Managing an Epidemic Using Compartmental Models and Measure Differential Equations

Abstract

Classical epidemiological models can be complemented with probabilistic ones to describe the dynamic of virus mutations. Here controlled ordinary differential equations are coupled with evolution equations for measures, called measure differential equations, to capture both the population dynamics and the virus ones. Optimal controls are then used to design non-pharmaceutical interventions minimizing the health and economic cost of a pandemic. Simulations show the capability of the model to capture observed waves and the effectiveness of optimization strategies.

Appendices

Appendix 1: Measure Differential Equations

In this section, we recall the basic definitions for the theory of MDEs introduced in [37]. These results are provided specifically for probability measures on $R^n$ with compact support and absolutely continuous with respect to the Lebesgue measure. These probability measures will be used as such when coupled with a classical epidemiological problem later. For the sake of simplicity, we will present the theory on a Euclidean space $\mathbb {R}^n$ and for measures with compact support, but most results are valid for Polish spaces $(X,d)$ and measures with finite moments. We refer the reader to [43] for details. We indicate by $\mathcal {M}(X)$ the set of positive Borel measures with finite mass and compact support on X. Given two measures $\mu$, $\nu \in \mathcal {M}(X)$, the set of transference plans from $\mu$ to $\nu$ is denoted by $P(\mu ,\nu )$, and for $\tau \in P(\mu ,\nu )$ we define the p-cost of $\tau$ as $J_p(\tau )=\left (\int _{X^2} d(x,y)^p\, d\tau (x,y)\right )^{\frac {1}{p}}$. The p-Wasserstein distance between $\mu$ and $\nu$ is given by:

$$\displaystyle \begin{aligned} {} W^X_p(\mu,\nu)=\inf_{\tau\in P(\mu,\nu)} J_p(\tau). \end{aligned} $$

(23)

We indicate by $P^{opt}(\mu ,\nu )$ the set of transference plans realizing the infimum in (23).

Since our measures will change mass in time, we need to consider the generalized Wasserstein distance. The generalized Wasserstein distance is defined for measures with different mass combining the total variation and the classical Wasserstein distances.

Definition 10

Given $\mu ,\nu \in \mathcal {M}(\mathbb {R})$ we set:

$$\displaystyle \begin{aligned} {} W^g(\mu,\nu):=\inf_{\tilde\mu,\tilde\nu\in\mathcal{M},\,|\tilde\mu|=|\tilde\nu|}|\mu-\tilde\mu|+|\nu-\tilde\nu|+W(\tilde\mu,\tilde\nu). \end{aligned} $$

(24)

For properties and results, we refer the reader to [38, 39]. We are ready to define the operator $\mathcal {W}^g$:

Definition 11

Fix $V_i \in \mathcal {M}(T\mathbb {R}^n)$, $i=1,2$, and set $\mu _i = \pi _1 \#V_i$. We define

$$\displaystyle \begin{aligned} {} \begin{aligned} \mathcal{W}^g (V_1,V_2) := & \inf \left\{ \int_{T\mathbb{R}^n\times T\mathbb{R}^n} |v-w| \ dT(x,v,y,w) \colon \right.\\ & \left.T\in P(\tilde{V}_1,\tilde{V}_2) \text{ with } , \tilde{V}_i {\leq} V_i, \tilde{\mu}_i {\,=\,} \pi_1 \# \tilde{V}_i, \pi_{13}\# T{\in} P^{opt}(\tilde{\mu}_1,\tilde{\mu}_2) \right.\\ & \left. \text{ where } (\tilde{\mu}_i, \tilde{\nu}_i) \text{ is a minimizer in }\text{(24)} \right\}. \end{aligned} \end{aligned} $$

(25)

A measure vector field (briefly MVF) on $\mathcal {M}(\mathbb {R}^n)$ is a map $V: \mathcal {M}(\mathbb {R}^n)\to \mathcal {M}(T\mathbb {R}^n)$ with $\pi _1\# V[\mu ]=\mu$, and the corresponding measure differential equation (MDE) is given by:

$$\displaystyle \begin{aligned} {} \dot\mu=V[\mu]. \end{aligned} $$

(26)

and the Cauchy problem by:

$$\displaystyle \begin{aligned} {} \dot\mu=V[\mu],\qquad \mu(0)=\mu_0. \end{aligned} $$

(27)

A solution to (27) is a map $\mu :[0,T]\to \mathcal {M}(\mathbb {R}^n)$ such that $\mu (0)=\mu _0$, and

$$\displaystyle \begin{aligned} {} \frac{d}{dt}\int_{\mathbb{R}^n} f(x)\,d\mu(t)(x) = \int_{T\mathbb{R}^n} (\nabla f(x)\cdot v)\ dV[\mu(t)](x,v). \end{aligned} $$

(28)

We omit here the details about the needed regularity for (28) to make sense and refer the reader to [37]. Existence of solutions to (27) is guaranteed if the following assumptions hold true:

(H:bound):

V  is support sublinear: $\exists C>0$ s.t. for every $\mu \in \mathcal {M}(X)$:

$$\displaystyle \begin{aligned} \sup_{(x,v)\in Supp(V[\mu])} |v|\leq C \left( 1 + \sup_{x\in Supp(\mu)} |x|\right). \end{aligned}$$

(H:cont):

$V:\mathcal {M}(\mathbb {R}^n)\to \mathcal {M}(T\mathbb {R}^n)$ is continuous (for the topologies of $W^{\mathbb {R}^n}$ and $W^{T\mathbb {R}^n}$.)

Solutions can be constructed directly using Euler-type approximations called lattice approximate solutions (briefly LAS). Given $N\in {\mathbb N}$ set $\Delta _N =\frac {1}{N}$, $\Delta ^v_N=\frac {1}{N}$, and $\Delta ^x_N=\Delta ^v_N\Delta _N=\frac {1}{N^2}$, which represents the discretization of time, velocity, and space. The solution will consist of Dirac deltas centered at the lattice points $x_i$ of ${\mathbb Z}^n/(N^2)\cap [-N,N]^n$ and $v_j$ of ${\mathbb Z}^n/N\cap [-N,N]^n$. Define the discretization operator:

$$\displaystyle \begin{aligned} {} {\mathcal A}^x_N(\mu)=\sum_i m^x_i(\mu) \delta_{x_i} \end{aligned} $$

(29)

with $m^x_i(\mu )=\mu (x_i+Q)$ and $Q=([0,\frac {1}{N^2}[)^n$. An MVF can be approximated by:

$$\displaystyle \begin{aligned} {} {\mathcal A}^v_N(V[\mu])= \sum_i \sum_j m^v_{ij}(V[\mu])\ \delta_{(x_i,v_j)} \end{aligned} $$

(30)

where $m^v_{ij}(V[\mu ])=V[\mu ](\{(x_i,v):v\in v_j+Q'\})$ and $Q'=([0,\frac {1}{N}[)^n$.

It is immediate to get:

$$\displaystyle \begin{aligned} W({\mathcal A}^x_N(\mu),\mu)\leq {\sqrt{n}}\, \Delta^x_N,\qquad W^{T\mathbb{R}^n}({\mathcal A}^v_N(V[\mu]),V[\mu])\leq {\sqrt{n}}\, \Delta^v_N. \end{aligned}$$

Given V  satisfying (H:bound), (27), $T>0$, and $N\in {\mathbb N}$, we define LAS $\mu ^N:[0,T]\to \mathcal {P}_c(\mathbb {R}^n)$ by recursion. First $\mu _0^N={\mathcal A}^x_N(\mu _0)$, and then:

$$\displaystyle \begin{aligned} {} \mu^N_{\ell+1}=\mu^N((\ell+1)\Delta_N)=\sum_i \sum_j m^v_{ij}(V[\mu^N(\ell\Delta_N)])\ \delta_{x_i+\Delta_N\, v_j}. \end{aligned} $$

(31)

$$\displaystyle \begin{aligned} {} \mu^N(\ell\Delta_N+t)=\sum_{ij} m^v_{ij}(V[\mu^N(\ell\Delta_N)])\ \delta_{x_i+t\, v_j}. \end{aligned} $$

(32)

By construction, for $\mu _0$ with $Supp(\mu _0)\subset B(0,R)$ and $\ell$ such that $\ell \Delta _N\leq T$, we have [37]:

$$\displaystyle \begin{aligned} Supp(\mu^N_\ell) \subset B\left(0,e^{C_N T} (R_N+1)-1\right), \end{aligned} $$

(33)

where $C_N=C+\frac {\sqrt {n}}{N}$ and $R_N=R+\frac {\sqrt {n}}{N^2}$.

Generalizations of such results to the case of coupled ODE-MDE are given in [23].

1.1 MDE for Finite Speed Diffusion

As shown originally in [24], the MDE framework allows modeling diffusion with finite speed, which we use for virus variant dynamics. The diffusion speed is regulated by assigning an increasing map $\varphi :[0,1]\to \mathbb {R}$ and defining a MVF $V_\varphi$ as follows. First, define:

$$\displaystyle \begin{aligned} {} J_\varphi(x)=\begin{cases} \delta_{\varphi(F_\mu(x))}&\mbox{ ~if }F_\mu(x^-)=F_\mu(x),\\ \\[2mm] \frac{\varphi\#\left(\chi_{[F_\mu(x^-),F_\mu(x)]}\lambda\right)}{F_\mu(x)-F_\mu(x^-)}&\mbox{ ~otherwise,} \end{cases} \end{aligned} $$

(34)

where $F_\mu (x)=\mu (]-\infty ,x])$ is the cumulative distribution of $\mu$, $\#$ indicates push-forward, and $\lambda$ is the Lebesgue measure. We set

$$\displaystyle \begin{aligned} {} V_\varphi[\mu]=\mu\otimes_x J_\varphi(x), \end{aligned} $$

(35)

where $\otimes _x$ is defined by $\int _{\mathbb {R}^2} \phi (x,v) \ d(\mu \otimes _x J_\varphi (x)) = \int _{\mathbb {R}} \int _{\mathbb {R}} \phi (x,v)\ dJ_\varphi (x)\ d\mu (x)$. Formally, the mass of $\mu$ at x moves with speed $\varphi (F_\mu (x))$. For example, choosing $\varphi (\alpha )=\alpha -\frac {1}{2}$, the solution starting from a Dirac delta centered at 0 is given by $\mu (t)=\frac {1}{t}\chi _{[-\frac {t}{2},\frac {t}{2}]} d\lambda$. We can use $V_\varphi$ for a SIR-type model as follows. First set $F_I(x)=\frac {I(]-\infty ,x])}{I(\mathbb {R})}$, which is the normalized cumulative distribution of infected up to a given x value, so that $F_I(-\infty )=0$ and $F_I(+\infty )=1$. The dynamics for I is thus given by the MDE:

$$\displaystyle \begin{aligned} \dot{I}=V_\varphi[I]+ \frac{S}{N} \beta(\alpha) I -\gamma(\alpha) I \end{aligned}$$

where $V_\varphi [\cdot ]$ is given by (35); thus the term $V_\varphi [I]$ moves the ordered masses of I with speed prescribed by $\varphi$ representing the appearance of new variants within the population of infected individuals.

Appendix 2: Proof of Theorem 5

We first state a Lemma, whose proof is entirely similar to Lemma 3 in [23], so we omit the proof.

Lemma 12 (Uniform Boundedness)

Let $\mu ^N_\ell$ and $x_\ell ^N$ be as in Definition 4. Under the assumptions of Theorem 5, for every $N, \ell \in \mathbb {N}^{+}$ the measure $\mu ^N_\ell$ has uniformly bounded mass and support, and $x^N_\ell$ is uniformly bounded in norm.

Similarly, the next lemma can be proved as Lemma 4 in [23], so we omit the proof.

Lemma 13

Under the assumption of Theorem 5, there exists $(x, \mu ) \colon [0, T] \to \mathbb {R}^{m} \times \mathcal {M}(\mathbb {R}^n)$ such that $\mu ^N \rightharpoonup \mu$ and $x^N \to x$ for $N \to \infty$, where $\mu ^N$ and $x^N$ are as in Definition 4. Moreover, the limit curve $(x, \mu )$ is a weak solution to system (1) in the sense of Definition 2.

Now we are ready to prove Theorem 5.

Proof

For the first statement of the theorem, notice that properties (4) and (5) for Euler-LAS approximations are consequences of Lemmas 12 and 13. Properties (6) and (7) for Euler-LAS approximations can be verified as in the proof of Theorem 1 of [23]. Moreover, using the same approach of using finite sum of Dirac masses, we can prove the existence of a semigroup verifying the same properties. Thus we pass to verify property (iv) of Definition 3.

Fix two controls $u(\cdot ), v(\cdot )\in {{\mathcal U}}$, and let $(\mu ^u_{N,\ell },x^u_{N,\ell })$, resp. $(\mu ^v_{N,\ell },x^v_{N,\ell })$, be the Euler-LAS approximate solutions associated to $u(\cdot )$, resp. $v(\cdot )$. Thanks to assumption (OM:ELAS), we have

$$\displaystyle \begin{aligned} \mathcal{W}^g(\mu^u_{N,\ell},\mu^v_{N,\ell})\leq K_2 \int_0^{\ell\Delta_N}|u(t)-v(t)|\,dt+O(\Delta_N). \end{aligned}$$

The case $u=v$ proves that there exists a unique limit of the measure part of the Euler-LAS approximation. Moreover, from (OM:Lip-g) we get:

$$\displaystyle \begin{aligned} \|g(x^u(t),\mu^u(t),u(t))-g(x^v(t),\mu^v(t),v(t))\| \leq \end{aligned}$$

$$\displaystyle \begin{aligned} L \left( |x^u(t)-x^v(t)|+\int_0^{\ell\Delta_N}|u(t)-v(t)|\,dt+ |u(t)-v(t)|) \right) \end{aligned}$$

where $(x^u,\mu ^u)$, resp. $(x^v,\mu ^v)$, denote a limit of the Euler-LAS approximations for control $u(\cdot )$, resp. $v(\cdot )$. Using the contraction map** principle (see Theorem 3.2.1 of [8]), we obtain that both components of Euler-LAS limit are unique and (8) hold true. □

Uniqueness results for the semigroup can be obtained via the concept of Dirac germ, the latter prescribing small-time evolution of finite sums of Dirac masses. Such results can be carried out for the coupled controlled ODE-MDE. Indeed the strong (OM:ELAS) assumption implies unique limits of Euler-LAS, which in turn define a Dirac germ. For the sake of space, we will not report the details on this uniqueness result for semigroup.