Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 59, pp. 1–13. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.59 OPTIMAL SWITCHING OF VACCINATION FOR AN INFECTIOUS DISEASE MODEL SHRADDHA SALWAHAN, SYED ABBAS, ABDESSAMAD TRIDANE Abstract. We study a switched SIHR (Susceptible Infected Hospitalized Re- covered) model for infectious diseases. The aim is to find the most effective switching signal to manage the disease’s effects as effectively as maintaining a continuous vaccination program. A nonlinear, non-convex optimal control problem of the switched system is converted into a linear and convex opti- mal control problem by applying the theory of moments and semi-definite programming. Finally, some numerical simulations are performed to compare continuous vaccination programs and optimal switching of vaccination control. 1. Introduction For many decades and even centuries, differential equations have served as pow- erful tools for modelling a wide range of complex phenomena. These models can be based on either ordinary differential equations (ODEs) or partial differential equa- tions (PDEs). Some examples include modelling the spread of infectious diseases [15], analyzing the impact of criminal behaviour on society [19], studying diseases such as Alzheimer’s [2, 9], and applications in engineering and fluid dynamics [22]. In this article, we specifically focus on a mathematical model related to controlling the spread of infectious diseases. Because of the spread of several infectious diseases, mathematical modelling of these diseases has gained much attention in the past few decades. Modeling in- fectious diseases and even epidemics helps study all kinds of diseases in different situations. There is a vast amount of literature on this. Numerous mathematical elements can be incorporated to enhance disease modeling. For instance, one can include delay, which can be time-dependent or state-dependent, etc., or even im- pulse, which can be instantaneous or non-instantaneous, and many more. Some of the articles and books for the references can be [4, 7, 27, 30]. The primary objective of approaching the study of infectious diseases in this manner is to achieve several vital aims: controlling disease transmission through various interventions, forecast- ing disease spread, or merely comprehending the dynamics of infectious diseases to enhance our readiness for future scenarios. The most prevalent strategies for man- aging an infectious disease involve vaccination and isolation. The optimal control of these interventions provides an optimum manner in which the disease can be 2020 Mathematics Subject Classification. 34D20, 34A38, 92B05. Key words and phrases. Stability analysis; switched system; optimal vaccination. ©2024. This work is licensed under a CC BY 4.0 license. Submitted August 26, 2024. Published October 8, 2024. 1 2 S. SALWAHAN, S. ABBAS, A. TRIDANE EJDE-2024/59 controlled under a particular objective function. Since there is much literature on optimal control of infectious diseases, some references specific to optimal control by vaccination are [5, 10, 11, 21]. A switched system is a type of hybrid system in which the system switches be- tween several subsystems depending on some conditions or a switching signal. They have vast applications in engineering [6]. Switched systems can also be employed in the mathematical modeling of infectious diseases. This approach is particularly valuable for emulating real-world scenarios through mathematical models, which can be challenging otherwise, for example, a limited stockpile of drugs and vac- cines and the effect of seasonality in the disease transmission [16, 26]. Impulses can also be included in a switched infectious disease model [25]. Optimal control of a switched system can also be an interesting application to disease modeling [28, 31]. In a switched system, determining the optimal control for the system becomes a challenging problem when the sequence of subsystems or the switching signal is not known in advance. This problem can be divided into two main parts, the first of which involves identifying the optimal instants to switch between subsystems, and this in itself is a complex task [29, 8]. Although numerically expensive, some methods like dynamic programming and gradient descent are used to evaluate the optimal switching signal [1]. An innovative approach to this challenge is studied in [17]. This novel method relies on leveraging the theory of moments for global polynomial optimization using semi-definite programming to determine the optimal switching instants. Several research articles discussing this approach include [3, 13, 23]. This article employs the methodology described in [3, 17]. The application of this method to modeling and controlling infectious diseases is also discussed in [20, 23]. Notably, both articles do not consider the population of critically ill or hospitalized individuals. While in [23], the treatment category is treated equiva- lently to hospitalization; this article assumes that individuals receiving treatment or undergoing isolation are within the infected group, with only critically ill indi- viduals hospitalized. Vaccination is considered the only control intervention. The goal is to minimize the infected and hospitalized populations. To assess the impact of limited vaccination, specifically vaccinations administered at optimal switching instants, the numerical simulations are compared against continuous vaccination scenarios. This article is structured as follows: The model is formulated in section 2. Section 3 covers the evaluation of equilibrium points and the Basic Reproduction Number. Section 4 defines the model as a switched system and then outlines the primary problem, which is the Switched Optimal Control Problem (SOCP). By using the Lagrange polynomial, the SOCP is transformed to its polynomial equivalent of the optimal control problem (PEOCP) in section 5. Finally, in section 6, the moments and moment matrix are discussed briefly, and by using the moments approach, a semi-definite program (SDP) is formulated. Section 7 illustrates the results with a numerical example. 2. Model formulation The objective of this study is to examine the effect of vaccination as a primary intervention to control the spread of disease. We work on the SIHR model with EJDE-2024/59 OPTIMAL SWITCHING OF VACCINATION 3 critically ill individuals admitted to hospitals. S′ = µ− βSI − µS − λϵS, I ′ = βSI − αI − γ1I + δH − (µ+ µ1)I, H ′ = αI − δH − γ2H − (µ+ µ2)H, R′ = γ1I + γ2H − µR+ λϵS, (2.1) with the initial condition at t = t0 as S(t0), I(t0), H(t0), and R(t0). The nonlinear SIHR epidemiological model includes four non-negative state variables S(t), I(t), H(t), and R(t) that are defined as Susceptible, Infected, Hospitalized, and Recov- ered respectively. Here, S(t) represents the number of individuals susceptible to the infection (i.e., they are not infected yet). The individuals who get infected move to the infected compartment at rate β. Individuals in this compartment spread the disease. All infected individuals undergoing treatment or are quarantined are con- sidered in this compartment. From this compartment, only the critically ill humans are shifted to hospitals at rate α. The recovered individuals move to the recovered compartment from the infected compartment at rate γ1 and from the hospitalized compartment at rate γ2. Death due to the disease is at rate µ1 in the infected com- partment and µ2 in the hospitalized compartment. µ is the natural mortality rate. The parameter λ is the vaccination rate, and ϵ is the control parameter, described later in the article. Figure 1 describes the flow diagram of the model (2.1). Figure 1. Conceptual flow diagram of the SIHR model. 3. Equilibria and basic reproduction number The disease-free equilibrium point is, E1 = ( µ λϵ+ µ , 0, 0, λϵ λϵ+ µ ) . The endemic equilibrium is denoted as, E2 = (S∗, I∗, H∗, R∗). Next, using the next-generation matrix [24], to calculate the basic reproduction number (R0). Let the population be grouped in k number of compartments. Consider xi as the population in each compartment. Then, a general model describing the movement of humans between the com- partments can be written as ẋi = Fi(x)− Vi(x), where x = (x1, . . . , xk) T . 4 S. SALWAHAN, S. ABBAS, A. TRIDANE EJDE-2024/59 We define Fi as the influx of infected population, and Vi denotes the outflow of population from the ith compartment. Vi can be further written as, Vi = V+ i − V− i . Furthermore, since each function represents the transfer of individuals, it is important to consider x ≥ 0, Fi ≥ 0, V+ i ≥ 0, V− i ≥ 0, for i = 1, . . . , k. If E1 is the equilibrium point and F and V are the k × k matrices defined as, F = [ ∂Fi ∂xj (E1) ] , V = [ ∂Vi ∂xj (E1) ] , then Basic reproduction is defined as R0 = ρ(FV −1). (3.1) We calculate the basic reproduction number R0 for the model (2.1) by using the method given above. We have F = [ βSI + δH 0 ] , V = [ (α+ γ1)I + (µ+ µ1)I δH + γ2H + (µ+ µ2)H − αI ] , where F and V are the column vectors of Fi and Vi, respectively. Then, by consid- ering the notation defined above, we obtain F = [ βS δ 0 0 ] , V = [ α+ γ1 + µ+ µ1 0 −α δ + µ+ µ2 + γ2 ] . Therefore, the Basic reproduction number is R0 = µβ (λϵ+ µ)(α+ γ1 + µ+ µ1) + δα (α+ γ1 + µ+ µ1)(δ + µ+ µ2 + γ2) . 4. SIHR model as switched system and optimal control problem Let y = (S, I,H,R)⊤. Then we can write model (2.1) as a switched system, ẏ(t) = gω(t)(y(t)), where gi : R4 → R4 is the ith vector field and ω : [t0, T ] → S is the switching signal. The switching signal ω(t) is both a time-dependent function and a piece-wise constant. We assume t0 = 0, so the interval becomes [0, T ] and initial conditions are y(0) = (S(0), I(0), H(0), R(0)⊤. Here ϵ will be the control and switching parameter, i.e., ϵ = 0, 1. Case ϵ = 0 means that there is no vaccination. In real life, it can mean that there is no stock of vaccination doses or that there is no vaccination drive. Case ϵ = 1 signifies the presence of vaccination as an intervention. Each mode in the switching system is associated with a particular subsystem. It is defined as, ẏ(t) = gi(y(t)) for i ∈ S = {0, 1}. The subsystem i = 0 corresponds to the case when ϵ = 0 and subsystem i = 1 means the case when ϵ = 1. Hence, the system switches between two subsystems according to the presence of vaccination control. Before further analysis, we shall assume some conditions. Since the switched system consists of a duplet of a finite sequence of nodes and switching time t0 = 0 < t1 < · · · < tn = T , then: • There are no jump discontinuities in the state. • There are no infinite switching accumulation points. The main goal of this work is to study the impact of optimal switching instants with respect to an objective function. Let us assume the objective functional to be J = ∫ T t0=0 {I(t) +H(t)}dt. EJDE-2024/59 OPTIMAL SWITCHING OF VACCINATION 5 Let the running cost be denoted by Lω(t) ,i.e., Lω(t)(t, y(t)) = I(t) +H(t). Hence, L0 = I(t) + H(t) and L1(t, y(t)) = I(t) + H(t). Therefore, the switched optimal control problem (SOCP) can be written as min ω(t) J(t0, T, y(t), ω(t)), (4.1) subject to ẏ(t) = gω(t)(y(t)), where ω(t) ∈ S = {0, 1}. (4.2) 5. Polynomial equivalent optimal control problem (PEOCP) We will first define the above SOCP (4.1) and (4.2) in the form of a continuous system, which is not a switched system with a control variable. A polynomial expression is built, which can mimic the behaviour of the switched system. Like the approach used in [17], Lagrange polynomials transform the switched system into a continuous one. The control variable shall be denoted by ξ, and ξ ∈ ∆ = {ξ ∈ R| ϕ(ξ) = 0} where, ϕ(ξ) = ξ(1− ξ). (5.1) According to the kth Lagrange polynomial pk(ξ), if k = 0, 1, then p0(ξ) = (1− ξ), p1(ξ) = ξ. (5.2) From the results in [17], we can write system (4.2) as ẏ(t) = G(y, ξ) = g0(y)p0(ξ) + g1(y)p1(ξ). (5.3) Similarly, the running cost can be written as L(y(t), ξ) = L0(t, y(t))p0(ξ) + L1(t, y(t))p1(ξ). (5.4) Therefore (4.1) becomes J = ∫ T t0=0 L(y(t), ξ)dt. Therefore, we can formulate the PEOCP as min ξ∈∆ J(t0, T, y(t), ξ), (5.5) subject to ẏ(t) = 1∑ k=0 gk(y)pk(ξ). (5.6) The problem becomes non-convex because the set ∆ is non-convex. To overcome it, we shall redefine the PEOCP using the moments technique. 6. Semi-definite relaxation using moments approach In this section, we will first discuss the moments and localizing matrices and then move on to the semi-definite program for the PEOCP. We will cover the topic of moments and moment matrices briefly. For more details, one can refer to [20, 13, 14] and the references within. The smallest closed set Q ⊂ Rn such that ν(Rn\Q) = 0 for a measure ν on Rn is defined as the support of ν, expressed as supp(ν), and the measure ν is said to be supported by Q if supp(ν) ⊂ Q. 6 S. SALWAHAN, S. ABBAS, A. TRIDANE EJDE-2024/59 If ν is a probability measure supported by Q ⊂ R and R[x]n is the space of polynomials of degree at most n and in variable x, then the ith moment of ν is defined as zi = ∫ Q xi ν(dx). It means for a polynomial l(x) ∈ R[x]n written as l(x) = ∑n i=0 lix i,∫ Q l(x) ν(dx) = n∑ i=0 lizi. Now for probability measure ν, if z = {zi}2si=0 is a sequence of moments, then moment matrix Ms(z) is defined as a (s+ 1)× (s+ 1) matrix with (i, j) the entry as zi+j , 0 ≤ i, j ≤ s. Hence, the matrix becomes, Ms(z) =  z0 z1 z2 . . . zs z1 z2 z3 . . . z(s+1) ... ... ... . . . ... zs z(s+1) z(s+2) . . . z2s  . For the polynomial l(x) and sequence of moments z defined above the Localising matrix can be defined as an (s + 1) × (s + 1) symmetric matrix with its (i, j)th entry as Ms(lz)(i, j) = n∑ k=0 lkzi+j+k. One can read [23, 12] for examples and other details. From the results discussed in [20, 23] for control variable ξ ∈ ∆, and the constraint ϕ(ξ) = 0 or ϕ(ξ) ≥ 0 and −ϕ(ξ) ≥ 0. We will denote ϕ1 = ϕ(ξ) and ϕ2 = −ϕ(ξ). The following two lemmas help formulate the semi-definite program. Lemma 6.1. If z = {zi}2si=0 is a sequence of moments of probability measure ν supported by Q, then Ms(z) ⪰ 0. Lemma 6.2. Assume that the polynomials ϕ1(ξ) and ϕ2(ξ) satisfies d 1 ϕ = ⌈deg(ϕ1)/2⌉ = 1 and d2ϕ = ⌈deg(ϕ2)/2⌉ = 1. Consider the sequence z = {zi}2si=0 of moments of probability measure ν supported by set Q1 = {ν ∈ R|ϕ1 ≥ 0}. Then Ms−d1 ϕ (ϕ1z) ⪰ 0, i.e, Ms−1(ϕ1z) ⪰ 0 and similarly Ms−1(ϕ2z) ⪰ 0. Proof. Letting h ∈ R[ξ](s−1), that is h(ξ) = ∑s−1 j=0 hjξ j , then h⊤M(s−1)(ϕ1z)h = s−1∑ α=0 s−1∑ β=0 hαhβM(s−1)(ϕ1z)(α, β) = s−1∑ α=0 s−1∑ β=0 hαhβ [ 0 + zα+β+1 − zα+β+2 ] = s−1∑ α=0 s−1∑ β=0 hαhβ [ ∫ Q1 ξα+β+1 ν(dξ)− ∫ Q1 ξα+β+2 ν(dξ) ] EJDE-2024/59 OPTIMAL SWITCHING OF VACCINATION 7 = ∫ Q1 (h(ξ))2ϕ1(ξ)ν(dξ) ≥ 0. Hence M(s−1)(ϕ1z) ⪰ 0. Similarly, we can consider measure µ supported by set Q2 = {µ ∈ R|ϕ2 ≥ 0} along with its corresponding sequence of moments and proceeding as above, it is not difficult to prove that M(s−1)(ϕ2z) ⪰ 0. □ Suppose that ∆ is a Borel subset of Rn and P (∆) is the space of probability measures ν which has support contained in Γ. Then we can say that min ξ∈∆ J = min ν∈P (∆) ∫ ∆ Jν(dξ). (6.1) The proof of (6.1) can be found in [13] and [12]. Hence the minimization problem (5.5) becomes min ξ∈∆ J(t0, T, y(t), ξ) = min ν∈P (∆) ∫ ∆ J ν(dξ) = min ν∈P (∆) ∫ ∆ ∫ T t0=0 L(y, ξ) dt ν(dξ). Then minν∈P (∆) ∫ ∆ ∫ T t0=0 L(y, ξ) dt ν(dξ) is equal to min ν∈P (∆) ∫ ∆ ∫ T t0=0 L0(t, y(t))p0(ξ) + L1(t, y(t))p1(ξ) dt ν(dξ) = min ν∈P (∆) ∫ ∆ ∫ T t0=0 {(L0 + L1)ξ + L0} dt ν(dξ) = min z∈Z ∫ T t0=0 {(L0 + L1)z1 + L0z0} dt = min z∈Z ∫ T t0=0 1∑ k=0 1∑ i=0 Lk(t, y(t))ckizidt, where Z is the space of moments i.e., Z = {z = {zi}|zi = ∫ ∆ ξiν(dξ), ν ∈ P (∆)}. Also, c00 = 1, c01 = −1, c10 = 0, and c11 = 1. Now, by using the approach in [17], a semi-definite program (SDP) of relaxation order s ≥ 1 can be formulated as SDPs: min z∈Z ∫ T t0=0 1∑ k=0 1∑ i=0 Lk(t, y(t))ckizi dt, Ms(z) ⪰ 0, Ms−1(ϕ1z) ⪰ 0, Ms−1(ϕ2z) ⪰ 0, ẏ(t) = 1∑ k=0 1∑ i=0 gk(y)ckizi. (6.2) 8 S. SALWAHAN, S. ABBAS, A. TRIDANE EJDE-2024/59 So, the SDP for the lowest order of relaxation is for s = 1. which is SDP1: min z∈Z ∫ T t0=0 1∑ k=0 1∑ i=0 Lk(t, y(t))ckizi dt, M1(z) ⪰ 0, M0(ϕ1z) ⪰ 0, M0(ϕ2z) ⪰ 0, ẏ(t) = 1∑ k=0 1∑ i=0 gk(y)ckizi. (6.3) 7. Numerical simulations We perform some numerical simulations to support the analysis presented above. The parameter values under consideration are not specific to any particular disease (table 1), and research articles that influence them are [23, 18]. The reproduction number is calculated to be R0 = 2.05 when there are no vaccination strategies, i.e., ϵ = 0 and R0 = 0.24 when there is continuous vaccination, i.e., ϵ = 1. The method followed for the numerical simulation is similar to [23]. Table 1. Parameters used Parameter Value Parameter Value β 0.99 δ 0.1 µ 0.033 γ1 0.2 µ1 = µ2 0.0833 γ2 0.1 α 0.2 λ 0.5 S(0) 0.79 I(0) 0.16 H(0) 0.05 R(0) 0 The primary aim of this research study is to reduce the number of people in both the infected and hospitalized compartments. This objective has been chosen be- cause it is more cost-effective to promote recovery through vaccination than treating individuals with specific medications after infection. Furthermore, hospitalization costs significantly exceed the expense of a vaccination dose. Here, we minimize the objective function by finding the optimal number of switches of vaccination so that we get similar behavior to continuous vaccination and the disease is in control even when R0 > 1. In Figure 2, it is illustrated that the susceptible population is the least in the presence of continuous vaccination. We can conclude that most of the population recovers from vaccination and will never be infected. Conversely, in case of no vaccination, the susceptible population is maximum, which means that all may get infected in the future. For the optimal switching of vaccination control, the susceptible population is close to that in the case of continuous vaccination. In Figures 3 and 4, the time series of the population in the infected and hospital- ized compartments are the same for both continuous vaccination and vaccinating optimally. Since R0 > 1 in case of no vaccination control, the infected and hospi- talized compartment population is stable at the endemic equilibrium. Finally, in Figure 5, the recovered population is maximum for the case of constant vaccination EJDE-2024/59 OPTIMAL SWITCHING OF VACCINATION 9 0 10 20 30 40 50 60 70 80 90 Time 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 s u s c e p ti b le p o p u la ti o n Optimal Vaccination No Vaccination Continuous vaccination Figure 2. Time evolution of model (2.1) when R0 > 1. 0 10 20 30 40 50 60 70 80 90 Time 0 0.05 0.1 0.15 0.2 0.25 In fe c te d p o p u la ti o n Optimal Vaccination No Vaccination Continuous vaccination Figure 3. Time evolution of model (2.1) when R0 > 1. and minimum for no vaccination. Finally, Figure 6 depicts the optimal switching signal. The optimal switching signal indicates that initiating vaccination strategies at the onset of an epidemic is critical for controlling the disease, and the availability of vaccines (duration of vaccination campaigns) can be gradually reduced over time. Based on all the figures related to the numerical simulation, it can be concluded that even when the reproduction number exceeds one and the availability of vac- cines is limited, the disease can still be effectively controlled through the optimal use of vaccination. The effect of switching vaccination strategies optimally, while not identical, is nearly as effective as continuous vaccination of the population. 10 S. SALWAHAN, S. ABBAS, A. TRIDANE EJDE-2024/59 0 10 20 30 40 50 60 70 80 90 Time 0 0.02 0.04 0.06 0.08 0.1 0.12 H o s p it a liz e d p o p u la ti o n Optimal Vaccination No Vaccination Continuous vaccination Figure 4. Time evolution of model (2.1) when R0 > 1. 0 10 20 30 40 50 60 70 80 90 Time 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R e c o v e re d p o p u la ti o n Optimal vaccination No vaccination Continuous vaccination Figure 5. Time evolution of model (2.1) when R0 > 1. 8. Conclusion This research article studies a simple SIR model with hospitalization as a differ- ent compartment. In light of the global experience with a pandemic and the lessons learned from being ill-prepared, it is imperative to take proactive measures for the future. Various interventions have been devised to manage the disease, but in this article, we focus solely on vaccination control. It is important to note that vaccina- tion doses can be limited based on the available stockpile within a city or country, making them potentially unavailable at times. Hence, we considered vaccination as a switched intervention. By using the strategy in [17], we solve the switched control problem. Implementing vaccination at the initial stages of disease spread is crucial, as this is when infections escalate rapidly. As time progresses, the period of effective control diminishes, and the intervals between vaccination drives can be increased. To conclude, optimal switching and Optimal control of the control inter- ventions can help control the spread of disease even when its reproduction number is greater than 1. EJDE-2024/59 OPTIMAL SWITCHING OF VACCINATION 11 0 10 20 30 40 50 60 70 80 90 Time 0 0.2 0.4 0.6 0.8 1 1.2 (t ) Figure 6. 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Teng; Pulse vaccination delayed SEIRS epidemic model with saturation incidence. Applied Mathematical Modelling, 32 (2008), no. 7, 1403-1416. [31] F. Zhu, P. J. Antsaklis; Optimal control of hybrid switched systems: A brief survey. Discrete Event Dynamic Systems, 25(2015), 345-364. Shraddha Salwahan School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Kamand (H.P.) - 175005, India Email address: ss14.salwahan@gmail.com EJDE-2024/59 OPTIMAL SWITCHING OF VACCINATION 13 Syed Abbas School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Kamand (H.P.) - 175005, India Email address: sabbas.iitk@gmail.com, abbas@iitmandi.ac.in Abdessamad Tridane Department of Mathematical Sciences, United Arab Emirates University, Al Ain P.O. Box 15551, UAE Email address: a-tridane@uaeu.ac.ae 1. Introduction 2. Model formulation 3. Equilibria and basic reproduction number 4. SIHR model as switched system and optimal control problem 5. Polynomial equivalent optimal control problem (PEOCP) 6. Semi-definite relaxation using moments approach 7. Numerical simulations 8. Conclusion Acknowledgments. References