2.1 SIR model

The (normalised) SIR model, see for example [17, Ch. 2] and [18], is described by the following ordinary differential equations: (1) (2) (3) with t ∈ [t₀, ∞[ denoting time, measured in days. The state variable S(t) ∈ [0, 1] denotes the proportion of individuals at time t that are susceptible to the disease; I(t) ∈ [0, 1] denotes the proportion that are infected with the disease and are infective (that is, can infect susceptibles); and R(t) ∈ [0, 1] denotes the proportion that have been infected and removed, in the sense that they cannot re-infect susceptibles. “Removal” may encapsulate recovery and immunisation from the disease, death from the disease, quarantine measures, vaccination, etc.

We assume that the birth and death rates are zero. In other words, we assume that the disease is relatively short-lived such that the effects of demographic change can be ignored. Therefore, assuming an initial state satisfying S(t₀) + I(t₀) + R(t₀) = 1, all integral curves of the system remain in the set: (4) for all t ≥ t₀. Let denote the constant population size, and let , and denote the number of susceptibles, infectives and recovered individuals, respectively. Let , the contact rate, denote the average number of contacts a person makes per unit time that is sufficient for disease transmission (this contact can be with infectives and/or other susceptibles). Then βI is the average number of contacts with infectives a susceptible makes per unit time, and βIs is the number of new infectives per unit time. Thus, βIS is the proportion of new infectives per unit time. It is assumed that recoveries occur according to a Poisson process with arrival rate . See [17, Ch. 2], [18] and [13] for a more in-depth discussion of this model’s derivation.

In conclusion, β may be interpreted as the average number of people a member of the population comes into contact with over one day, and is the average number of days an infective individual remains infective.

2.2 SEIR model

This model is described by the following ODEs: (5) (6) (7) (8) t ∈ [t₀, ∞[, with the additional compartment E(t) ∈ [0, 1] denoting exposed individuals. This is a general model for diseases where individuals only become infective after a non-negligible period of time from exposure. It is assumed that an individual’s evolution from being exposed to being infective follows a Poisson process with arrival rate . Thus, is the average number of days it takes an exposed individual to become infective.

As before, we assume that the total population remains constant and that the SEIR model is normalised. Thus, all solutions remain in the (redefined) set (9) for all t ∈ [t₀, ∞[.

2.3 On set-based methods applied to epidemiology

Now we present an introductory overview of how we intend to apply set-based methods to the management of epidemics. The subsequent sections will present rigorous treatment of the ideas.

Consider the SIR model, (1)–(3), and impose a hard infection cap, I(t) ≤ I_max, for all time. Moreover, assume that the contact rate, β, may be changed over time within some bounds, representing social distancing measures. That is, assume β(t)∈[β_min, β_max], 0 < β_min ≤ β_max where β_min is the minimal contact rate (and thus represents the maximal social distancing) and β_max is the maximal or nominal contact rate (and may thus represent minimal social distancing).

Consider the SEIR model, (5)–(8), and also impose a hard infection cap, I(t) ≤ I_max, for all time. Assume that in addition to the contact rate, β, the parameter γ may also be changed over time within some bounds, i.e., γ(t) ∈ [γ_min, γ_max], 0 < γ_min ≤ γ_max. This may represent quarantine or vaccination measures.

The first problem we want to address is the following:

Problem 1. For the SIR model, supposing the parameter γ is perfectly known (SIR prefect model), how should the contact rate β(t), be specified over time such that the infection cap, I_max, is never breached? For the SEIR model, supposing the parameter η is perfectly known (SEIR prefect model), how should the contact rate β(t) and the removal rate γ(t) be specified over time such that the infection cap is never breached?

Our approach to solving this problem will be to find the admissible set, , which, for the SIR model, will correspond to all states from which there exists an input β satisfying β(t)∈[β_min, β_max] for all t ∈ [t₀, ∞[ and such that the infection cap, I_max, is never breached. For the SEIR model, will correspond to all states for which inputs β(t) ∈ [β_min, β_max] and γ(t) ∈ [γ_min, γ_max] exist such that the infection cap is never breached. Then the contact rate β, as well as the removal rate γ for the SEIR model, will be deduced depending on the location of the state with respect to the computed sets. Moreover, the complement of the admissible set, , is the set of states for which the infection cap cannot be maintained, regardless of the interventions. Thus, from this set the problem has no solution and if the state is located here, either the infection cap needs to be relaxed, or the maximal allowable social distancing measures (β_min), and/or maximal quarantine rate for the SEIR model (γ_max), need to be strengthened.

The set is obtained by constructing its boundary, which will be seen to be made of two parts, the usable part, a subset included in the boundary of the state constraints, and the barrier, entirely contained in the interior of the state constraint set (see the general presentation of Section 4, and further results and comparisons for the SIR and SEIR perfect models in Sections 7 and 8).

The second problem concerns also the SIR and SEIR perfect models:

Problem 2. For the SIR perfect model, we want to determine the set of initial states, (S(t₀), I(t₀), R(t₀)), from which the infection cap I_max will be maintained for all times and for any function β: [t₀, ∞[→[β_min, β_max], γ being perfectly known. For the SEIR model, the problem consists also in maintaining the infection cap I_max for all times and any pair of functions (β, γ) with β: [t₀, ∞[→[β_min, β_max] and γ: [t₀, ∞[→[γ_min, γ_max], η being perfectly known.

These sets are called the Maximal Robust Positively Invariant (MRPI) sets associated to the SIR and SEIR perfect model respectively (see their general definitions 3.2 and 3.3). Their construction, again via the construction of their boundary, consisting, as before, of a usable part and a barrier, is presented in Sections 5, 7 and 8).

The third problem concerns the SIR and SEIR imperfect models:

Problem 3. Suppose now that, for the SIR model, the value of the parameter γ is not known. All that we know is that γ ∈ [γ_min, γ_max]. We also assume that an intervention strategy has been pre-designed (SIR imperfect model). Find the states from which this intervention strategy will maintain the infection cap regardless of the error in the parameter γ, for all t ∈ [t₀, ∞[. For the SEIR model, we pose the same problem with unknown parameter η ∈ [η_min, η_max] in place of γ and with predesigned intervention strategies and (SEIR imperfect model).

Its solution will consist in constructing the MRPI set associated to the SIR (resp. SEIR) imperfect model (see Sections 6, 7 and 8).

Table 1 summarises the status of the parameters and controls for each problem.

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Table 1. Summary of known/unknown parameters and controlled/uncontrolled inputs.

https://doi.org/10.1371/journal.pone.0257598.t001

The main mathematical tool to solve these three problems is sketched in the next section.

4.1 Admissible set for perfect SIR model

We assume that the SIR model is perfect (that is, γ is known perfectly) and that β is a controllable time-varying input, which may, for example, model social-distancing measures to halt the disease’s spread. We impose hard constraints on the input and state: I(t) ≤ I_max and β(t) ∈ [β_min, β_max] for all t ∈ [t₀, ∞[, where I_max ∈ [0, 1] is the hard infection cap, and , with 0 < β_min ≤ β_max, are the minimal and maximal contact rates, respectively. Recall that the bound β_max may be interpreted as the nominal contact rate in absence of the disease, and β_min as the maximal social distancing.

Because R has no influence on S or I we can study the two-dimensional subsystem: (19) (20) (21) with t ∈ [t₀, ∞[. To ease our notation we will sometimes label the state , the control , the control constraint set , the single constraint function , and the set , where Π is given by (4). We also denote by .

Applying condition (12), the usable part is given by the condition which yields (β_min S − γ)I_max ≤ 0, or . Therefore the usable part is the set .

Focusing now on the barrier , let us label tangent points to G₀ by , where denotes the time at which G₀ is reached, as explained in Theorem 3.1. Invoking ultimate tangentiality, (17), we get: Dg(z₁, z₂) = (0, 1) and . Thus: (22) and along with the constraints z₂ = I_max and z₁ + z₂ ≤ 1, we get that the single tangent point must satisfy (β_min z₁ − γ)I_max = 0, or , with , one of the end point of the usable part.

The adjoint system, (14), reads: (23)

From the Hamiltonian minimisation condition, (15), we get: (24) from where we get: (25) where denotes the input associated with the barrier trajectories. Thus, at the point z, we have , which immediately implies, by continuity of the adjoint, that in some nonempty interval, that is, for , ϵ > 0.

Proposition 4.1. The input , given by (25), associated with the barrier of the admissible set of the perfect SIR model, (19)–(21) only switches at isolated points in time.

Proof. Suppose that λ₂(t) − λ₁(t) = 0 over an interval of time , with arbitrary and ϵ > 0. Then, we would have identically in this interval, i.e., according to (23),

We deduce that γλ₂(t) = γλ₁(t) = 0 over this interval. Therefore, the adjoint would identically vanish, which is impossible according to Theorem 3.1, hence the proposition.

Remark 4.1. As is known from optimal control, an input with bang-bang structure may exhibit other complicated behaviour, such as the “Fuller phenomenon”. Though we cannot make definite statements concerning the existence or nonexistence of such behaviours along barrier trajectories, they were not observed in any of the numerical examples in this paper.

The next Proposition states that barrier curves associated with the admissible set of the perfect SIR model always evolve backwards into G₋.

Proposition 4.2. Consider the constrained perfect SIR model, (19)–(21). The barrier is constituted by the curve generated by (25) with (23), that ends tangentially to G₀ ∩ Π at the point with , and is contained in G₋ at least for all , ϵ > 0.

Proof. Let denote the mapping over some open interval , with ϵ > 0. We have shown that the barrier curve has to be such that vanishes at as well as its first derivative, which is equal to . Moreover, β must remain constant, equal to β_min, in the interval . Let us compute ’s left second derivative:

Using the fact that , , and , this simplifies to:

Therefore, using Taylor’s expansion formula, in an interval for a sufficiently small ϵ, which proves that the barrier curve remains in G₋ in this interval.

Remark 4.2. Note that the point of the barrier satisfying I = 0 is an equilibrium point: , . Therefore the unique solution from (S(t₀), 0) is reduced to this point.

In fact, the whole axis I = 0 may be seen to be included in since every point of this axis is an equilibrium and thus satisfies the constraint forever and belongs to ∂Π, the boundary of Π (see also Section 7.1). It is remarkable that the points in this axis, though in the barrier, do not satisfy the conditions of Theorem 3.1 since they do not cross G₀ (recall that Theorem 3.1 is only necessary).

Finally, the integral curve of the system S(t) = 0, , , i.e. the semi-open line segment {(0, I)∣I ∈ ]0, I_max]} also lies in and hence in , neither satisfies the conditions of Theorem 3.1 since it does not intersect G₀ tangentially. It results that is the union of the set made by the integral curves satisfying Theorem 3.1 and of a set of special curves that do not intersect G₀ tangentially.

4.2 Admissible set for perfect SEIR model

We now focus on the model (5)–(8) and assume that β and γ are controllable inputs representing social distancing measures, and quarantining/isolation measures of infectives, respectively. We assume that η is perfectly known. Thus, we consider the three-dimensional constrained system: (26) (27) (28) (29) with t ∈ [t₀, ∞), 0 < I_max ≤ 1, 0 < β_min ≤ β_max, and 0 < γ_min ≤ γ_max. We will sometimes label the state , the controllable input , the control constraint set , , and , where Π is given by (9).

As we did with the perfect SIR model, we first construct the usable part given, using (12), by the condition which yields γ = γ_max and , S ∈ [0, 1 − I_max], or

Now, the barrier curves of must satisfy the ultimate tangentiality condition, (17), to get: and along with the constraints z₃ = I_max and 0 ≤ z₁ + z₂ + z₃ ≤ 1, we obtain the set of tangent points: (30)

Thus, unlike with the perfect SIR model, we now have a line segment in the boundary of where the admissible set’s barrier curves intersect G₀ tangentially. The adjoint system, (14), reads: (31) and condition (15) gives the input associated with the admissible set’s barrier curves: (32)

Since , we immediately see that . Moreover, we see from (31) that and thus, at , we get , implying that for all , ϵ > 0.

We have the following results concerning integral barrier curves.

Proposition 4.3. Consider the constrained perfect SEIR model, (26)–(29). Along any barrier integral curve satisfying (31) and (32) and such that z₁ ≠ 0 in (30), the inputs and , given by (32), only switch at isolated points in time.

Proof. Following the same arguments as in Proposition 4.1, suppose λ₂(t) − λ₁(t) = 0 over some interval of time, , , . Then, we would have for all , which would imply λ₂(t) − λ₃(t) = 0 for all . This would also imply that λ₃(t) = 0 for all since for all . Moreover, noting that , we would have for all t ∈ I. Thus, λ₁(t) = λ₂(t) = λ₃(t) = 0 for all , which is impossible according to Theorem 3.1.

Suppose now that λ₃(t) = 0 over some nonempty interval . Then we would have over this interval, which would imply that S(t)(λ₂(t) − λ₁(t)) = 0, provided that S(t)≠0, over this interval. But if S(t) = 0 over some interval of time, according to the uniqueness of the solution of (26)–(29), this would imply that S(t) = 0 for all times and in particular , which is excluded by assumption. Therefore λ₃(t) = 0 over some nonempty interval would imply λ₂(t) − λ₁(t) = 0 over this interval, which we have established is impossible in the first part of the proof.

The next proposition identifies which parts of are associated with barrier curves that evolve backwards into G₋. To now numerically construct the admissible set one only needs to focus on these tangent points.

Proposition 4.4. Consider the constrained perfect SEIR model, (26)–(29). The barrier is constituted by the integral curves satisfying (31) and (32), that end tangentially to G₀ at a point of , defined by (30), for which (33) and is contained in G₋ at least for all , ϵ > 0.

Proof. As in Proposition 4.2, the result follows from considering the inequality for points on . First, evaluating at , thanks to (30) and (31), we get . Therefore, λ₂(t) − λ₁(t) is decreasing before vanishing at , which proves that λ₂(t) − λ₁(t)>0 in an interval , with ϵ > 0 suitably chosen. We immediately deduce that β(t) = β_min in the same time interval . We also get, using the fact that , that γ = γ_max in this interval (possibly with a smaller ϵ). Thus . Moreover, in this interval, we must have which proves that z₁ must satisfy and, thanks to the definition of , , hence (33). The same Taylor’s expansion as in Proposition 4.2, implies that the barrier curves evolve backwards into G₋ provided that (33) holds.

Remark 4.3. Similar to Remark 4.2, the set {(S, E, I):S ∈ [0, 1], E = 0, I = 0} is a set of equilibrium points for the SEIR model and clearly a subset of . Moreover, any integral curve of the system initiating from the set {(S, E, I):S = 0, E = 0, I ∈ [0, I_max]} results in S(t)≡0, E(t)≡0, I(t)≤I_max for all t ∈ [t₀, ∞[, because γ(t)>0. Thus, as in the SIR case, the barrier is the union of curves that satisfy the necessary conditions of Theorem 3.1 and this special subset that does not intersect G₀ tangentially.

5.1 MRPI for perfect SIR model

We consider the same setting as that of Subsection 4.1, the constrained two-dimensional system (19)–(21), and describe the system’s MRPI set.

We first remark that the usable part , according to condition (13), is given by which yields .

Now, carrying out the analysis with the ultimate tangentiality condition, (18), and Hamiltonian maximisation condition (16), we note that the only difference with respect to (22) is that we maximize with respect to β in place of minimizing. Thus, we find that the sole tangent point is located at with . The adjoint equation being the same as (23), the input associated with the MRPI barrier curve is given by: (34) which also only switches at isolated points in time, the proof being exactly the same as that of Proposition 4.1.

The following proposition is the analogue of Proposition 4.2. Its proof follows exactly the same lines and is left to the reader.

Proposition 5.1. Consider the constrained perfect SIR model, (19)–(21). The barrier is constituted by the curve generated by (34) with (23), that ends tangentially to G₀ ∩ Π at the point with , and is contained in G₋ at least for all , ϵ > 0.

The reader may also verify that Remark 4.2 also applies to the barrier of the MRPI set.

5.2 MRPI for perfect SEIR model

We consider the same setting as that of Subsection 4.2: the constrained three-dimensional system (26)–(29) and describe the system’s MRPI set. The analysis follows exactly the same lines as in Subsection 4.2, the operation of maximisation replacing the minimisation one. We therefore only sketch the results.

The usable part is given by

Therefore,

Concerning the barrier, using conditions (16) and (18), we identify the line segment of potential tangent points: (35) where the subscript in refers to the “perfect” model case.

The adjoint system being described by (31), the input associated with the MPRI’s barrier curves are thus given by: (36)

These inputs may only switch at isolated points in time provided that z₁ ≠ 0 in (35), the proof following exactly the same arguments as the proof of Proposition 4.3. The next proposition is the analogue of Proposition 4.4, its proof also following the same lines of reasoning.

Proposition 5.2. Consider the constrained perfect SEIR model, (26)–(29). The barrier is constituted by the integral curves satisfying (31) and (36), that end tangentially to G₀ at a point of , defined by (35), for which (37) and is contained in G₋ at least for all , ϵ > 0.

As before, the reader may verify that Remark 4.3 applies to the MPRI for the perfect SEIR model as well.

6.1 MRPI for imperfect SIR model

We again consider the normalised SIR model, but now assume that the model parameter γ is not perfectly known and that the intervention strategy, a simple affine feedback strategy, is given by: (38) where the level of social distancing depends on the proportion of infective individuals, has been pre-designed and implemented. Thus, the system is now: (39) (40) (41) with t ∈ [t₀, ∞), and 0 < γ_min ≤ γ_max. Again, to ease our notation, we will sometimes label the state , the uncertain model parameter , the parameter set , and .

The adjoint Eq (14) reads: (42) where . Invoking conditions (16) we find the input associated with the MRPI barrier curve, given by: (43)

We again have a result concerning singular barrier curves.

Proposition 6.1. Consider the constrained imperfect SIR model, (39)–(41). Along any barrier integral curve satisfying (42) and (43) and such that z₁ ≠ 0 and 2β_min < β_max, the input , given by (43), only switches at isolated points in time.

Proof. As before, assume that λ₂(t) = 0 in some open interval . Then, indeed . The reader may confirm that if 2β_min < β_max, then α(I)≠0. Thus, if also z₁ ≠ 0, we see that if λ₂(t) = 0 over an interval , then λ₁(t) = λ₂(t) = 0 on , hence the contradiction.

From condition (18) the sole tangent point is located at , with , which we derive from the fact that .

Again, the barrier curve evolves backwards into G₋:

Proposition 6.2. Consider the constrained imperfect SIR model, (39)–(41). The barrier is constituted by the curve generated by (43) with (42), that ends tangentially to G₀ ∩ Π at the point , with , and is contained in G₋ at least for all , ϵ > 0.

Proof. The reader may easily verify that and . Thus, since and , and since γ(t) is constant in a suitable interval before , i.e. in this interval, we have and . The conclusion follows by the same Taylor’s expansion argument as in Proposition 4.2.

6.2 MRPI for imperfect SEIR model

Similar to the previous case, we consider the system: (44) (45) (46) (47) with t ∈ [t₀, ∞[, 0 < η_min ≤ η_max. The inputs and are assumed to be pre-designed by the simple affine feedback strategies: (48) and (49) which may be interpreted as the fact that we increase social distancing and increase the rate of quarantining if the infective number increases.

For this setting we identify: , the uncertain modelling parameter , the uncertain parameter set , , and .

As before, the reader may easily verify that the set of potential tangent points is: (50) where the subscript M_I in denotes the “imperfect” case. The adjoint, (14), reads: (51) where , and α(I) is as in Subsection 6.1.

It can be verified that the input associated with barrier curves of the MRPI is given by: (52)

Proposition 6.3. Consider the constrained imperfect SEIR model, (44)–(47). Along any barrier integral curve satisfying (51) and (52) and such that z₁ ≠ 0 and the determinant for all I ∈ [0, I_max], the input , given by (52), only switches at isolated points in time.

Proof. Assume, as before, that λ₃(t) − λ₂(t) = 0 in a given interval of time . We immediately deduce that and . Differentiating once more this latter expression yields . Thus, if the above determinant does not vanish for all I ∈ [0, I_max] and if S ≠ 0 we arrive at λ₁ = λ₂ = λ₃ = 0, the desired contradiction.

The proof of the next proposition follows the same lines as before and is left to the reader.

Proposition 6.4. Consider the constrained imperfect SEIR model, (44)–(47). The barrier is constituted by the curves generated by (52) with (51), that end tangentially to G₀ ∩ Π at a point of , defined by (50), with and is contained in G₋ at least for all , ϵ > 0.

7.1 SIR model

Proposition 7.1. Consider the constrained perfect SIR model (19)–(21). We have the following:

• if and only if ,
• , if and only if and ,
• , if and only .

Consider the constrained imperfect SIR model (39)–(41), under the feedback (38). We have the following:

• if and only if ,
• if and only if .

Proof. The constraint set G_Π is the polyhedron: and its boundary is the union of the four faces

Indeed, for every choice of β_min, β_max, I_max.

Assume that (53)

We will have if the vector field points towards the interior of G_Π along ∂G_Π for all values of β_min, β_max, I_max satisfying (53), i.e. (54) (55) (56) (57)

The last three inequalities (55)–(57) are trivially satisfied and (54) results from (53), which proves that (53) implies and thus also . The converse, namely that implies (53), follows from the fact that (54) is valid only if (53) holds.

We proceed in the same way to prove that if and , up to the fact that, in (54)–(57), the maximisation with respect to β is replaced by the minimisation with respect to β.

The third bullet point results from the fact that none of the two previous cases hold, which prevents from having an equality of or to G_Π.

The last two bullet points are obtained as the first and third bullets by changing f in and maximising with respect to γ ∈ [γ_min, γ_max].

Proposition 7.1 provides easily checkable inequalities of the system parameters that may be used to determine whether the sets are trivial or not. For example, focusing on the perfect SIR model, if then G_Π is invariant for any choice of input and one need not worry about ever breaching the infection cap, as long as the state initiates inside G_Π. However, this is not the case when , and the barrier of the MRPI must be computed by integrating the coupled differential equations of the system and the adjoint ones. If then the MRPI and the admissible set are proper subsets of G_Π. In this case there are parts of G_Π from which the infection cap will definitely be breached and, starting from or , we must compute the corresponding barrier to derive an adapted intervention strategy.

7.2 SEIR model

We now summarise the analogous results for the perfect and imperfect SEIR models.

Proposition 7.2. Consider the constrained perfect SEIR model (26)–(29). We have the following:

• if and only if η(1 − I_max) − γ_min I_max ≤ 0,
• , if and only if η(1 − I_max) − γ_min I_max > 0 and η(1 − I_max) − γ_max I_max ≤ 0,
• , if and only if η(1 − I_max) − γ_max I_max > 0.

Consider the constrained imperfect SEIR model (44)–(47) under the feedback (48) and (49). We have the following:

• if and only if η_max(1 − I_max) − γ_max I_max ≤ 0,
• if and only if η_max(1 − I_max) − γ_max I_max > 0.

Proof. We follow the same arguments as in the proof of Proposition 7.1. For the SEIR model (26)–(29), the set G_Π is the polyhedron and

We thus prove that is equivalent to the first bullet point inequality, i.e. (58) by showing that the vector field (recall that η is fixed) points towards the interior of G_Π along ∂G_Π for all values of β ∈ [β_min, β_max], γ ∈ [γ_min, γ_max] provided that I_max satisfies (58). We thus evaluate the scalar product of f with the normal to ∂G_Π at each of its edges. and and and and

Therefore, f points towards the interior of G_Π along ∂G_Π for all values of β and γ, or equivalently , if and only if (58) holds.

We proceed in the same way to prove that if and only if η(1 − I_max) − γ_min I_max > 0 and η(1 − I_max) − γ_max I_max ≤ 0, up to the fact that the maximisation with respect to β and γ is replaced by the minimisation with respect to β and γ.

As in the proof of Proposition 7.1, the third bullet point results from the fact that none of the two previous cases hold, which prevents from having an equality of or to G_Π.

Finally, the last two bullet points are obtained as the first and third bullets by changing f in and maximising with respect to η ∈ [η_min, η_max].

8.1 SIR examples

8.1.1 Perfect SIR example.

First, consider the perfect SIR system, (19)–(21), with γ = 0.5, β_min = 0.6, β_max = 0.8 and I_max ∈ {0.02, 0.15, 0.4}. Thus, this (hypothetical) disease has an average recovery rate of 2 days and, on average, an individual nominally comes into contact (sufficiently to catch the disease) with 0.8 other people per day. We have chosen these parameters to demonstrate the different types of possible sets.

For I_max = 0.4 we see from Proposition 7.1 that . Thus, with such a large cap on the infection numbers the entire space G_Π is robustly invariant. For I_max = 0.15 we see from Proposition 7.1 that . To now obtain the barrier associated with the MRPI, we integrate the system backwards in time from the tangent point utilising the input (34). We stop integrating once the trajectory intersects the boundary of G_Π because the set Π is positively invariant (recall the discussions in Subsections 2.1 and 2.2.) For I_max = 0.02 we see from Proposition 7.1 that . To obtain the barriers of the admissible set and MRPI we integrate the system backwards in time from the tangent points and utilising the input (34) and (25), respectively. See Fig 1 for details.

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Fig 1. Sets for perfect SIR model example.

Projections of the admissible and MRPI sets onto the S − I axes are shown. If the infection cap is large enough the entire constrained state space G_Π is robustly invariant (top figure), otherwise the sets are subsets of G_Π.

https://doi.org/10.1371/journal.pone.0257598.g001

It turns out that the input associated with the admissible set is saturated at β_min, and the one associated with the MRPI is saturated at β_max. Moreover, we can see that, going backwards, the barrier curves always evolve into G₋, as is expected from Propositions 4.2 and 5.1.

To now use the sets in the management of an epidemic the intervention strategy may be chosen according to the location of the state. A possible strategy could be:

• If , let β(t) = β_max,
• If , let β(t) = β_min,
• If , let .

To clarify, if the state is located in the MRPI then it is guaranteed that the infection cap can always be maintained, and the nominal contact rate, β_max, may be allowed. The same is true if the state is located in the interior of the admissible set, but this may only be possible for some period of time. If the state reaches the admissible set’s barrier, , then maximal social distancing must be imposed, thus β(t) = β_min, and the infection cap will be reached in finite time, but never breached. If the state reaches the admissible set’s usable part, , then some freedom in the social distancing is allowed. In fact, we may choose β(t) such that , which yields . Lastly, the set should always be avoided, as from here the infection cap is guaranteed to be breached regardless of β. Fig 2 shows the result of this switching law being applied from an arbitrary initial condition, , for the case where I_max = 0.02.

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Fig 2. Solution with the switching law.

The state trajectory is shown in the top figure (dashed curve). The middle plot is of the intervention strategy, β, and the right plot is the proportion of infectives over time.

https://doi.org/10.1371/journal.pone.0257598.g002

8.1.2 Imperfect SIR example.

Consider the imperfect SIR model, (39)–(41), with the simple feedback strategy as in (38) and the constants I_max = 0.2, β_min = 0.6, β_max = 0.8, and γ_min = 0.3, γ_max = 0.5. Thus, the recovery rate for this hypothetical disease may vary. We see from Proposition 7.1 that , and so we find the barrier of the system’s MRPI according to the analysis in Subsection 6.1, see Fig 3. It turns out that is saturated at γ_min all along the MRPI’s barrier. We also show a few solutions of the system from the initial state with γ ∈ [0.3, 0.5], emphasising that the infection cap can be maintained with this feedback regardless of the modelling uncertainty.

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Fig 3. MRPI for imperfect SIR model example.

We also show ten simulations from an initial state with randomly sampled γ ∈ [0.3, 0.5].

https://doi.org/10.1371/journal.pone.0257598.g003

Remark 8.1. In epidemiology it is common to want an intervention strategy that renders the basic reproduction number, , less than one in order to eradicate the disease. However, this is not required to maintain the infection cap using the introduced sets. In fact, in the examples of Subsection 8.1, for all mentioned values of β and γ.

8.2 SEIR examples

8.2.1 Perfect SEIR example.

Consider the perfect SEIR model, (26)–(29), with the constants β_min = 0.8, β_max = 1, , , and I_max ∈ {0.3, 0.4}. From Proposition 7.2 we see that for I_max = 0.4. Thus, we sample final tangent points along the line segment for which (see Proposition 5.2) and integrate backwards with the appropriate inputs, as specified in Subsection 5.2, terminating the integration if the curves intersect the infection cap, or the plane . This produces the red curves in Fig 4, running along the MRPI’s barrier.

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Fig 4. Sets for perfect SEIR model, with I_max ∈ {0.3, 0.4}.

Shown are curves running along the barriers of the MRPI (in red) and the admissible set (in blue).

https://doi.org/10.1371/journal.pone.0257598.g004

For I_max = 0.3 we see that . Thus, for this case we find both and , integrating backwards from and with the appropriate inputs as given in Subsections 5.2 and 4.2.

The inputs and are saturated at β_max and γ_min (resp. β_min and γ_max) for the barrier of the MRPI (resp. the barrier of the admissible set). As was observed in the examples for the SIR model, if I_max is large enough the entire set G_Π is robustly invariant, and if I_max is small enough, the sets and become nontrivial.

8.2.2 Imperfect SEIR example.

Now consider the imperfect SEIR model, (44)–(47) with the parameters β_min = 0.8, β_max = 1, , , , and I_max = 0.1, and the feedback (48) and (49). From Proposition 7.2, we have . Thus, we integrate backwards from points in for which (see Proposition 6.4) with the input as specified in Subsection 6.2.

An unexpected result for this set is that the input associated with barrier curves, , switches from η_min to η_max at some point along the barrier (these points are indicated on the figure). To explain this, we can interpret η as an input with the goal of maximising I over time. For this example it is optimal for it to remain at η_min in order to build up the proportion of exposed individuals to some threshold when it switches to η_max, which then results in a large increase in the proportion of infectives. This is the worst-case change in η that can be expected. The computed set is shown in Fig 5.

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Fig 5. MRPI for imperfect SEIR model.

Going backwards in time, the input associated with the barrier, , switches from η_max (blue curves) to η_min (red curves) at the black dots.

https://doi.org/10.1371/journal.pone.0257598.g005