The basic reproduction number of discrete SIR and SEIS models with periodic parameters

Seasonal fluctuations have been observed in many infectious diseases. 
Discrete epidemic models with periodic epidemiological parameters are formulated and studied 
to take into account seasonal variations of infectious diseases. 
The definition and the formula of the basic reproduction number $R_0$ are given by following the framework in [1,2,3,4,5]. 
Threshold results for a general model are obtained which show that the magnitude of $R_0$ determines 
whether the disease will go extinct (when $R_0 1$) in the population. 
Applications of these general results to discrete periodic SIR and SEIS models are demonstrated. 
The disease persistence and the existence of the positive periodic solution are established. 
Numerical explorations of the model properties are also presented via several examples 
including the calculations of the basic reproduction number, conditions for the disease extinction or persistence, 
and the existence of periodic solutions as well as its stability.

1. Introduction. Periodic fluctuations are known to be a common phenomenon in outbreaks of many infectious diseases. For diseases such as malaria, diarrhea, and tuberculosis, seasonality plays an important role in the disease dynamics [6,7]. For childhood diseases, contact rates vary seasonally due to the opening and closing of schools [8,9,10]. Periodic birth rates in some populations are also evidenced [11,12,13]. It is natural to introduce periodic parameter functions in epidemiological models for disease dynamics that can be influenced by seasonally fluctuating factors.
Periodic epidemic models have been considered by many researchers [14,15,16,17,18,19]. There has been a growing interest in the study of discrete time epidemiological models. Allen et al [20,21,22] and Castillo-Chavez et al [23,24,25] studied deterministic and stochastic discrete SI, SIR, and SIS models. Other studies involving discrete epidemiological models include Zhou and Paolo [26](age-structured SIS model); Zhou and Ma [27](age-structured SIS models with immigration); Li and Wang [28](stage-structured model); and Franke and Yakubu [29](seasonal variation in environment). One of the reasons for this upsurge of interest in discrete time models is that these types of models have advantages in modeling infectious disease due to the fact that epidemic data are usually collected in discrete time units, which would make it more convenient to use discrete-time models [30].
One of the most important concepts in epidemiological modeling is the basic reproduction number, denoted by R 0 , which is defined as the expected number of secondary cases produced by a typical infective individual in a completely susceptible population during his/her entire period of infection [1]. The quantity of R 0 is often used to assess how likely an infectious disease can spread in a population [2]. Diekmann et al [1], van den Driessche and Watmough [2] have considered continuous models and presented general approaches for computing R 0 . Allen and van den Driessche [4] have investigated a large class of discrete-time epidemic models and illustrated the method for computing R 0 using the next generation matrix approach.
For a class of continuous epidemic or population models with periodic coefficients, the definition and threshold property of the basic reproduction number have been investigated [3,14,16,31]. The biological interpretation and the threshold property of R 0 can be found in these studies. As in the case of continuous-time models, it will be very helpful to define a parallel quantity, R 0 , for discrete epidemic models with periodic parameter functions and provide a method for computing R 0 . Bacaër [5] has defined the basic reproduction number for periodic matrix population models, and presented an explicit formula for R 0 . Particularly, some inequalities that connect the growth rate λ and the basic reproduction number R 0 is established in [5].
The central ideal in defining the basic reproduction number for an epidemic model is to find an expression for new infections generated by a typical infectious individual. The general framework and procedures for doing this include: (1) rearranging the order of model equations and separating new infection terms from stage transition terms; (2) linearizing the model at the disease-free state; (3) defining a linear operator and finding its spectral radius of the linear operator. It is often the case that the spectral radius of the linear operator is the basic reproduction number for the epidemic model. For autonomous continuous ordinary differential equation models or autonomous difference equation models, the linear operator is an n × n matrix, and it is relatively easier to derive an explicit formula for the basic reproduction number R 0 , or at least, to give an explicit expression for the linear operator or matrix. However, for continuous epidemic models with periodic parameters, the operator is usually a linear integral operator defined on a space of continuous and periodic functions, in which case, it is not easy to obtain an explicit formula for the basic reproduction number R 0 . The common approach for these periodic models is to calculate R 0 numerically.
In this paper, we define the basic reproduction number R 0 for the discrete periodic epidemic models in a general setting. We consider a linear operator matrix L (the spectral radius of L is R 0 ), present the explicit formula for R 0 , and establish the threshold result. In our applications, we also consider the persistence of the disease when R 0 > 1 and the existence of positive periodic solution. Particularly, it is shown that R 0 > 1 is the condition for persistence as well as the existence of the periodic solution.
The paper is organized as follows. Some properties of a linear periodic difference system are discussed at the end of the introduction section. In section 2, we present a general discrete periodic model and outline the process in which the basic reproduction number R 0 is defined. An explicit formula of R 0 is presented and the stability of the disease free periodic state is studied. In section 3, the periodic discrete SIR and SEIS models are discussed to demonstrate the application of our general results. The sufficient conditions for the persistence and the existence of the periodic solution of those periodic discrete SIR and SEIS models are given. The global stability of the periodic solution for each of the models is investigated numerically. Discussions and concluding remarks are included in section 4.
In the remainder of this section, we present some results for linear periodic difference systems. Consider the following ω-periodic difference system (ω is a natural number): where u(t) = (u 1 (t), u 2 (t), · · · , u n (t)) τ (n ∈ Z + ), and A(t) is an n × n matrix whose entries are function of t satisfying A(t) = A(t + ω). Using the same notation as in [32], we call U (t, s) (t ≥ s, s = 0, 1, · · · ) the evolution operator of system (1), that is, U (t, s) = A(t − 1)A(t − 2) · · · A(s), and U (s, s) = I, where I is the identity matrix. We have two lemmas on the spectral radius of U (t, s).
From those two lemmas, we know that the spectral radius of U (ω, 0) determines the stability of the solution of system (1). The solution of the system is globally asymptotically stable if ρ(U (ω, 0) < 1, and unstable if ρ(U (ω, 0) > 1.
2. The basic reproduction number and the stability of the disease free periodic state. Consider an infectious disease with seasonal fluctuation spreading in a population. We group individuals into n epidemiological compartments. Let x = (x 1 , x 2 , · · · , x n ), with x i ≥ 0 (i = 1, 2, · · · , n) denoting the number of individuals in the ith compartment. According to their epidemiological status we sort these compartments so that the first m (m < n) compartments correspond to infected individuals, and the remaining n − m compartments correspond to uninfected individuals. Let F i (t, x) be the number of newly infected individuals of the ith compartment at time t, V + i (t, x) be the number of individuals transferring into compartment i by all other means (e.g., disease progression, immigration, etc.) at time t, and V − i (t, x) be the number of individuals transferring out of compartment i (e.g., death, recovery, etc.) at time t. The disease transmission model is represented by the following non-autonomous difference system: x(t)), i = 1, 2, · · · , n, t = 0, 1, · · · , (2)

HUI CAO AND YICANG ZHOU
. Assume there exists a positive invariant set X ⊆ R + n := {(x 1 , x 2 , · · · , x n ) ∈ R n | x i ≥ 0, i = 1, 2, · · · , m}, such that f (t, x(0)) ≥ 0 and f (t, x(0)) ∈ X for any x(0) ∈ X , with f (t, x(t)) = (f 1 (t, x(t)), f 2 (t, x(t)), · · · , f n (t, x(t))) τ . This assumption is reasonable since the total population is usually bounded due to the resource limitation, and we study the solution dynamics of model (2) with the initial values located in the state space X . Further, we assume that model (2) has a unique disease free periodic state x 0 (t) ∈ X , with x 0 i (t) ≥ 0, m + 1 ≤ i ≤ n for all t = 0, 1, · · · , and there exists at least a j, m + 1 ≤ j ≤ n, such that x 0 j (t) > 0. The disease free periodic state x 0 (t) may be an equilibrium or a period solution, depending on the parameters in the model.
Following the framework and notations used in [2,3,4], we make following assumptions: x) are all nonnegative and continuous on R n + and continuously differentiable with respect to x; (A2) There is a natural number ω > 0 such that for each 1 ≤ i ≤ n, the functions x 0 (t)) = 0 for i = 1, 2, · · · , m. These assumptions are based on epidemiological interpretations as well as mathematical requirements. (A1) is the natural assumption for the uniqueness and existence of solutions. (A2) describes a periodic environment (due to seasonal variations in climate and social activities). (A3) comes from the fact that there is no transfer of individuals out of the compartment if a compartment is empty. (A4) represents that the incidence of infection for the uninfected compartment is zero. (A5) implies that the population will remain free of disease if it is free of disease at the beginning.
The Jacobian matrix D x f (t, x 0 (t)) of f (t, x) at the disease free periodic state . D x F(t, x 0 (t)) and D x V(t, x 0 (t)) can be partitioned as where F (t) and V (t) are the m × m matrices, C(t) is an (n − m) × (n − m) matrix, and T (t) is a (n − m) × m matrix defined by Furthermore, F (t) and V (t) are non-negative matrices. Following the setting of [4] for difference systems, we call F (t) and V (t) the fertility matrix and the transition matrix, respectively.
Let U C (t, s) and U V (t, s) (t ≥ s) be the evolution operators of the linear ωperiodic systems respectively. The internal evolution of individuals in the infectious compartments is dissipative due to deaths and movements. The loss of infective members from natural mortalities and disease-induced mortalities may lead exponential decays [3]. So we can assume It follows from Lemma 1.2 that the zero solutions of y(t+1) = C(t)y(t) and u(t+1) = V (t)u(t) are asymptotically stable if assumption (A6) holds. Under those assumptions, we can define the basic reproduction number for the discrete periodic epidemic model. Let the population be near the disease-free periodic state x 0 (t) of (2), and let φ(s), ω-periodic in s, be the initial distribution of infectious individuals. Then F (s)φ(s) is the distribution of new infections produced by the infected individuals who were introduced at time s. Given t ≥ s, then U V (t, s + 1)F (s)φ(s) gives the distribution of these infected individuals who were newly infected at time s and remain in the infected compartments at time t. It follows that is the distribution of accumulative new infectious at time t produced by all those infected individuals φ(s) introduced at previous time to t.
Let C ω be the order Banach space of all ω-periodic discrete functions from N to R m , equipped with the maximum norm · . Let C + ω = {φ ∈ C ω | φ(t) ≥ 0, ∀t = 0, 1, · · · } be the positive cone in C ω . Then we can define a linear operator L : Obviously, the operator L is continuous and compact on C ω , and L is positive in the sense that L(C + ω ) ⊂ C + ω . We call L the next infection operator. The basic reproduction number of model (2) is defined to be the spectral radius of L, The similar linear operator (Lφ where is a nonnegative positive number, and E is the m × m matrix with each element being 1. Then V (t) is non-negative and irreducible for each t ∈ N . Let U V (t, s) be the evolution operator of the linear system u(t + 1) = V (t)u(t). The linear operator L can be defined by replacing U V (t, s + 1) in (4) with U V (t, s + 1). The corresponding basic reproduction number is R 0 = ρ(L ). Following the idea in [3], we can have the lemma.
Since ρ(L) = ρ(L), we use the expression in [5] directly to obtain R 0 = ρ(M ), where M is the product of matrices M F and M −1 V defined by , and The equivalent definition of R 0 = ρ(M ) and the expression of matrix M given in [5] has the advantage that reduces the computation of the spectral radius of the linear operator L (orL) to the computation of an eigenvalue of the matrix M . The can also be written as a product of the two matrices, e.g., in the case ω = 2, For the constant environment, [4] shows that In order to explain our result in the constant environment is consistent with [4], we take and V is an non-negative matrix, Perron-Frobenius theorem [34] implies that R 0 is the eigenvalue with maximum modular. That is, R 0 is the spectral radius of , which is consistent with [4].
It is natural to expect that the basic reproduction number R 0 can characterize the dynamics of epidemiological models. The similar threshold result (see [2,3,4]) is given in our next theorem.
The last equivalent conclusion can be obtained directly from those first two. Finally, from the fact ρ(C(t)) < 1 and the linearized matrix at x 0 (t) we obtain that x 0 (t) is asymptotically stable (unstable) if ρ(U F +V (ω, 0)) < 1 (ρ(U F +V (ω, 0)) > 1). The condition ρ(U F +V (ω, 0)) < 1 (ρ(U F +V (ω, 0)) > 1) is equivalent to R 0 < 1 (R 0 > 1). Theorem 2.3 shows that the magnitude of R 0 determines the stability of the disease-free periodic state x 0 (t). Furthermore, R 0 = 1 may be the threshold value for the global stability of the disease-free periodic state, the persistence of the disease, and the existence of the periodic solution of model (2). This conjecture is verified for the discrete periodic SIR and SEIS models in next section. 3. Application examples. In this section, the discrete periodic SIR and SEIS models are studied to illustrate the application of our results. Similar to the discrete epidemic model without the effect of seasonal fluctuation [21,23,24,25], we assume that susceptible individuals become infected with nonlinear probability The main focus in this section is on the global stability of the disease-free periodic state, the persistence, and the existence of the positive periodic solution of those models. The implicit expression of the basic reproduction number R 0 is given in the simple case ω = 2 for the SIR model. The calculation of R 0 for large ω can be done numerically.
3.1. SIR model. Let S(t), I(t), and R(t) denote the numbers of individuals in the susceptible, infectious and recovery compartments at time t, respectively. The discrete periodic SIR model is where Λ is the constant recruitment of the population, p is the probability of survival after a time unit, γ is the probability that an infectious individual gets recovered. N (t) is the total number of population at time t, that is, N (t) = S(t) + I(t) + R(t), and N (t) satisfies The fact 0 < p < 1 and the last equation in (11) implies that N * = Λ 1−p is the unique equilibrium of (11), and N * is globally asymptotically stable, i.e., for any solution N (t) of (11) with positive initial value, lim t→∞ N (t) = N * holds.
We study model (10) in the following compact, positively invariant set Let x = (I, S, R) τ , and The SIR model (10) has a unique disease free state P 0 1 = 0, The threshold result of model (10) is given in the following theorem.
Theorem 3.1. The disease free equilibrium state of model (10) is globally asymptotically stable if R 0 < 1. P 0 1 is unstable and the disease persists if R 0 > 1. Furthermore, there exists at least one positive periodic solution of model (10) if R 0 > 1.
The limiting system theory implies that model (10) is persistent when R 0 > 1 and has a positive periodic solution.
We use following equivalent system to investigate the stability of the periodic solution of model (10) with ω = 2.
As an application example we take Λ = 1000, p = 0.994, γ = 1/8, and β 1 = 0. The positive periodic state, therefore, is asymptotically stable. Next, we illustrate our theoretical results on disease extinction or persistence by numerical simulation. The same parameter values are taken in the simulation, except for ω = 12 and β(t) = β c (0.01 + 0.0005 cos πt 6 ), with β c = 5 or β c = 60. In the case where β c = 5, we have R 0 = 0.38154 < 1, and the disease-free state, P 0 1 (0, 166666.67, 0) is globally asymptotically stable. The solution curves of the infectious individuals I(t) with the initial values (60, 200000, 120000) and (100, 150000, 120000) are given in the top plot of Fig 1. We observe that solutions with positive initial values will converge to the disease-free state quickly.
In the case where β c = 60, we obtain that R 0 = 4.57851 > 1, and the diseasefree equilibrium state P 0 1 (0, 166666.67, 0) is unstable. Theorem 3.1 implies that the disease will persist in the population, and model (10)  The childhood disease which transmits from one child to the other is quasiinstantaneous, and it is governed by periods of school terms and holidays. Therefore, we can use epidemic model with periodic parameters to describe the transmission of childhood disease, such as measles, mumps, chickenpox, rubella, or whooping cough. Measles is an infectious disease suitable for SIR model with periodic parameters. The periodic SIR models has been formulated to describe the transmission of measles [6]. So we can use model (10) to describe the transmission of measles.  (10) 3.2. SEIS model. Let us consider the transmission of infectious diseases with an exposed/latent stage. We assume that individuals can't obtain the lifelong immunity when they get recovered. The population is divided into the susceptibles, the exposed/latent, and the infectious. Let S(t), E(t), and I(t) denote the numbers of individuals in the susceptible, exposed/latent, and infectious compartments at time t, respectively. The discrete SEIS model in the constant environment was formulated and studied by [39]. We study the dynamical behavior of the discrete SEIS model in the seasonal environment by introducing periodic parameters. The model is where Λ, p, γ and β(t) have the same interpretation as that in SIR model (10). α(t) is the progression rate of the exposed/latent individuals becoming infectious, and α(t) > 0 is a continuous and periodic function with period ω. From epidemiological interpretation and mathematical requirement, we assume that (E(0), I(0), S(0)) ∈ Ω 2 = (E, I, S) ∈ R 3 The equation for the total population, N (t) = S(t) + E(t) + I(t), implies that . The domain Ω 2 is a compact, positively invariant set of (20). Let x = (E, I, S) τ , and

THE BASIC REPRODUCTION NUMBER OF DISCRETE MODELS 51
The SEIS model (20) has a unique disease free equilibrium state P 0 2 = 0, 0, The linearization at the disease free equilibrium state yields F (t) = 0 pβ(t) 0 0 , , T (t) = (0, −pβ(t) + pγ) τ and C(t) = p. It is easy to verify that (20) satisfies (A1)-(A6). The basic reproduction number, R 0 , of (20) is given by the spectral radius of the F (1), ..., F (ω − 1)), and with I is an 2 × 2 identity matrix. When ω = 2, the straight forward calculation gives the explicit formula where The stability of disease free equilibrium state, the persistence of disease, and the existence of the positive periodic solution of model (20) is given in Theorem 3.2.
Theorem 3.2. The disease free equilibrium state P 0 2 of (20) is globally asymptotically stable if R 0 < 1. The disease-free periodic state P 0 2 is unstable and the disease persists if R 0 > 1. Furthermore, there exists a periodic solution if R 0 > 1.
In the case where R 0 > 1, we assume that the total population is N * , and consider the limiting system of model (20): The limiting system (22) possesses the same dynamical property as that of the original system (20) [36,37].
In the case of β c = 20, we obtain that R 0 = 0.55031 < 1, and the disease free equilibrium state, P 0 2 (0, 0, 166666.67) is globally asymptotically stable. Two solutions of (20) with initial values (50000, 80000, 100000) and (19600,26375,120720) are shown in Fig 2. The curves in the top plot represent the numbers of exposed individuals E(t), and the curves in the bottom plot show the numbers of infectious individuals I(t). We observe that solutions with positive initial values will converge to the disease free equilibrium state quickly.   (20) In the case of β c = 50, we obtain that R 0 = 1.37578 > 1, and the disease free equilibrium state P 0 2 (0, 0, 166666.67) is unstable. Theorem 3.2 implies that the disease will persist in the population, and model (20) has a positive periodic solution. The numerical simulations demonstrate that the positive periodic solution may be globally asymptotically stable (see Fig 3). The positive periodic solution and other two solutions with the initial values (100, 10000, 80000) and (50000, 80000, 100000) are shown in  Tuberculosis (TB) is an infectious disease spreading worldwide. TB has a very long latent period, which may last for several months, even decades. The SEIS or SEIR models are often used to describe the transmission of TB. In addition, according to the monthly reporting data about new case of TB from the Chinese Center for Disease Control and Prevention (China CDC), we found an obvious seasonal variation of TB incidence in China. The periodic SEIS and SEIT models have formulated to study the TB infection in China [7]. Therefore, model (20) can be used to describe the transmission of TB with seasonal fluctuation.
4. Concluding remarks. The discrete epidemic models with periodic parameters or parameter functions are more realistic in describing disease dynamics under the influence of seasonal fluctuation. We define the basic reproduction number for a large class of discrete periodic epidemic model. Although R 0 is defined by the radius of a linear operator, it is also the eigenvalue of a specific matrix. It is easy to calculate R 0 when the values of the model's parameters or parameter functions are provided. The possibility of deriving an explicit formula of R 0 is one of the advantages over continuous periodic models. Under natural and general assumptions, the threshold theorems are established: the disease free state is asymptotically stable if R 0 < 1, and it is unstable if R 0 > 1.
After the general frame work and theory are established, discrete periodic SIR and SEIS models are discussed as application examples to demonstrate our method and results. The specific expression of R 0 for SIR and SEIS models with period ω = 2 are presented. The global stability conditions of the disease free state, the persistence, and the existence of the positive periodic solution are established for those two models. Numerical simulations are done to demonstrate the calculation of R 0 , the disease extinction, or the persistence, the existence and the stability of the periodic solutions. The numerical simulation provide us with possibility of investigating the dynamical behavior of any model after parameter values are determined.
The limitation of the paper is that we can not give the specific expression R 0 as a function of the model parameters though the explicit formula for the matrix M F and M V is given. The inverse M −1 V , the product M F M −1 V , and the spectral radius M F M −1 V (i.e., the eigenvalue of M F M −1 V ) will be very complicated for the model with more infectious compartments and long period ω. The expression of the positive periodic solution is also very difficult, though the expression of a positive periodic solution can be obtained by the iteration, theoretically. The iteration might be too complicated to manage even with computer software.