Dynamics of epidemic models with asymptomatic infection and seasonal succession

In this paper, we consider a compartmental SIRS epidemic model with asymptomatic infection and seasonal succession, which is a periodic discontinuous differential system. The basic reproduction number $\mathcal{R}_0$ is defined and valuated directly for this model, and the uniformly persistent of the disease and threshold dynamics are obtained. Specially, global dynamics of the model without seasonal force are studied. It is shown that the model has only a disease-free equilibrium which is globally stable if $\mathcal{R}_0\le 1$, and as $\mathcal{R}_0>1$ the disease-free equilibrium is unstable and the model has an endemic equilibrium, which is globally stable if the recovering rates of asymptomatic infective and symptomatic infective are close. These theoretical results provide an intuitive basis for understanding that the asymptomatic infective individuals and the disease seasonal transmission promote the evolution of epidemic, which allow us to predict the outcomes of control strategies during the course of the epidemic.


Introduction. Since Kermack and McKendrick
proposed the classical deterministic compartmental model (called SIR model) to describe epidemic outbreaks and spread, mathematical models have become important tools in analyzing the spread and control of infectious diseases, see [1,2,5,9,11,12,20,21,27] and references therein. The number of infected individuals used in these models is usually calculated via data in the hospitals. However, some studies on influenza show that some individuals of the population who are infected never develop symptoms, i.e. being asymptomatically infective. The asymptomatically infected individuals will not go to hospital but they can infect the susceptible by contact, then progress to the recovered stage, see for instance [3,14,22]. Hence, using the data from hospitals to mathematical models to assess the epidemic will underestimate infection risks.
On the other hand, seasonality is very common in ecological and human social systems (cf. [26]). For example, variation patterns in climate are repeated every year, birds migrate according to the variation of season, opening and closing of schools are almost periodic, and so on. These seasonal factors significantly influence the survival of pathogens in the environment, host behavior, and abundance of vectors and non-human hosts. A number of papers have suggested that seasonality plays an important role in epidemic outbreaks and the evolution of disease transmissions, see [4,6,8,9,16,17,19,21,28]. However, it is still challenging to understand the mechanisms of seasonality and their impacts on the dynamics of infectious diseases.
Motivated by the above studies on asymptomatic infectivity or seasonality, we develop a compartmental model with asymptomatic infectivity and seasonal factors in this paper. This model is a periodic discontinuous differential system. We try to establish the theoretical analysis on the periodic discontinuous differential systems and study the dynamics of the model. This will allow us to draw both qualitative and quantitative conclusions on the effect of asymptomatic infectivity and seasonality on the epidemic.
The rest of the paper is organized as follows. In section 2, we formulate the SIRS model with asymptomatic infective and seasonal factors, then discuss the existence and regularity of non-negative solutions for this model. In section 3, we define the basic reproduction number R 0 for the model, give the evaluation of R 0 and investigate the threshold dynamics of the model (or the uniformly persistent of the disease). It is shown that the length of the season, the transmission rate and the existence of asymptomatic infection affect the basic reproduction number R 0 . In section 4, we study the global dynamics of the model ignoring seasonal factor. We prove that there is a unique disease-free equilibrium and the disease always dies out when R 0 ≤ 1; while when R 0 > 1 there is an endemic equilibrium which is global stable if the recovering rates of asymptomatic infective and symptomatic infective are close. A brief discussion is given in the last section.
2. Model formulation. In this section, we first extend the classic SIRS model to a model which incorporates with the asymptomatic infective and seasonal features of epidemics, and then study the regularity of solutions of the model.
Because there are asymptomatically infectious and symptomatically infectious individuals in the evolution of epidemic, the whole population is divided into four compartments: susceptible, asymptomatically infectious, symptomatically infectious and recovered individuals. More precisely, we let S, I a , I s and R denote the numbers of individuals in the susceptible, asymptomatic, symptomatic and recovered compartments, respectively, and N be the total population size. Let R + = [0, +∞), Z + be the set of all nonnegative integers, and ω > 0 be given as the period of the disease transmissions. In addition to the assumptions of the classical SIRS model, we list the following assumptions on seasonal factors, asymptomatic infectivity and symptomatic infectivity.
(A1) Due to the opening and closing of schools or migration of birds, each period of the disease transmission is simply divided into two seasons with high and low transmission rates, which are called high season J 2 and low season J 1 , respectively. The seasonality is described by a piecewise constant function with high transmission rate β 2 in J 2 and low transmission rate β 1 in J 1 , respectively, where J 1 = [mω, mω + (1 − θ)ω) and J 2 = [mω + (1 − θ)ω, (m + 1)ω). Here m ∈ Z + , and 0 < θ < 1 which measures the fraction of the high season to the whole infection cycle.
(A2) There are two classes of infective individuals: asymptomatically infective ones and symptomatically infective ones. Both of them are able to infect susceptible individuals by contact. A fraction µ of infective individuals proceeds to the asymptomatically infective compartment and the remainder (i.e. a fraction 1 − µ of infective individuals) goes directly to the symptomatically infective compartment. And the asymptomatically infective and symptomatically infective individuals recover from disease at rate r a and r s , respectively. (A3) The symptomatically infective individuals will get treatment in hospital or be quarantined. Hence, the symptomatic infective individuals reduce their contact rate by a fraction α.
Based on these assumptions, the classical SIRS model can be extended to the following system where N (t) = S(t) + I a (t) + I s (t) + R(t), all parameters d, α, σ, µ, r a and r s are nonnegative, and Parameters β 2 and β 1 are the rates of contact transmission of the disease in high season and low season respectively for which β 2 ≥ β 1 ≥ 0. Besides, d is both the birth rate and death rate, α, 0 ≤ α ≤ 1, is the fraction of the symptomatically infective individuals reducing their contact rate with susceptible, the fraction of infective individuals becoming asymptomatically infective µ satisfies 0 ≤ µ ≤ 1, parameter σ is the rate of recovered population loss of the immunity and reentering the susceptible group, and r a and r s are the rates of asymptomatic infective and symptomatic infective recovering with immunity, respectively. Note that the transmission rate β(t) is a piecewise function or a step function. This idea has appeared in some literatures such as [19,24,15,10].
From the biological point of view, we focus on the solutions of system (2.1) with initial conditions Hence, N (t) = S 0 + I a0 + I s0 + R 0 , which is a constant for almost all t ∈ R + . Since the total population does not change by the assumption, we let S(t) + I a (t) + I s (t) + R(t) ≡ N for almost all t ∈ R + . Therefore, system (2.1) with the initial conditions ( Clearly, the right hand side of system (2.3) is not continuous on the domain R + ×D 0 . We claim that the solution of system (2.3) exists globally on the interval R + = [0, +∞) and is unique.
Theorem 2.1. For any P 0 ∈ D 0 , system (2.3) has a unique global solution ϕ(t, P 0 ) = (S(t, P 0 ), I a (t, P 0 ), I s (t, P 0 )) in R + , which is continuous with respect to t and all parameters of this system. Moreover, ϕ(t, P 0 ) ⊆ D 0 for any t ∈ R + and the solution ϕ(t, P 0 ) is differentiable with respect to P 0 , where some derivatives are one-sided if P 0 is on the domain boundary.
Proof. Assume that ϕ(t, P 0 ) is a solution of system (2.3). We first consider the two systems in the domain R + × R 3 + , i = 1, 2, respectively. It is clear that for each i the solution of system (2.5) exists and is unique on its maximal interval of existence, and the solution of system (2.5) is differentiable with respect to the initial value P * by the fundamental theory of ordinary differential equations.
Note that the bounded closed set D 0 in R 3 + is a positive compact invariant set of system (2.5) since the vector field of system (2.5) on the boundary ∂D 0 of D 0 is directed toward to the interior of D 0 or lies on ∂D 0 , where + , S + I s + I a = N }. Therefore, the solution of system (2.5) exists globally for any P * ∈ D 0 ⊂ R 3 + , and these solutions are in D 0 for all t > 0.
It follows that the solution ϕ(t, P 0 ) for t ≥ 0 of system (2.3) can be determined uniquely by induction. For simplicity, we let s m = (m − 1)ω and and ϕ(t, P 0 ) can be written as follows.
where u m and v m are determined by letting This implies that the solution ϕ(t, P 0 ) of system (2.3) exits globally in R + and is unique for any P 0 ∈ D 0 , and it is continuous with respect to t and all parameters. By the expression (2.7), it is easy to see that the solution ϕ(t, P 0 ) lies in D 0 for all t ≥ 0 and ϕ(t, P 0 ) is differentiable with respect to P 0 . The proof is completed. Theorem 2.1 tells us that system (2.3) is ω-periodic with respect to t in R + × D 0 , and it suffices to investigate the dynamics of its associated period map P on D 0 for the dynamics of system (2.3), where 3. Basic reproduction number and threshold dynamics. In epidemiology, the basic reproduction number (or basic reproduction ratio) R 0 is an important quantity, defined as the average number of secondary infections produced when an infected individual is introduced into a host population where everyone is susceptible. It is often considered as the threshold quantity that determines whether an infection can invade a new host population and persist. Detailedly speaking, if R 0 < 1, the disease dies out and the disease cannot invade the population; but if R 0 > 1, then the disease is established in the population. There have been some successful approaches for the calculations of basic reproduction number for different epidemic models. For example, Diekmann et al [7] and van den Driessche and Watmough [23] presented general approaches of calculating R 0 for autonomous continuous epidemic models. And for periodic continuous epidemic models, Wang and Zhao in [25] defined the basic reproduction number. To our knowledge, there is no theoretic approach to calculate the basic reproduction number for periodic discontinuous epidemic models such as system (2.3). In this section, we use the idea and some notations given in [25] to define and calculate the basic reproduction numbers for system (2.3), and discuss the uniform persistence of the disease and threshold dynamics.
We define X to be the set of all disease-free states of system (2.3), that is Clearly, the disease-free subspace X is positively invariant for system (2.3). It can be checked that the period map P(P 0 ) in X has a unique fixed point at (N, 0, 0), which is a unique disease-free equilibrium (N, 0, 0) of system (2.3), denoted by E 0 . We now consider a population near the disease-free equilibrium E 0 . For simplicity, we let x = (S, I a , I s ) T , and for i = 1, 2 set Then the linearized system of (2.3) at E 0 can be rewritten as System (3.1) is a piecewise continuous periodic linear system with period ω in t ∈ R + . In order to determine the fate of a small number of infective individuals introduced into a disease-free population, we first extend system (3.1) from t ∈ R + to t ∈ R, and introduce some new notations. When t ∈ ∪ +∞ m=−∞ (J 1 ∪ J 2 ) = (−∞, +∞), we set I(t) = (I a (t), I s (t)) T , and where Clearly, F(t) is a 2 × 2 piecewise continuous periodic matrix with period ω in R, and it is non-negative. And which is cooperative in the sense that the off-diagonal elements of −V are nonnegative.
Let Y (t, s), t ≥ s, be the evolution operator of the linear system where d, r a and r s are positive numbers. We denote · 1 the 1-norm of vectors and matrices. Thus, there exist K > 0 and κ > 0 such that From the boundedness of F(t), i.e. F(t) 1 < K 1 , it follows that there exists a constant K 1 > 0 such that We now consider the distribution of infected individuals in the periodic environment. Assume that I(s) is the initial distribution of infected individuals in infectious compartments. Then F(s)I(s) is the distribution of new infections produced by the infected individuals who were introduced at time s. Given t ≥ s, then Y (t, s)F(s)I(s) is the distribution of those infected individuals who were newly infected at time s and still remain in the infected compartments at time t. Thus, the integration of gives the distribution of cumulative new infections at time t produced by all those infected individuals introduced at times earlier than t.
Let C ω = C(R, R 2 ) be the ordered Banach space of ω-periodic continuous functions from R to R 2 , which is equipped with the norm · c , and the generating positive cone It can be checked that the linear operator L is well defined. Proof. Since Y (t, s) = e −V (t−s) and F(t) is a nonnegative bounded matrix, we get that L(C + ω ) ⊂ C + ω . This implies that the linear operator L is positive. We now prove the continuity of L. For each t ∈ R, we have which implies that L is continuous and uniformly bounded since ∞ j=0 e −κωj is convergent.
In the following we prove the compactness of L. We first claim that LI(t) is equicontinuous. Consider I(t) ∈ C ω and ∀t 1 , t 2 ∈ [0, ω] with t 1 < t 2 . Then This implies that {(LI)(t)} are equicontinuous. According to Ascoli-Arzela theorem, we know that L is compact. The proof of this lemma is completed.
L is called the next infection operator, and the spectral radius of L can be defined as the basic reproduction number (or ratio) Following [25], we consider how to calculate R 0 and whether the basic reproduction ratio (or number) R 0 characterizes the threshold of disease invasion, i.e., the disease-free periodic solution (N, 0, 0) of system (2.3) is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1.
It is clear that the disease-free periodic solution (N, 0, 0) of system (2.3) is locally asymptotically stable if all characteristic multipliers of periodic system (3.1) are less than one, and it is unstable if at least one of the characteristic multipliers of periodic system (3.1) is greater than one. By straightforward calculation, we obtain that the characteristic multipliers of periodic system (3.1) consist of e −(d+σ)ω and the eigenvalues of the following matrix Note that e −(d+σ)ω < 1 because d + σ > 0. Therefore, all characteristic multipliers of periodic system (3.1) are less than one if and only if the largest eigenvalue of , is less than one (i.e. ρ(Φ F −V (ω)) < 1), and at least one of the characteristic multipliers of periodic system (3.1) is greater than one if and only if ρ( On the other hand, it is easy to check that all assumptions (A2)-(A7) in [25] are valid for system (3.1) except the assumption (A1). Using the notations in [25], we define a matrix V ε = V − εP , here P = 1 1 1 1 and ε is a very small positive number. Thus, −V ε is cooperative and irreducible for each t ∈ R. Let Y ε (t, s) be the evolution operator of the linear system (3.3) with V replaced by V ε . For some small ε 0 , as ε ∈ [0, ε 0 ) we can define the linear operator L ε by replacing Y (t, s) in (3.5) with Y ε (t, s) such that the operator L ε is positive, continuous and compact on C ω . Let R ε 0 := ρ(L ε ) for ε ∈ [0, ε 0 ). By the proof of Theorem 2.1, we know that the solutions of the following system are continuous with respect to all parameters. Thus, where Φ F −Vε (ω) is the monodromy matrix of system (3.7), and Φ F −V (ω) is the monodromy matrix of system (3.7) as ε = 0. According to the continuity of the spectrum of matrices, we have From Lemma 3.1, we use the similar arguments in [25] to the two linear operator L ε and L, and obtain that lim ε→0 R ε 0 = R 0 . We now easily follow the arguments in [25] to characterize R 0 . Let W λ (t, s), t ≥ s be the fundamental solution matrix of the following linear periodic system where the parameter λ ∈ (0, +∞). Consider an equation of λ ρ(W λ (ω, 0)) = 1. (3.8) Then R 0 can be calculated as follows. Note that ρ(W 1 (ω, 0)) = ρ(Φ F −V (ω)). Using similar arguments in [25], we can prove that the basic reproduction ratio (or number) R 0 can characterize the threshold of disease invasion.
To save space, the proofs of the above theorems are omitted. From Theorem 3.3, we can see that R 0 is a threshold parameter for local stability of the disease-free periodic solution (N, 0, 0). We next show that R 0 is also a threshold parameter for dynamics of system (2.3) in D 0 .   Thus, the auxiliary system of (3.9) is İ a (t) = µβ(t)N (I a + αI s ) − (d + r a )I a , which is a periodic linear discontinuous system with period ω. The periodic map associated with system (3.10) is defined by Φ F −V (ω), which is a linear continuous map. When R 0 < 1, we have ρ(Φ F −V (ω)) < 1, which implies that (0, 0) is a globally asymptotically stable solution of system (3.10).
Note that systems (3.9) and (3.10) are cooperative. Using the similar arguments in [18], we can prove that the comparison principle holds. Hence, lim t→+∞ (I a (t), I s (t)) = (0, 0). So, for arbitrarily small constant ε > 0, there exists T > 0 such that I a (t)+αI s (t) < ε as t > T . From the first equation of system (2.3), we havė On the other hand, S(t) ≤ N in D 0 , which admits lim t→+∞ S(t) = N.
In the following, we show that the disease is uniformly persistent when R 0 > 1.  in the domain R 3 + . By the formula (3.6), we let β 1 = β 2 and obtain the basic reproduction number R 0 of system (4.1) as follows.
which is consistent with the number calculated using the approach of basic reproduction number in [7] and [23]. From the expression (4.2), we can see that there is still the risks of infectious disease outbreaks due to the existence of asymptomatic infection even if all symptomatically infective individuals have been quarantined, that is, α = 0. This provides an intuitive basis for understanding that the asymptomatically infective individuals promote the evolution of epidemic.
In the following we study the dynamics of system (4.1). By a straightforward calculation, we obtain the existence of equilibria for system (4.1).
Lemma 4.2. The disease-free equilibrium E 0 (N, 0, 0) of system (4.1) in R 3 + is asymptotically stable if R 0 < 1; E 0 (N, 0, 0) is a saddle-node with one dimensional center manifold and two dimensional stable manifold if R 0 = 1; and E 0 (N, 0, 0) is a saddle with two dimensional stable manifold and one dimensional unstable manifold if R 0 > 1.
Proof. A routine computation shows that the characteristic polynomial of system (4.1) at E 0 is where a 0 = (d + r a )(d + r s )(1 − R 0 ), It is clear that −(d + σ) < 0 is always one root of (4.3). We consider three cases, R 0 < 1, R 0 = 1 and R 0 > 1, to discuss the other roots of (4.3). If R 0 < 1, then a 1 < 0 and a 0 > 0 by βN µ d+ra < R 0 and βN α(1−µ) d+rs < R 0 . Thus, three roots of (4.3) have negative real parts, which leads to the local asymptotical stability of the disease-free equilibrium E 0 .
Summarized the above analysis, we complete the proof of this lemma.
From Lemma 4.1 and Lemma 4.2, we can see that system (4.1) undergoes saddlenode bifurcation in a small neighborhood of E 0 (N, 0, 0) as R 0 increases passing through R 0 = 1.
About the endemic equilibria, we have the following local stability.
From Lemma 4.2 and Lemma 4.3, we can see that R 0 is the threshold quantity of local dynamics of system (4.1) in R 3 + . By Theorem 2.1, we only need to consider system (4.1) for its global dynamics in D 0 . The following theorems will show that R 0 is also the threshold quantity of global dynamics of system (4.1) in D 0 . Theorem 4.4. If R 0 ≤ 1, then the disease-free equilibrium E 0 (N, 0, 0) of system (4.1) is globally asymptotically stable in D 0 .
The proof of this theorem can be finished by constructing a Liapunov function L(S, I a , I s ) = I a (t) + d + r a d + r s αI s (t) in D 0 . For the sake of simplicity, we omit the details.
Proof. Let I = I a + αI s , N 1 = S + I a + I s . Then under the assumption r a = r s = r, system (4.1) in D 0 can be written as Thus, equilibrium E 1 (S * , I * a , I * s ) of system (4.1) becomes equilibriumẼ 1 (S * , I * , N * 1 ) of system (4.8) andẼ 1 is locally asymptotically stable, where I * = I * a + αI * s , N * 1 = S * + I * a + I * s . Applying a typical approach of Liapunov functions, we define and construct a Liapunov function of system (4.8) where arbitrary constants ν 1 > 0, ν 2 = ν 1 βS * /μ and ν 3 = ν 1 σ/r. Note that g( Note that the only compact invariant subset of the set {(S, I, N 1 ) : dV1(S,I,N1) dt = 0} is the singletonẼ 1 (S * , I * , N * 1 ) inD 0 . Consequently, we can conclude that E 1 (S * , I * a , I * s ) is globally asymptotically stable and attracts all orbits of system (4.1) in D 0 except E 0 (N, 0, 0). From Theorem (4.6) and the continuity of solutions with respect to parameters r a and r s , we obtain the following results.
5. Discussion. In this paper, we established a compartmental SIRS epidemic model with asymptomatic infection and seasonal factors. In our model, we divided the period of the disease transmission into two seasons. In fact, it can be divided into n seasons for any given n ∈ Z + . Compared with continuous periodic systems, our piecewise continuous periodic model can provide a straightforward method to evaluate the basic reproduction number R 0 , that is to calculate the spectral radius of the matrix Φ F −V (ω) = e (F2−V )θω e (F1−V )(1−θ)ω . It is shown that the length of the season, the transmission rate and the existence of asymptomatic infective affect the basic reproduction number R 0 , and there is still the risks of infectious disease outbreaks due to the existence of asymptomatic infection even if all symptomatically infective individuals have been quarantined, that is, α = 0. This provides an intuitive basis for understanding that the asymptomatically infective individuals and the disease seasonal transmission promote the evolution of the epidemic. And theoretical dynamics of the model allow us to predict outcomes of control strategies during the course of the epidemic.