conditions are simultaneously satisfied:

  1. 1.

    there is at least one sequence \(\chi _{\mathrm {p}}[\beta ]\ni p_n\rightarrow p_{\mathrm {opt}}\) (in measure).

  2. 2.

    for any \(p\in \chi _{\mathrm {p}}[\beta ]\), \(\mathbb {E}[p]\ge \mathbb {E}[p_{\mathrm {opt}}]\).

Whenever \(p_{\mathrm {opt}}\) is a function (as discussed in Sect. 3.3), Definition 4 means simply that there is a sequence of preventive strategies that converge to \(p_{\mathrm {opt}}\) and that \(p_{\mathrm {opt}}\) minimizes the vaccination effort in the closure of the set \(\chi _{\mathrm {p}}\).

3.2 Rational vaccination

In this subsection, we study a population of rational individuals and how their decisions influence the disease dynamics.

For each focal individual the probability of getting the disease is assumed to depend on the disease incidence for each time step. The following lemma shows how this probability can be computed from the model.

Lemma 6

The probability that a susceptible non-vaccinated individual at time t gets the disease between times t and \(t+{\varDelta }t\), for \({\varDelta }t\) sufficiently small, is given by \(\beta (t)I(t){\varDelta }t+\mathrm {o}\left( {\varDelta }t\right) \).

Proof

All susceptible non-vaccinated individuals are in the category \(\mathbf S \). From time t to time \(t+{\varDelta }t\), \(\beta (t)I(t)S(t){\varDelta }t\) individuals will be infected, \(\mu S(t){\varDelta }t\) will die and the remainder \(S(t)-\left[ \mu S(t)+\beta (t)I(t)S(t)\right] {\varDelta }t\) will be in the class S at time \(t+{\varDelta }t\). Therefore, the probability to be infected from times t to \(t+{\varDelta }t\), given that he/she did not die, is given by

$$\begin{aligned} \frac{\beta (t)I(t)S(t){\varDelta }t}{S(t)-\left[ \mu S(t)+\beta (t)I(t)S(t)\right] {\varDelta }t+\beta (t)I(t)S(t){\varDelta }t}=\beta (t)I(t){\varDelta }t+\mathrm {o}\left( {\varDelta }t\right) , \end{aligned}$$

where we used \((1-\mu {\varDelta }t )^{-1}=1+\mathrm {o}\left( {\varDelta }t\right) \).

See Fig. 2, for a schematic representation of this reasoning.

Fig. 2
figure 2

Transitions of non-vaccinated individuals from state S at time t out by death (down arrow) and to states S and I at time \(t+{\varDelta }t\). Note that indications in the arrows are for the total number of individuals leaving state S during interval \({\varDelta }t\)

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For the voluntary vaccination, we consider that a rational individual will (not) vaccinate him/herself if the risk of the disease times the probability to get the disease, given the overall strategy of the population, is larger than (respectively, small than) the risk of the vaccine. If both risks are the same, any strategy is equally advantageous. A fully informed rational individual will access, in the beginning of the season, the probability to get the disease, using all available epidemiological data, and decides his/her personal strategy as the strategy that minimizes the joint risk, i.e., the risk of the disease times the probability to get it (conditional to no vaccination), plus the risk of the vaccine (conditional to vaccination), during the next season.

We start by defining the set of immunity provider strategies, i.e., the set of strategies for which a focal rational individual will decide to be not vaccinated.

Definition 5

Let \(\beta \in C([0,T])\), \(\beta (0)=\beta (T)\) be given. For a given vaccination strategy p, assume the existence of a persistent endemic solution \((S_1,I_1)\). Let \(r_{\mathrm {d}}>0\) and \(r_\mathrm {v}>0\) be the risks of the disease and of the vaccination, respectively. We define \(r{:}{=}\frac{r_\mathrm {v}}{r_\mathrm {d}}\). We say that p is a herd immunity provider strategy if \(\beta (t)I_1[p](t)< r\) for all t. We call \(\chi _{\mathrm {h}}=\chi _\mathrm {h}[\beta ]\) the set of all herd immunity provider strategies, i.e.

$$\begin{aligned} \chi _{\mathrm {h}}=\left\{ p \in C_+([0,T])\ \text {with}\ p(0)=p(T),\ \text {and}\ \beta (t)I_1[p](t)<r, \text {for all } t \in [0,T]\right\} . \end{aligned}$$

If there is no endemic solution, we define \(\chi _{\mathrm {h}}=\left\{ p \in C_+([0,T]) \text { with}\ p(0)=p(T)\right\} \).

Note that, from the definition, it is clear that any preventive strategy is also herd immunity provider, i.e., \(\chi _{\mathrm {p}}\subset \chi _{\mathrm {h}}\).

Finally, we will define the Nash-equilibrium strategy as the strategy that minimizes the joint risk for every individual, given the strategy of all other individuals.

Definition 6

Let \(\beta \in C([0,T])\), \(\beta (0)=\beta (T)\) be given. Let us consider a population with strategy \(p\in C_+([0,T])\), and a focal individual that uses vaccination strategy \(p_*\in \mathsf {RM}_+\). Let \(\mathsf {P}\) and \(\mathsf {P}_*\) be the cumulative distributions, associated to p and \(p_*\), respectively. Assume that the focal individual is susceptible at time \(t=0\), and therefore the probability to be susceptible at a later time t is given by \(\mathrm {e}^{-\int _0^t\mathrm {d}P_*-\mu t}\). The joint (disease and vaccination) risk during one season (i.e, the probability that something bad—disease or reaction to the vaccine—happens in one season, times the associated risks) is given by

$$\begin{aligned} \rho [p_*,p]= & {} r_\mathrm {d}\int _0^T\beta (t)I[p](t)\mathrm {e}^{-\int _0^t\mathrm {d}\mathsf {P}_*-\mu t}\mathrm {d}t+r_\mathrm {v}\int _0^T\left( 1-\mathrm {e}^{-\int _0^t\mathrm {d}\mathsf {P}_*-\mu t}\right) \mathrm {d}t\\= & {} -r_\mathrm {d}\int _0^T\left( r-\beta (t)I[p](t)\right) \mathrm {e}^{-\int _0^t\mathrm {d}\mathsf {P}_*-\mu t}\mathrm {d}t+r_\mathrm {v}T. \end{aligned}$$

Given a strategy p, a rational individual will choose a strategy \(p_*\) such that for every strategy \(p'\in \mathsf {RM}_+\)

$$\begin{aligned} \rho [p_*,p]\le \rho [p',p]. \end{aligned}$$

We say that \(p_{\mathrm {Nash}}\in \mathsf {RM}_+\) is a Nash strategy if for any sequence \(p_n\in C_+([0,T])\), such that \(p_n\rightarrow p_{\mathrm {Nash}}\) and for every strategy \(p'\in \mathsf {RM}_+\),

$$\begin{aligned} \limsup \left( \rho [p_{\mathrm {Nash}},p_n]-\rho [p',p_n]\right) \le 0. \end{aligned}$$

If \(p_{\mathrm {Nash}}\) is a function, the above definition simplifies to the assertion that \(\rho [p_{\mathrm {Nash}},p_{\mathrm {Nash}}]\le \rho [p',p_{\mathrm {Nash}}]\) for every strategy \(p'\in \mathsf {RM}_+\).

We finish the subsection stating the existence theorem for both optimal and Nash-equilibrium strategies. In general terms, for \(\beta \in C([0,T])\), with \(\beta (0)=\beta (T)\), we prove that there is at least one optimal vaccination strategy and at least one Nash vaccination strategy. These strategies may not be functions, but measures. This implies that, after rewriting System (1) in the form \(X'={\varGamma }(t,X)\), the function \({\varGamma }:\mathbb {R}_+\times {\varDelta }^2\rightarrow \mathbb {R}^2\) is a Charatheodory function (i.e., measurable in the first variable and continuous in the second) and therefore there is a (weakest) topology which guarantees existence of solutions of the differential equations and gives continuous dependence for each initial data point. See (Aliprantis and Border 1999; Heunis 1984) for further details. The proof of the existence theorem below will be postponed to “Appendix 2”.

Theorem 1

Assume \(\beta \in C([0,T])\), with \(\beta (0)=\beta (T)\). Then, there is at least one optimal vaccination strategy \(p_{\mathrm {opt}}[\beta ]\) and at least one Nash vaccination strategy \(p_{\mathrm {Nash}}[\beta ]\).

3.3 Vaccination strategies for regular transmission functions

Despite the fact that we cannot guarantee a priori existence of optimal and Nash strategies as functions, we will provide precise conditions for which \(p_{\mathrm {opt}}\) and/or \(p_{\mathrm {Nash}}\) are functions. In particular, we derive explicit formulas for the optimal and Nash strategies for sufficiently regular transmission functions \(\beta \), with some extra technical conditions. In the end, we discuss vaccination strategies when \(\beta \) is discontinuous (in particular of bounded variation).

We start by finding an explicit formula for \(p_{\mathrm {opt}}\) in some special cases.

Theorem 2

Consider system (1). Assume that

$$\begin{aligned} \beta '(t)\ge -(\mu +\alpha )\beta (t)\left( \frac{\beta (t)}{\gamma +\mu }-1\right) . \end{aligned}$$
(4)

Then

$$\begin{aligned} p_{\mathrm {opt}}(t)=(\mu +\alpha )\left( \frac{\beta (t)}{\gamma +\mu }-1\right) +\frac{\beta '(t)}{\beta (t)}\ \end{aligned}$$
(5)

is an optimal strategy.

Proof

First note that \(p_{\mathrm {opt}}(t)\ge 0\) if and only if \(\beta (t)\) satisfies Eq. (4) for all \(t\ge 0\).

We divide the proof in several steps:

Step 1: We start by using Lemma 4 to show that \(\beta (0)S[p_{\mathrm {opt}}](0)=\gamma +\mu \). Indeed, let \(p=p_{\mathrm {opt}}\) and therefore

$$\begin{aligned} S_0[p_{\mathrm {opt}}](0)= & {} \frac{(\mu +\alpha )\int _0^T\mathrm {e}^{-\frac{\mu +\alpha }{\gamma +\mu }\int _s^T\beta (\tau )\mathrm {d}\tau -\log \frac{\beta (T)}{\beta (s)}}\mathrm {d}s}{1-\mathrm {e}^{-\frac{\mu +\alpha }{\gamma +\mu }\int _0^T\beta (\tau )\mathrm {d}\tau }}\\= & {} \frac{\gamma +\mu }{\beta (T)}\times \frac{\int _0^T\frac{\mathrm {d}\ }{\mathrm {d}s}\mathrm {e}^{-\frac{\mu +\alpha }{\gamma +\mu }\int _s^T\beta (\tau )\mathrm {d}\tau }\mathrm {d}s}{1-\mathrm {e}^{-\frac{\mu +\alpha }{\gamma +\mu }\int _0^T\beta (\tau )\mathrm {d}\tau }} =\frac{\gamma +\mu }{\beta (0)}. \end{aligned}$$

Step 2: Now, we show that for any \(t>0\), \(\beta (t)S_0[p_{\mathrm {opt}}](t)=\gamma +\mu \). Using Eq. (1a) with \(S(t)=S[p](t)\), \(I(t)=I[p](0)=0\) and \(p=p_{\mathrm {opt}}\), we find

$$\begin{aligned} \left( \beta (t)S(t)\right) '= & {} \beta '(t)S(t)+\beta (t)S'(t)=\beta '(t)S(t)+\beta (t)(\mu +\alpha )\\&-\left[ \beta (t)(\mu +\alpha )\left( \frac{\beta (t)}{\gamma +\mu }-1\right) +\beta '(t)\right] S(t)-\beta (t)(\mu +\alpha ) S(t)\\= & {} \beta (t)(\mu +\alpha )\left( 1-\frac{\beta (t)S(t)}{\gamma +\mu }\right) . \end{aligned}$$

We conclude that \(\beta (t)S(t)=\gamma +\mu \) is the unique solution of the last equation with the initial condition found in the first step.

Step 3: Let \(\mathsf {P}_i\) be the cumulative distribution associated to \(p_i\), \(i=1,2\). We will prove now that if \(\int _s^T\mathrm {d}\mathsf {P}_1\ge \int _s^T\mathrm {d}\mathsf {P}_2\), \(s\in [0,T)\) and \(\int _0^T\mathrm {d}\mathsf {P}_1>\int _0^T\mathrm {d}\mathsf {P}_2\), then \(S_0[p_1](t)<S_0[p_2](t)\). For simplicity, we write \(S_i=S_0[p_i]\), \(i=1,2\). From Lemma 4, it is clear that \(S_1(0)<S_2(0)\). Furthermore,

$$\begin{aligned} \left( S_1-S_2\right) '+p_1(t)(S_1-S_2)+(\mu +\alpha )(S_1-S_2)=-(p_1-p_2)S_2\le 0. \end{aligned}$$

After rewriting the last equation, we find that

$$\begin{aligned} \frac{\mathrm {d}\ }{\mathrm {d}t}\left[ \mathrm {e}^{\int _0^t(p_1+\mu +\alpha )(\tau )\mathrm {d}\tau }(S_1-S_2)\right] =-\mathrm {e}^{\int _0^t(p_1+\mu +\alpha )(\tau )\mathrm {d}\tau }(p_1-p_2)S_2\le 0, \end{aligned}$$

and conclude that \(S_1(t)<S_2(t)\) for all t.

Step 4: We now show that \(\mathbb {E}[p_{\mathrm {opt}}]=(\mu +\alpha )(1-\langle S_0[p_{\mathrm {opt}}]\rangle )\). Indeed, let \(p_n\) be a sequence such that \(\int _s^T\mathrm {d}\mathsf {P}_n>\int _s^T\mathrm {d}\mathsf {P}_{\mathrm {opt}}\) for any \(s\ge 0\), where \(\mathsf {P}_n\) and \(\mathsf {P}_{\mathrm {opt}}\) are the cumulative distributions associated to \(p_n\) and \(p_{\mathrm {opt}}\), respectively. Assume, furthermore, that \(p_n\rightarrow p_{\mathrm {opt}}\) as measure. We use \(S_n=S_0[p_n]\): therefore \(S_n(t)<S_0[p_{\mathrm {opt}}](t)=\frac{\gamma +\mu }{\beta (t)}\) and then \(p_n\in \chi _{\mathrm {p}}\). Furthermore, \(0=\langle S_n'\rangle =\mu +\alpha -\mathbb {E}[p_n]-(\mu +\alpha )\langle S_n\rangle \) and \(\mathbb {E}[p_n]=(\mu +\alpha )(1-\langle S_0[p_n]\rangle )\). We take \(n\rightarrow \infty \) and use the continuity of \(S_0\) in \(p_n\). This finishes this step.

Step 5: Finally, we prove that for any \(p\in \chi _{\mathrm {p}}\), \(\mathbb {E}[p]>\mathbb {E}[p_{\mathrm {opt}}]\). From \(S_0[p](t)<\frac{\gamma +\mu }{\beta (t)}=S_0[p_{\mathrm {opt}}](t)\), we conclude that \(\langle S_0[p]\rangle <\langle S_0[p_{\mathrm {opt}}]\rangle \) and therefore \(\mathbb {E}[p]>(\mu +\alpha )(1-\langle S_0[p_{\mathrm {opt}}]\rangle )=\mathbb {E}[p_{\mathrm {opt}}]\).

Now, we show that for a non-constant seasonal epidemics, it is always better to consider the natural fluctuations, also at the level of the vaccination campaign. This result goes along with Agur et al. (1993). See also the discussion in Onyango and Müller (2014).

Corollary 1

Let \(\beta \) be a non-constant periodic function, and assume that the optimal strategy \(p_{\mathrm {opt}}[\beta ]\) is given by Eq. (5). Then, \(\mathbb {E}\left[ p_{\mathrm {opt}}[\beta ]\right] <\mathbb {E}\left[ p_{\mathrm {opt}}\left[ \langle \beta \rangle \right] \right] \).

Proof

We use the classical harmonic/arithmetic mean inequality, i.e., \(\left\langle \frac{1}{\beta }\right\rangle ^{-1}\le \langle \beta \rangle \), with equality if and only if \(\beta \) is constant. Hence,

$$\begin{aligned} \mathbb {E}\left[ p_{\mathrm {opt}}\left[ \beta \right] \right]&=(\mu +\alpha )\left( 1-\langle S_0\left[ p_{\mathrm {opt}}\left[ \beta \right] \right] \rangle \right) =(\mu +\alpha )\left( 1-\left\langle \frac{\gamma +\mu }{\beta }\right\rangle \right) \\&\le (\mu +\alpha )\left( 1-\frac{\gamma +\mu }{\langle \beta \rangle }\right) =(\mu +\alpha )\left( 1-S_0\left[ p_{\mathrm {opt}}\left[ \langle \beta \rangle \right] \right] \right) \\ {}&=\mathbb {E}\left[ p_{\mathrm {opt}}\left[ \langle \beta \rangle \right] \right] , \end{aligned}$$

with equality if and only if \(\beta \) is constant.

Corollary 2

In the optimal vaccination case, the total number of vaccinations in a single season do not exceed the number of newborns plus the number of individuals that lost their immunity during the previous season, i.e, assuming the worst case scenario, where everyone got the disease or was vaccinated, i.e., \(\mathbb {E}[p_{\mathrm {opt}}]<\mu +\alpha \).

Proof

Consider \(p_0\) the vaccination strategy defined in Lemma 5. Then

$$\begin{aligned} \mathbb {E}[p_{\mathrm {opt}}]\le \mathbb {E}[p_0]=\frac{1}{T}\int _0^Tp_0(t)S_0[p_0](t)\mathrm {d}t=\frac{p_0(\mu +\alpha )}{p_0+\mu +\alpha }<\mu +\alpha . \end{aligned}$$

The preceding corollary shows that it is possible to prevent outbreaks without vaccinating the entire population, which means that it is still possible to attain the herd immunity effect in this more restricted framework of preventive strategies.

We finish this section by stating an explicit formula for \(p_{\mathrm {Nash}}\) in some special cases.

Theorem 3

Assume \(r<\inf _{t\in [0,T]}\beta (t)\),

$$\begin{aligned}&\frac{\beta '(t)}{\beta (t)}\le \gamma +\mu \le \frac{\beta '(t)}{\beta (t)}+\beta (t)\qquad \text {and} \end{aligned}$$
(6)
$$\begin{aligned}&\frac{\mathrm {d}\ }{\mathrm {d}t}\left[ \mathrm {e}^{(r+\alpha -\gamma )t}\frac{\mathrm {d}\ }{\mathrm {d}t}\left( \frac{\mathrm {e}^{(\gamma +\mu )t}}{\beta (t)}\right) \right] \le \mathrm {e}^{(r+\mu +\alpha )t}\left( \mu +\alpha -\frac{\alpha r}{\beta (t)}\right) . \end{aligned}$$
(7)

Then, the strategy given by

$$\begin{aligned} \begin{array}{ll} &{} p_{\mathrm {Nash}}(t)\\ &{}\quad =\displaystyle \frac{\beta ^2(t)}{(\gamma +\mu )\beta (t)-\beta '(t)}\left[ \mu +\alpha +(\gamma +\mu )\frac{\beta '(t)}{\beta ^2(t)}-2\frac{(\beta '(t))^2}{\beta ^3(t)}+\frac{\beta ''(t)}{\beta ^2(t)}-\frac{\alpha r}{\beta (t)}\right] \\ &{}\qquad -[r+\mu +\alpha ]\ \end{array} \end{aligned}$$
(8)

is a Nash-equilibrium strategy.

Proof

We show that \(p_{\mathrm {Nash}}\ge 0\), \(S_1(t),I_1(t)\in [0,1]\), \(\forall t\) and \(\beta (t)I_1[p_{\mathrm {Nash}}](t)=r\) for all \(t\ge 0\). Initially, let us define

$$\begin{aligned} S_1(t)=\frac{\gamma +\mu }{\beta (t)}-\frac{\beta '(t)}{\beta ^2(t)}=\mathrm {e}^{-(\gamma +\mu )t}\frac{\mathrm {d}\ }{\mathrm {d}t}\left( \frac{\mathrm {e}^{(\gamma +\mu )t}}{\beta (t)}\right) . \end{aligned}$$

Note that Eq. (6) guarantees that \(S_1(t)\in [0,1]\) for all t. Also, the assumption on r implies that \(I(t)\in [0,1]\) for all t. Furthermore,

$$\begin{aligned} p_{\mathrm {Nash}}(t)=\frac{\mu +\alpha -S_1'(t)-\alpha I_1(t)}{S_1(t)}-(r+\mu +\alpha ). \end{aligned}$$

With this definition, note that

$$\begin{aligned} -\beta (t)S_1I_1=S_1'+p_{\mathrm {Nash}}S_1-(\mu +\alpha )(1-S_1)+\alpha I_1=-rS_1 \end{aligned}$$

and therefore \(\beta (t)I_1[p_{\mathrm {Nash}}](t)=r>0\). Finally, we use Eq. (7) to prove that \(p_{\mathrm {Nash}}\ge 0\). From Definition 6, we conclude that \(\rho [p_*,p_{\mathrm {Nash}}]\) does not depend on \(p_*\) and therefore \(p_{\mathrm {Nash}}\) is a Nash-equilibrium.

Note that the definition of \(S_1\) was motivated by plugging \(I(t)=r/\beta (t)\) into Eq. (1b); in particular, this choice of I(t) cancels the first term of \(\rho \) on Definition 6.

3.4 Comparisons

Now, we compare the vaccination effort associated with the two extreme vaccination strategies.

Proposition 1

Assume that \(p_{\mathrm {Nash}}\not \in \chi _{\mathrm {p}}\). Then, \(\mathbb {E}[p_{\mathrm {Nash}}]<\mathbb {E}[p_{\mathrm {opt}}]\).

Proof

We use that \(0=\langle \beta I_1 S_1\rangle -(\gamma +\mu )\langle I_1\rangle \) and \(0=(\mu +\alpha )(1-\langle S_1\rangle )-\mathbb {E}[p_{\mathrm {Nash}}]-\alpha \langle I_1\rangle -\langle \beta I_1S_1\rangle \) to conclude that \(\mathbb {E}[p_{\mathrm {Nash}}]=(\mu +\alpha )(1-\langle S_1\rangle )-(\alpha +\mu +\gamma )\langle I_1\rangle < (\mu +\alpha )(1-\langle S_1\rangle )\). Assume \(I_1(t)\ne 0\) for some t, and consequently \(I_1(t)\ne 0\) for all t. Therefore

$$\begin{aligned} S_1[p_{\mathrm {Nash}}](t)=\frac{\gamma +\mu }{\beta (t)}+\frac{I_1'[p_{\mathrm {Nash}}](t)}{\beta (t) I_1[p_{\mathrm {Nash}}](t)}=\frac{\gamma +\mu }{\beta (t)}+\frac{I'_1[p_{\mathrm {Nash}}](t)}{r}\ , \end{aligned}$$

and finally \(\langle S_1[p_{\mathrm {Nash}}]\rangle = (\gamma +\mu )\langle \beta ^{-1}\rangle =\langle S_0[p_{\mathrm {opt}}]\rangle \) and \(\mathbb {E}[p_{\mathrm {opt}}]=(\mu +\alpha )(1-\langle S_0\rangle )\).

Corollary 3

Let \(p_{\mathrm {Nash}}\) and \(p_{\mathrm {opt}}\) be the Nash and optimal vaccination strategies associated to a given transmission parameter \(\beta \), respectively. Assume further that conditions in Theorems 2 and 3 are satisfied. Then \(\mathbb {E}[p_{\mathrm {opt}}]-\mathbb {E}[p_{\mathrm {Nash}}]=r(\gamma +\mu +\alpha )\langle \beta ^{-1}\rangle >0\).

Proof

From Eq. (1), we have \(\mathbb {E}[p_{\mathrm {opt}}]=\langle p_{\mathrm {opt}}S_0\rangle =(\mu +\alpha )(1-\langle S_0\rangle )\), where \(\langle S_0\rangle =(\gamma +\mu )\langle \beta ^{-1}\rangle \). Furthermore,

$$\begin{aligned} \mathbb {E}[p_{\mathrm {Nash}}]=\langle p_{\mathrm {Nash}}S_1\rangle = (\mu +\alpha )(1-\langle S_1\rangle )-\langle \beta I_1S_1\rangle -\alpha \langle I_1\rangle . \end{aligned}$$

We also have that

$$\begin{aligned} \left\langle S_1\right\rangle =\left\langle \left( \frac{I_1'}{\beta I_1}+\frac{\gamma +\mu }{\beta }\right) \right\rangle =\frac{1}{r}\langle I_1'\rangle +(\gamma +\mu )\langle \beta ^{-1}\rangle =(\gamma +\mu )\langle \beta ^{-1}\rangle . \end{aligned}$$

On the other hand, \(\langle \beta S_1I_1\rangle =r(\gamma +\mu )\langle \beta ^{-1}\rangle \). Furthermore, \(\langle I_1\rangle =r\langle \beta ^{-1}\rangle \). Finally,

$$\begin{aligned} \mathbb {E}[p_{\mathrm {Nash}}]&=(\mu +\alpha )\left( 1-(\gamma +\mu )\langle \beta ^{-1}\rangle \right) -r(\gamma +\mu )\langle \beta ^{-1}\rangle -\alpha r\langle \beta ^{-1}\rangle \\&=\mathbb {E}[p_{\mathrm {opt}}]-r(\gamma +\mu +\alpha )\langle \beta ^{-1}\rangle . \end{aligned}$$

In this work, instead of minimizing the vaccination effort in the set of strategies with \(\mathcal {R}_0\) below one, we minimized the vaccination effort in a subset \(\chi _p\), which we have called the set of preventive strategies. Still, we can prove the following result.

Proposition 2

\(\mathcal {R}_0[p_{\mathrm {opt}}]\le 1\).

Proof

The inequality follows immediately from the Definitions 3 and 4 and Eq. (2). \(\square \)

4 Examples

4.1 The constant case

Let us consider \(\beta (t)=\beta _0\), for all t.

If \(\beta _0\le \gamma +\mu \), we have that \(S_0(t)=1\) for all t, and therefore \(\{p_0{:}{=}p(t)=0,\forall t\}\in \chi _{\mathrm {p}}\). As \(\mathbb {E}[0]=0\), we conclude that \(p_{\mathrm {opt}}(t)=0\). As there is no endemic solution, we conclude that \(p_{\mathrm {Nash}}(t)=0\).

Now, assume \(\beta _0>\gamma +\mu \) and assume additionally that \(r>0\) is small. From Theorem 2, \(p_{\mathrm {opt}}=(\mu +\alpha )(S_0[p_{\mathrm {opt}}]^{-1}-1)=(\mu +\alpha )\left( \frac{\beta _0}{\gamma +\mu }-1\right) \) which coincides with the optimal strategy in the traditional sense of \(\mathcal {R}_0[p_{\mathrm {opt}}]= 1\). The first condition on Theorem 3 is trivially satisfied and the second one is satisfied whenever

$$\begin{aligned} r\le r_*{:}{=}\frac{(\mu +\alpha )(\gamma +\mu )}{\gamma +\mu +\alpha }\left( \frac{\beta _0}{\gamma +\mu }-1\right) . \end{aligned}$$

In this case, \(I_1[p_{\mathrm {Nash}}]=\frac{r}{\beta _0}\ne 0\), and therefore \(0=I_1[p_{\mathrm {Nash}}]'=\) \(I_1[p_{\mathrm {Nash}}](\beta _0S_1[p_{\mathrm {Nash}}]-\gamma -\mu )\) and therefore \(S_1[p_{\mathrm {Nash}}]=\frac{\gamma +\mu }{\beta _0}\). Furthermore \(0=S_1[p_{\mathrm {Nash}}]'(t)=\mu +\alpha -\alpha I_1[p_{\mathrm {Nash}}]-\beta _0I_1[p_{\mathrm {Nash}}]S_1[p_{\mathrm {Nash}}]-p_{\mathrm {Nash}}S_1[p_{\mathrm {Nash}}]-(\mu +\alpha ) S_1[p_{\mathrm {Nash}}]\). Finally,

$$\begin{aligned} p_{\mathrm {Nash}}\!=\!(\mu \!+\!\alpha )\left( \frac{\beta _0}{\gamma \!+\!\mu }\!-\!1\right) \!-\!r\left( 1\!+\!\frac{\alpha }{\gamma +\mu }\right) =p_{\mathrm {opt}}-r\left( 1+\frac{\alpha }{\gamma +\mu }\right) <p_{\mathrm {opt}}, \end{aligned}$$

i.e., the rational level of vaccination will not be able to eliminate the disease as previously shown in (Bauch and Earn 2004). It is clear that both \(p_{\mathrm {opt}}\) and \(p_{\mathrm {Nash}}\) are admissible.

4.2 The sinusoidal case

For the sinusoidal case, with \(\beta (t)=\beta _{0}(1+\varepsilon \cos t)\) we can provide precise results. First, note that the condition (4) in Theorem 2 is

$$\begin{aligned} -\beta _0\varepsilon \sin t\ge -(\mu +\alpha )\beta _0(1+\varepsilon \cos t)\left( \frac{\beta _0(1+\varepsilon \cos t)}{\gamma +\mu }-1\right) , \end{aligned}$$

or, equivalently

$$\begin{aligned} \varepsilon \sin t\le \frac{(\mu +\alpha )\beta _0}{\gamma +\mu }(1+\varepsilon \cos t)^2-(\mu +\alpha )(1+\varepsilon \cos t). \end{aligned}$$

If

$$\begin{aligned} \varepsilon \le (\mu +\alpha )\left( \frac{\beta _0}{\gamma +\mu }-1\right) -\varepsilon (\mu +\alpha )\left( \frac{2\beta _0}{\gamma +\mu }-1\right) , \end{aligned}$$

condition (4) will be satisfied for every t. It follows that last equation is true if

$$\begin{aligned} \varepsilon \le \varepsilon _0{:}{=}\left( \frac{2(\mu +\alpha )\beta _0}{\gamma +\mu }+1-\mu -\alpha \right) ^{-1}(\mu +\alpha )\left( \frac{\beta _0}{\gamma +\mu }-1\right) . \end{aligned}$$

Now, assume \(\beta (t)=\beta _{0}(1+\varepsilon \cos t)\) with \(\varepsilon \le \varepsilon _0\). Note that

$$\begin{aligned} \langle S_0[p_{\mathrm {opt}}]\rangle =\frac{\gamma +\mu }{2\pi }\int _0^{2\pi }\frac{\mathrm {d}t}{\beta (t)}=\frac{\gamma +\mu }{2\pi \beta _0}\int _0^{2\pi }\frac{\mathrm {d}t}{1+\varepsilon \cos t}=\frac{\gamma +\mu }{\beta _0\sqrt{1-\varepsilon ^2}}, \end{aligned}$$

showing that the vaccination effort for the optimal solution

$$\begin{aligned} \mathbb {E}[p_{\mathrm {opt}}]=(\mu +\alpha )(1-\langle S_0\rangle ) \end{aligned}$$

decreases with the oscillation amplitude. Furthermore,

$$\begin{aligned}&p_{\mathrm {opt}}(t)\\&\quad =(\mu +\alpha )\left( \frac{\beta _0(1+\varepsilon \cos t)}{\gamma +\mu }-1\right) -\frac{\varepsilon \beta _0\sin t}{\beta _0(1+\varepsilon \cos t)}\\&\quad =(\mu +\alpha )\left( \frac{\beta _0}{\gamma +\mu }-1\right) +\varepsilon \left( \frac{(\mu +\alpha )\beta _0}{\gamma +\mu }\cos t-\sin t\right) +\mathcal {O}\left( \varepsilon ^2\right) \\&\quad =(\mu +\alpha )\left( \frac{\beta _0}{\gamma +\mu }-1\right) +\varepsilon \frac{\sqrt{(\mu +\alpha )^2\beta _0^2+(\mu +\gamma )^2}}{(\mu +\gamma )}\cos (t+\varphi )+\mathcal {O}\left( \varepsilon ^2\right) , \end{aligned}$$

where \(\varphi =\mathop {\mathrm {arctg}}\frac{\gamma +\mu }{\beta _0(\mu +\alpha )}\). It is clear that \(\langle \frac{\beta '}{\beta }\rangle =0\) and therefore \(\langle p_{\mathrm {opt}}\rangle \) is the same as in the constant case. However, the vaccination effort \(\langle p_{\mathrm {opt}}S_0[p_{\mathrm {opt}}]\rangle =(\mu +\alpha )\left( 1-\langle S[p_{\mathrm {opt}}]\rangle \right) =(\mu +\alpha )\left( 1-\frac{\gamma +\mu }{\beta _0\sqrt{1-\varepsilon ^2}}\right) \) is strictly smaller than in the constant case. Furthermore, to first order in \(\varepsilon \), the optimal vaccination strategy \(p_{\mathrm {opt}}\) lags behind the transmission rate by a phase shift of \(\varphi \). In particular, if the birth/mortality rate is very high (and therefore there is an extremely fast renewal of susceptible individuals), i.e., \(\mu \rightarrow \infty \), then \(\varphi \rightarrow \mathop {\mathrm {arctg}}\frac{1}{\beta _0}\). This means that the optimal vaccination time shift will depend on the average transmission rate. In the more realistic case \(\mu \rightarrow 0\), i.e, when the renewal is low, then \(\varphi \rightarrow \mathop {\mathrm {arctg}}\frac{\gamma }{\beta _0\alpha }\). This case is of much higher interest in practice as it models the case when the disease time scales (contact rate, recovery time, temporary immunity) are much smaller than the typical time scale of one generation of the population. Assuming \(\frac{\gamma }{\beta _0\alpha }\approx 1\) (i.e., immunity lasts for approximately \(\mathcal {R}_0\approx \alpha ^{-1}\) years), then \(\varphi \approx \frac{\pi }{4}\), i.e., in a seasonal epidemic, the peak of the vaccination rate, p, should be approximately 1.5 months before the transmission peak. Note that this is not the peak of the instantaneous number of vaccinations, pS, as illustrated in Fig. 3.

Now, we use Theorem 3 to obtain the Nash strategy. Condition (6) can be rewritten

$$\begin{aligned} -\varepsilon \sin t+\mathcal {O}\left( \varepsilon ^2\right)&= -\frac{\varepsilon \sin t}{1+\varepsilon \cos t}\le \gamma +\mu \le \beta _0+\varepsilon (\beta _0\cos t-\sin t)+\mathcal {O}\left( \varepsilon ^2\right) \\&=\beta _0+\varepsilon \sqrt{\beta _0^2+1}\cos (t+\tilde{\varphi })+\mathcal {O}\left( \varepsilon ^2\right) , \end{aligned}$$

for a certain \(\tilde{\varphi }\). This is true if, for example, \(\varepsilon \le \gamma +\mu \le \beta _0-\varepsilon \sqrt{\beta _0^2+1}\), in particular if \(\beta _0>\gamma +\mu \), and \(\varepsilon \) is small enough.

Equation (7) is equivalent to the non-negativeness of \(p_{\mathrm {Nash}}\), given by Eq. (8). Let us assume that the relative risk of the vaccination is low, i.e., \(r=\mathcal {O}(\varepsilon )\). After some extensive, but straightforward calculations, we conclude that

$$\begin{aligned}&p_{\mathrm {Nash}}[\beta ](t)\\&\quad =(\mu +\alpha )\left( \frac{\beta _0}{\gamma +\mu }-1\right) -r\left( \frac{\alpha }{\gamma +\mu }+1\right) \\&\quad +\varepsilon \frac{\beta _0}{\gamma +\mu }\sqrt{(\gamma +\mu )^2+1}\left( \frac{\mu +\alpha }{\gamma +\mu }\cos (t+\psi )-\frac{1}{\beta _0}\sin (t+\psi )\right) \\&\quad = (\mu +\alpha )\left( \frac{\beta _0}{\gamma +\mu }-1\right) -r\left( \frac{\alpha }{\gamma +\mu }+1\right) \\&\quad +\varepsilon \frac{1}{(\gamma +\mu )^2}\sqrt{(\gamma +\mu )^2+1}\sqrt{\beta _0^2(\mu +\alpha )^2+(\gamma +\mu )^2}\cos (t+\varphi +\psi ), \end{aligned}$$

where \(\tan \psi =\frac{1}{\gamma +\mu }\) and \(\tan \varphi =\frac{\gamma +\mu }{\beta _0(\mu +\alpha )}\). It is clear that if \(\frac{\beta _0}{\gamma +\mu }>1\), then, for \(\varepsilon \), r small enough, the Nash-equilibrium is given by \(p_{\mathrm {Nash}}>0\) as obtained in the previous equation. It is also clear that both \(p_{\mathrm {opt}}\) and \(p_{\mathrm {Nash}}\) are admissible at leading order.

Fig. 3
figure 3

Optimal and Nash strategies for the sinusoidal transmission with low and high vaccination risk, r. We use \(\mu =(80 T)^{-1}\), \(\gamma =52/T\), \(\alpha =2/T\), with \(T=2\pi \) and consider \(\beta (t)=26(1+0.3\cos (t))\), implying \(\frac{\gamma }{\beta _0\alpha }=1\) and \(\mathcal {R}_0=\frac{\langle \beta \rangle }{\gamma }=\pi \approx 3\). Left Both \(p_{\mathrm {opt}}\) (blue, continuous) and \(p_{\mathrm {Nash}}\) (red, dashed) oscillate in a synchronous way. The peak \(p_{\mathrm {opt}}\) is \(\pi /4\) before the peak of the transmission rate; the peak of \(p_{\mathrm {Nash}}\) is slightly before. Right Time dependent vaccination effort, in the two cases. Above small vaccination risk, \(r=0.005\); \(p_{\mathrm {Nash}}\) is higher than \(p_{\mathrm {opt}}\) in the beginning of the epidemic season and lower otherwise. Below high vaccination risk, \(r=0.1\); \(p_{\mathrm {opt}}>p_{\mathrm {Nash}}\) for all t, but the difference is larger when \(\beta \) is decreasing. For simplicity, we plot in all cases \(\beta (t)\) in gray dotted line (out of scale). The choice of parameters is inspired by influenza like epidemiological parameters: life expectancy of 80 years, recovery time of 1 week, and temporary immunity of 6 months, and \(\mathcal {R}_0\approx 3\) (cf. Goeyvaerts et al. 2015). In the absence of vaccination, there is only one stable attractor of the solution (S(t), I(t)) with period T (cf. Kuznetsov and Piccardi 1994)

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If \(\mu \rightarrow 0\), \(\beta _0\alpha /\gamma =\mathcal {O}(1)\), then \(\varphi \approx \frac{\pi }{4}\); this means that \(p_{\mathrm {Nash}}\) will peak shortly before \(p_{\mathrm {opt}}\); however amplitude oscillations are slightly larger for \(p_{\mathrm {Nash}}\). Finally, the difference between both vaccination efforts are given by

$$\begin{aligned} \mathbb {E}[p_{\mathrm {opt}}]-\mathbb {E}[p_{\mathrm {Nash}}]=\frac{r(\gamma +\mu +\alpha )}{2\pi \beta _0}\int _0^{2\pi }\frac{\mathrm {d}t}{1+\varepsilon \cos t}=\frac{r(\gamma +\mu +\alpha )}{\beta _0\sqrt{1-\varepsilon ^2}}. \end{aligned}$$

Figure 3 shows the peaks and oscillations of the optimal and Nash strategies, for the sinusoidal case, and how the difference between them depends on the vaccination risk.

In Fig. 4, we consider three different vaccination profiles for sinusoidal transmission: optimal vaccination, \(p(t)=p_{\mathrm {opt}}[\beta ]\); optimal vaccination in the case of average transmission rate, \(p=p_{\mathrm {opt}}[\langle \beta \rangle ]\); and optimal vaccination in the case of maximum transmission, \(p=p_{\mathrm {opt}}[\bar{\beta }]\) (used in Lemma 5). This example also illustrates the result of Corollary 1, namely, it shows that, for a given sinusoidal transmission rate \(\beta \), the effort associated to \(p_{\mathrm {opt}}[\langle \beta \rangle ]\) is larger than the effort associated to \(p_{\mathrm {opt}}[\beta ]\). However, \(p_{\mathrm {opt}}[\langle \beta \rangle ]\) does not prevent initial outbreaks. Additionally, we show that \(p_{\mathrm {opt}}[\bar{\beta }]\) prevents the initial outbreak but with a higher associated effort.

Fig. 4
figure 4

Prevention of initial outbreaks for different vaccination profiles. We consider all parameters, including the transmission rate, as in Fig. 3 and assume initial conditions given by \(S_0[p](0)-0.01\) and \(I(0)=0.01\), where \(S_0[p](0)\) is given by Lemma 4. We consider three different vaccination profiles: \(p=p_{\mathrm {opt}}[\beta ]\) (blue, continuous), \(p=p_{\mathrm {opt}}[\langle \beta \rangle ]\) (red, dashed) and \(p=p_{\mathrm {opt}}[\bar{\beta }]\) (green, dash-dot). Note that the optimal strategy in the case of the average transmission rate is unable to prevent initial outbreaks (however, \(I(t)\rightarrow 0\) when \(t\rightarrow \infty \)). Vaccination efforts \(\mathbb {E}[p_{\mathrm {opt}}[\bar{\beta }]]=0.1165> \mathbb {E}[p_{\mathrm {opt}}[\langle \beta \rangle ]=0.1098>\mathbb {E}[p_{\mathrm {opt}}[\beta ]]=0.1090\), respectively

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4.3 A critical case

Condition (4) in Theorem 2, provides a lower bound on the derivative of the transmission coefficient \(\beta \). In particular, if \(\beta \) is increasing, Theorem 2 provides (at least, in principle) one optimal strategy. However, to apply Theorem 2, it is important that \(\beta \) does not decrease instantaneously. In this section, we will study a critical example for optimal vaccinations, where \(\beta \) satisfies the critical condition \(\beta '=-(\mu +\alpha )\beta \left( \frac{\beta }{\gamma +\mu }-1\right) \) in \((0,2\pi )\) and is not differentiable at \(t=0\). As usual, \(\beta \) is periodic in \(\mathbb {R}\). Explicitly, we consider

$$\begin{aligned} \beta (t)= \frac{\gamma +\mu }{1-K\exp (-(\mu +\alpha )t))},\qquad t\in [0,2\pi )\ , \end{aligned}$$
(9)

where \(K\in (0,1)\) is a constant.

We consider a mollification of \(\beta \), \(\beta _\varepsilon \), such that \(\beta _\varepsilon \) is differentiable, satisfies the condition (4), and \(\beta _\varepsilon \rightarrow \beta \) when \(\varepsilon \rightarrow 0\) pointwise. Let \(p_\varepsilon {:}{=}p_{\mathrm {opt}}[\beta _\varepsilon ]\). It is clear that \(p_\varepsilon (t)\rightarrow 0\) for \(t\ne 0\), and

$$\begin{aligned} \int _0^{2\pi } p_\varepsilon \mathrm {d}t&\rightarrow (\mu +\alpha )2\pi \left( \frac{1}{\gamma +\mu }\langle \beta \rangle -1\right) \\ {}&=2\pi (\mu +\alpha )\left( \frac{1}{2\pi (\mu +\alpha )}\log \frac{\mathrm {e}^{2(\mu +\alpha )\pi }-K}{1-K}-1\right) \\&=\log \frac{\mathrm {e}^{2(\mu +\alpha )\pi }-K}{1-K}-2\pi (\mu +\alpha )=\log \frac{\mathrm {e}^{2(\mu +\alpha )\pi }-K}{1-K}+\log \mathrm {e}^{-2\pi (\mu +\alpha )}\\&=\log \frac{1-K\mathrm {e}^{-2(\mu +\alpha )\pi }}{1-K}=:{\varGamma }. \end{aligned}$$

Thus \(p_\varepsilon \rightarrow {\varGamma }\sum _{i\in \mathbb {Z}}\delta _{2\pi i}:=p_{\mathrm {opt}}\). From Eq. (5), it is clear that for \(t\in (0,2\pi )\), \(p_{\mathrm {opt}}[\lim \beta _\varepsilon ](t)=\lim p_{\mathrm {opt}}[\beta _\varepsilon ](t)\), and therefore \(p_{\mathrm {opt}}[\lim \beta _\varepsilon ]=\lim p_{\mathrm {opt}}[\beta _\varepsilon ]\) as measures. This shows that discontinuities in \(\beta \) will be associated to peak vaccinations. From Lemma 4, we have that

$$\begin{aligned} S[p_{\mathrm {opt}}](0)=\frac{(\mu +\alpha )\int _0^T\mathrm {e}^{-(\mu +\alpha )(T-s)-\frac{{\varGamma }}{2}-\frac{{\varGamma }}{2}\theta (-s)}\mathrm {d}s}{1-\mathrm {e}^{-{\varGamma }-(\mu +\alpha )T}}=\frac{\mathrm {e}^{-{\varGamma }/2}(1-\mathrm {e}^{-(\mu +\alpha )T})}{1-\mathrm {e}^{-{\varGamma }-(\mu +\alpha )T}}, \end{aligned}$$

where \(\theta (s)=0\) for \(s\le 0\) and \(\theta (s)=1\) for \(s>0\) is the Heaviside function. Furthermore

$$\begin{aligned} \int S[p_\varepsilon ](t)p_\varepsilon (t)\mathrm {d}t\rightarrow {\varGamma }\mathrm {e}^{-{\varGamma }/2}\frac{1-\mathrm {e}^{-(\mu +\alpha )T}}{1-\mathrm {e}^{-{\varGamma }-(\mu +\alpha )T}}\le (\mu +\alpha )T, \end{aligned}$$

where we used that

$$\begin{aligned} x\mathrm {e}^{-x/2}\frac{1-\mathrm {e}^{-y}}{1-\mathrm {e}^{-x-y}}=x\frac{\mathop {\mathrm {sinh}}\frac{y}{2}}{\mathop {\mathrm {sinh}}\frac{x+y}{2}}\le y\ , \forall x,y>0. \end{aligned}$$

This last inequality follows from the convexity of the function

$$\begin{aligned} \mathop {\mathrm {sinh}}\alpha =\mathop {\mathrm {sinh}}\left( \frac{\beta }{\alpha +\beta }0+\frac{\alpha }{\alpha +\beta }(\alpha +\beta )\right) \le \frac{\beta }{\alpha +\beta }\mathop {\mathrm {sinh}}0+\frac{\alpha }{\alpha +\beta }\mathop {\mathrm {sinh}}\left( \alpha +\beta \right) , \end{aligned}$$

and therefore, for \(\alpha ,\beta > 0\),

$$\begin{aligned} \frac{\mathop {\mathrm {sinh}}\alpha }{\mathop {\mathrm {sinh}}(\alpha +\beta )}\le \frac{\alpha }{\alpha +\beta }\le \frac{\alpha }{\beta }. \end{aligned}$$

Optimal strategy for the critical case, using a relaxed version of the transmission function \(\beta (t)\), is illustrated in Fig. 5.

Fig. 5
figure 5

Optimal strategy for the critical case, with relaxed \(\beta (t)\). Left relaxed version of \(\beta (t)\) given by Eq. (9) with \(K=0.862\), and the other parameters as in Fig. 1, with \(T=2\pi \). We consider a sequence \(\beta _\varepsilon \) of differentiable functions, such that \(\beta _\varepsilon (t)=\beta (t)\) for \(t\in (\varepsilon ,2\pi -\varepsilon )\) and the points \(2\pi -\varepsilon \) and \(2\pi +\varepsilon \equiv \varepsilon \) are connected by a third order polynomial. Right a sequence of \(p_\varepsilon {:}{=}p_{\mathrm {opt}}[\beta _\varepsilon ]\); note that \(\left( p_\varepsilon \right) _{\varepsilon >0}\) resembles a delta-sequence

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5 Discussion

In this work, we consider a SIRS model with periodic transmission, where we introduce periodic vaccination of adults. We are naturally led to consider temporary immunity, as discussed in (Onyango and Müller 2014): in the case of diseases with life-long immunity and lifespan much longer than the period (\(\mu \sim \alpha \ll {1}/{T}\)) the vaccination is not affected by periodicity. For instance, measles has periodic transmission rate but, since it confers permanent immunity, periodic vaccination is not used. In this work we implicitly assume \(\mu \ll {1}/{T} \approx \alpha \).

We study the consequences of two extreme types of vaccination strategies: mandatory vaccination, where a certain predefined fraction of the population is vaccinated; and voluntary vaccination, where individuals can choose freely to be vaccinated or not, according to their risk perception. Classically, the objective is to minimize the vaccination effort while reducing the effective reproductive number below one, which guaranties long term disease elimination. Here, we choose to work with an alternative definition of optimal vaccination. We define a class of preventive vaccination strategies as vaccination profiles that, for any sufficiently small perturbation of the disease free state, the number of infectious individuals is monotonically decreasing, avoiding the occurrence of any epidemic event. This approach allows, for specific regular transmission functions \(\beta (t)\), the derivation of an analytical expression of the optimal strategy. In general, we prove the existence of an optimal strategy, in a suitably defined closure of the space of all preventive strategies, which minimizes the vaccination effort.

In this work, we extend the classical results by Bauch and Earn (2004) to periodic functions, based on a series of recent results on periodic diseases. We model human behaviour using classical economical theory, where individuals are assumed to be rational and fully informed. We define the set of vaccination strategies that provide herd immunity, for which the rational strategy of a given focal individual is not to be vaccinated. Finally, we prove the existence of a Nash vaccination strategy as the strategy that minimizes the joint risk for every individual, taking into account the strategy of all other individuals.

In general, both optimal and Nash strategies will not be functions but Radon measures. For specific forms of the transmission rate, we provide explicit formulas, which includes some important examples as constant or sinusoidal transmission functions.

There are several natural limitations of the work presented here. One first limitation is that we consider only the stationary solution of the System (1), but we never discuss the approach to this equilibrium. This is an important question, both in the study of ordinary differential equations (i.e., the study of the basin of attraction) and in evolutionary game theory, where the study of \(\omega \)-limits of conveniently defined dynamical equations is preferred to the static study of Nash-equilibria. In the non-stationary case, a rational decision will require the ability to forecast the evolution of the epidemic, i.e., rational decisions will depend on future decisions of the entire population and not only on the past decisions. This is mathematically described by the so called “mean field game theory” (Lasry and Lions 2007) and will be object of a future work.

Closely related ideas will also help us to solve one of the major gaps of the current work: the lack of a numerical method for finding Nash-equilibria solution when Theorem 3 fails. More precisely, the idea will be to develop a numerical method that allows constant update in individual decisions and, consequently, also at the population level. As discussed before, this will require, at the individual level, a certain expectation on the future evolution of the disease.

It is also important, and will be subject of a future work, to design a precise scheme, possibly numerical, that allows to go beyond Theorem 2. This will require the use of Optimal Control Theory. In fact, given p, it is possible to explicitly obtain the disease free solution S[p](t) (see Lemma 4) and therefore we need to minimize \(\int _0^Tp(t)S[p](t)\mathrm {d}t\) in \(\chi _{\mathrm {p}}[\beta ]\). (Equivalently, we may maximize \(\int _0^TS[p](t)\mathrm {d}t\) in the same set, as \(\langle pS_0\rangle =(\mu +\alpha )\left( 1-\langle S_0\rangle \right) \).) We also plan to compare \(p_{\mathrm {opt}}\) with the optimal solution in the approach in which the vaccination effort is minimized in the class of vaccination functions p such that \(\mathcal {R}_0\le 1\).

Furthermore, despite the simplicity of the periodic SIRS system (even with vaccination), solutions can be extremely complicated; even chaotic solutions may be present in such simple systems, cf. (Kuznetsov and Piccardi 1994). The coupling of the differential equations with human rational behaviour presented in this work only started the exploration of all this mathematical richness.