Given a vector x, the transpose is indicated by x ,x is the average of its entries, and diag(·) is a diagonal matrix with the argument on the diagonal. We use 0 and 1 to denote a vector or matrix of zeros and ones, respectively, of the appropriate dimensions. We define a directed graph G = (V, E, w), where V is the set of nodes, E ⊆ V × V is the set of edges, and w : E → R + is a function mapping directed edges to their weightings, with R + being the set of positive real values. Given G, we denote an edge from node i ∈ V to node j ∈ V by (i, j). We say node i ∈ V is a neighbor of node j ∈ V if and only if (i, j) ∈ E, and denote the neighbors of node j as N j . We denote the weighted adjacency matrix associated with G as A with the nonzero entry a ji indicating the strength of edge (i, j) as given by w.
Here we introduce the discrete-time networked SIR and SEIR models. In each case we assume that the virus spreads over G = (V, E, w) with adjacency matrix A. Each node in V can be interpreted as a single individual or a subpopulation and the states are interpreted as probabilities or proportions, respectively, referred to as levels from here on out.
For an SIR process spreading over G, node i's levels of susceptibility s k i , infection p k i , and recovery r k i evolve as
where k is the time step, h is the sampling parameter, and β i and γ i are node i's infection and recovery parameters, respectively. The SIR discrete-time model can be expressed in matrix form as follows
For this model to be well-defined we need the following assumptions. 
For the discrete-time SEIR model spreading over G with s k i = 1 − e k i − p k i − r k i , the exposed level of e k i and other states evolve as
where
β E i is infection parameter associated with the e k i state, σ i captures the rate at which the exposed become confirmed infected cases.
For transportation model we defině
where L is the set of transportation networks,ǎ l ij is the transportation network, andβ E,l i andβ l i are the corresponding infection rates, for the lth transportation network.
The SEIR discrete-time model can also be expressed in matrix form as follows
where S k = diag(s k i ), B E = diag(β E i ), and σ = diag(σ i ). For the discrete-time SEIR model to be well-defined we need the following assumptions. 
Proof. We prove this result by induction. By assumption, it holds for the base-case k = 0. We follow the proof by showing the induction-step, that is, assume s k i , e k i , p k i , r k i ∈ [0, 1] and s k i + e k i + p k i + r k i = 1, for all i ∈ [n], and we now show that this holds also for time-step k + 1. By Assumption 2 and (3a), s k+1 ≥ 
In this section we discuss the limiting behavior of the networked models from Section II.
For the matrix I + h diag(s k )BA − hγ, define its dominant eigenvalue as λ max .
) there existsk such that λ max < 1 for all k ≥k, and 5) there existsk, such that p k i converges linearly to 0 for all k ≥k.
Let λ M k max be the dominant eigenvalue of M k , where M k is defined as
Observe that z k+1 :=
max is monotonically decreasing as a function of k, 4) there exist ak such that λ M k max < 1 for all k ≥k, 5) there existsk, such that p k i converges linearly to 0 for all k ≥k and i ∈ [n].
Proof. We present the proof for each part of the theorem, starting with 1).
and k ≥ 0. Therefore, from (3a), we have s k+1 i ≤ s k i .
2) Since the rate of change of s k , −hdiag(s k ) B E Ae k + BAp k , is non-positive for all k ≥ 0 and s k is lower bounded by zero, by Lemma 2, we conclude that lim k→∞ s k exists. Therefore,
Similarly, we show that lim k→∞ p k
3) By assumption s 0 i > 0 for all i ∈ [n], and from the proof of Lemma 2 we can see that s k i > 0 for all i ∈ [n], k ≥ 0. Therefore, since we have that BA is irreducible, from (7), the matrix M k is irreducible, and non-negative by Assumption 2, for all finite k. Thus by the Perron-Frobenius Theorem for irreducible non-negative matrices we
increases when any entry increases [9, Theorem 2.7] and by 1) of this theorem, we have that
There are two possible equilibria: i) lim k→∞ s k = 0, and ii) lim k→∞ s k = s * = 0. We explore the two cases separately.
i) If lim k→∞ s k = 0,
Therefore, by Assumption 2, there exists ak such that λ M k max < 1 for all k ≥k. ii) If lim k→∞ s k = s * = 0, then, by 2), for any s 0 , e 0 , p 0 , r 0 the system converges to some equilibrium of the form (s * , 0, 0, 1 − s * ). Define k s := s k − s * and k p := z k − 0 2n .
By 1) and Lemma 2, respectively we know that k s ≥ 0 n and k p ≥ 0 2n for all k ≥ 0. Furthermore, we know that 
Let λ M * max be the maximum eigenvalue of
with corresponding normalized eigenvector w * , that is,
If λ M * max > 1, then the system in (10) is unstable. Therefore, by Lyapunov's Indirect Method, lim k→∞ k s , k p = (s * , 0 2n ), which is a contradiction. Now consider the case where λ M * max = 1. Definẽ
Then we can write M k = M * +M k , observe that all entries inM are non-negative. Using (9) and left multiplying the equation of k+1 p in (10b) by w * we get
Thus,
where the last inequality holds since all elements are non-negative. This contradicts that lim k→∞ z k = 0 2n , that is 2). Therefore, there exists ak such that λ M k max < 1 for all k ≥k. 5) Since, by 4), there exists ak such that λ M k max < 1 for all k ≥k, and we know that λ M k max = ρ(M k ) ≥ 0 by Assumption 2, we have
Therefore, for k ≥k, p k converges linearly to 0 n .
In this section we discuss estimating the parameters of the networked models from Section II using data. Note that these results are similar to the ones for the networked SIS model in [10] , [11] .
We first explore conditions for estimating the model parameters for the homogeneous model, that is, where every node has the same infection and healing parameters. In order to do so we define the following matrices:
Using the above matrices we construct Q =   Φ Γ   and, write (1b) and (1c) as
We find the least squares estimatesβ andγ by employing the pseudoinverse of Q. 
Proof. Using (16) and (17) we can write Q as follows
If the assumptions in (19) hold, a and b each have at least one element that is nonzero, thereforeQ has full column rank. Clearly D has full rank which implies that the rank for Q is equal to the rank ofQ i [12] . Therefore, there exists a unique solution to (18) using the inverse or pseudoinverse.
If one of the assumptions in (19) is not met, Q will have a nontrivial nullspace. Therefore, in that case, (18) does not have a unique solution.
Note that we do not need to know all s k , p k , r k but only the entries s k
To estimate the spreading parameters for the discrete-time SIR model from Section II-A in the heterogeneous case, we form the following matrices:
, and (20)
Using the above matrices we construct
For each i, we find the least squares estimatesβ i andγ i by using the pseudoinverse of Q i .
We now explore conditions for estimating the SEIR model parameters in the homogeneous case. In order to do so we define the following matrices:
and
Using the above matrices we construct
We find the least squares estimatesβ E ,β,σ, andγ using the pseudoinverse of Q E . 
Proof. Using (23)-(25), we can write Q E as follows
Since n > 1,Φ E = a E b E has at least two rows, and given that (27b) holds,Φ E has column rank equal to two. Moreover, if (27a) holds c and d each have at least one element that is nonzero. Thus,Q E has full column rank. Clearly D E has full rank which implies that the rank of Q E is equal to the rank ofQ E [12] . Therefore, there exists a unique solution to (26) using the pseudoinverse.
If one of the assumptions in (27a)-(27b) is not met, Q E will have a nontrivial nullspace. Therefore, in that case, 
and
Using the above matrices we construct
We find the least squares estimatesβ E i ,β i ,σ i , andγ i using the pseudoinverse of Q E i . 
where g k i (x) = s k i j∈Ni a ij x j which only uses the entries x j for which j ∈ N i . k (a) Network average exposed state. k (b) Network average infected state. k (c) Network average removed state. Fig. 1 : Simulation of a homogeneous SEIR system with its measured states, and the recovered states to show how well the recovered states captures the average state of the system.
Proof. Using (29)-(31), we can write Q E i as follows r k i = r k i +ε(r k i ) where ε(x i ) ∼ N (0, 0.008x i +0.00001). In order to emulate the difficulty of measuring the states at the beginning of an outbreak, we start measuring from k = 14, and recover the spread parameters by left multiplying (26) by the pseudo-inverse of Q E . The estimated statesê,p, andr are constructed using (3)-(4), the first set of measured statesẽ 14 ,p 14 , andr 14 , and the recovered spread parameters. In Figure 1 we show how well the average states are recovered compared to the average of the actual states, e, p, and r using the measured states to recover the spread parameters. The recovered spread parameters are βE ,β,σ,γ = (0.0398, 0.0602, 0.4000, 0.3000).
The error ofê,p, andr are 0.0190, 0.0186, and 0.0055, respectively.
In conclusion, the discrete time SIR and SEIR models are extended to capture virus spread on a network. The limiting behavior of each model is analyzed and sufficient conditions for estimating spread parameters from data are presented. The developed models are implemented in simulations. To extend this work and improve the performance of the model, we plan to incorporate asymptomatic transmission and apply the results to real data.