EFFECT OF MEDIA-INDUCED MODIFICATION OF TRAVEL RATES ON DISEASE TRANSMISSION IN A MULTIPLE PATCH SETTING∗

A general SIS epidemic model is formulated that incorporates media-induced modification of travel rates. Basic local properties of solutions to the model are established. In particular, it is shown that the basic reproduction number does not involve parameters related to the effect of media on travel. The general model is subsequently specialised to two-patch models, with two different scenarios regarding patch population size. Qualitative analyses show that the basic reproduction number acts as a sharp threshold between disease persistence and extinction. The concept of uniform weak persistence is used to prove the existence of an endemic equilibrium and disease uniform strong persistence under a certain condition. Numerical investigations are carried out to gain insight into the analytically tractable and intractable cases, highlighting the importance of considering not only the basic reproduction number but also other measures of disease severity.


Introduction
Communication in mass media has been sometimes employed as a tool in the effort to control and mitigate epidemics of emerging and re-emerging diseases [9,34,38]. Media inform the public of cases of infection during an epidemic and accordingly influence perceptions of the threat of infectious diseases [9,18,49]. Media coverage during the 2002-2003 severe acute respiratory syndrome (SARS) [11], the 2009 H1N1 influenza [27] and 2019 novel coronavirus disease (COVID-19) outbreak-s [10,12,13,19] delivered preventive health information, alerted the public to take precautionary measures (e.g., wearing masks, washing hands frequently, avoiding crowds) in relation to a disease outbreak. It has been shown to elicit individuals' positive behavior change [9,18,22,43]. This in turn reduces the frequency of potentially infecting contacts and helps lower the probability of disease transmission among the well-informed population.
The impact of media alerts on the propagation of communicable diseases has gained much attention and been studied extensively [9,18,22,34,38,43,49]. Recently, a number of mathematical models have been proposed to tackle this issue. However, existing modelling approaches focus on the effect of media-induced social distancing on disease transmission. For instance, [15,16,30,31,44,47] and references therein incorporated an effect of media coverage into epidemic models by adding a term that directly affects contact rates. Instead of employing different incidence function to describe media effect, [14,21,32,37,45,48] suggested adding a mass media compartment, reflecting the involvement of mass media in an epidemic. All mathematical models cited above assume that space is homogeneous and confine the population to a single location or population.
However, infectious diseases usually spread heterogeneously in space and time; integrating spatial heterogeneity into epidemic models provides a more accurate description of reality [2,4,7,8,28,35]. To explore the impact of media coverage in a heterogeneous environment, [24] formulated and studied a spatio-temporal SIS reaction-diffusion model, while [23,40] took individual travel into account, and proposed epidemiological models in a multiple patch setting. (A patch here refers to a geographical location; it can be a city, a region or a country.) Media coverage of an epidemic can affect individuals' travel intentions, resulting in a reduction of the volume of travel. The influence of media-induced modifications of travel rates between patches on the dynamics of disease propagation has not been the object of much attention. A disease outbreak generally does not result in complete interruption of travel between patches, but people tend to avoid areas where incidence of the disease is high; travellers are especially affected by diseaseinduced panic. One may naturally imagine that the reporting on a disease outbreak by media may amplify the degree of this sort of fear or panic and consequently lead to a drop of the number of travellers. During the 2002-2003 SARS outbreak, the World Health Organization (WHO) issued a recommendation advising visitors to China to consider postponing or cancelling their trips [11]; they issued a similar recommendation regarding travel to Toronto. Most overseas tourist groups to Beijing scheduled for April and May of 2003 were cancelled; in April, the city's foreign tourist arrivals dropped 59.9% compared with the same period in 2002 [11]. During the 2009 H1N1 pandemic, it was, for instance, observed that air traffic to Mexico was affected in the early stages of the epidemic [27].
On the modelling side, in mathematical epidemiology there are very few models with nonlinear movement rates, only ones with variable rates. Wang and Zhao [46] considered a two-patch epidemic model with the coefficients of movement rates being periodic. Dhirasakdanon et al. [17] formulated a general n-patch SEIRS host model, in which individuals may alter their travel rates in terms of the disease status of the host population, and established a sharp threshold to separate disease persistence from the extinction of small disease outbreaks.
In the same spirit, a metapopulation model for the spread of infection within and between patches is considered here to incorporate the adjustment of travel rates to media alerts and to study its resulting effect on the spatial and temporal spread of an infectious disease. A general SIS (susceptible-infective-susceptible) metapopulation model in a |P| patch setting is developed, integrating nonlinear media-induced modification of travel rates. Two scenarios in perspective of population size in a two patch setting are considered: in the first scenario, the population levels in both patches are high and they both possess proportional incidence of infection; in the second scenario, motivated by the work of Arino and McCluskey [5], Arino and Portet [6], Fromont et al. [20] and Zhao and Wang [46], the population in one patch is large and incidence is modelled using using proportional incidence, while in the other patch the population is small and mass action incidence is used. The following questions are addressed: Q1. Does media-induced modification of travel rates affect the epidemiology of a disease outbreak?
Q2. Is there any differentiation in effect of media-induced modification of travel rates for systems with different population sizes?
Q3. Does the population size of coupled patches play a role in facilitating the spread of an infectious disease?
Analytical analyses are conducted to derive a critical threshold of disease persistence and extinction, and to determine sufficient conditions under which an infectious disease is uniformly weakly and strongly persistent, respectively. Numerical simulations are subsequently performed to complement and further extend analytical results.

Assumptions
Consider the transmission of a disease conferring no immunity against reinfection that can be modelled using an SIS epidemic model (e.g., common cold or staphylococcus aureus). Set this in a metapopulation setting, i.e., assume that there is a set of locations called patches, with the dynamics in each patch described by an SIS model and movement of individuals between the patches; for more details on metapopulation models, see [2] and the references therein. The population in each patch is classified into two categories: individuals who are susceptible to the disease (S) and those who are infected with the disease and infectious to others (I). The simple SIS formalism allows to capture the main characteristics of disease propagation without the burden of additional equations and parameters. Further, this allows to focus on the effect of modification of travel rates between patches led by media coverage, since the dynamics of a classical SIS metapopulation model without media effects, or with effect of media-induced social distancing is well understood.
Let P be the set of patches. Individuals in patch p ∈ P move to patch q ∈ P at the per capita rate m qp ≥ 0, with m qp = 0 if movement from p to q is not possible. For simplicity of notation, we assume that for all p ∈ P, Before proceeding further, let us briefly discuss the model used in each patch in the absence of mobility. We make the following assumptions for the dynamics in patch p ∈ P.
H1. Birth occurs at the rate b p with all births in the susceptible compartment; death rate occurs at the per capita rate d p .
H2. Recovery occurs at the per capita rate γ p . There is no immunity, so upon recovery, individuals are immediately susceptible to the disease again.
H3. Disease transmission from infectious to susceptible individuals is described by a standard (proportional) incidence function β p S p I p /N p where β p is the effective contact rate in patch p.
The model in each patch, in the absence of movement between patches, then has the flow diagram showed in Figure 1. Media coverage may exert negative effects over the travel of individuals between patches. The following assumptions are made regarding these effects.

H4. Both susceptible and infectious individuals travel.
H5. Knowledge of the presence of disease leads to modification of the rate of travel between patches for all susceptible individuals. If m pq ≥ 0 is the rate of travel from patch q ∈ P to patch p ∈ P, then the revised travel rate for individuals takes the form where α p > 0 describes the insensitivity of response to knowledge of cases and σ p ∈ [0, 1] is the fraction of cases known in location p.

General model
For all p ∈ P, the general SIS metapopulation model with media-induced change of travel rates takes the form with non-negative initial conditions S p (0) > 0, I p (0) ≥ 0 for all p ∈ P and p∈P I p (0) > 0. 3. Mathematical analysis of the general model

Preliminaries
The vector field of (2.2) is locally Lipschitz so solutions with the prescribed initial conditions uniquely exist. Furthermore, (2.2) is well posed in the following sense. is invariant under the flow of (2.2), with S p remaining positive and I p staying non-negative for all p ∈ P. The total population converges to an equilibrium as t → ∞ and solutions are bounded.
Proof. Under the stated initial conditions S p (0) > 0, I p (0) ≥ 0 for all p ∈ P and p∈P I p (0) > 0, suppose, without loss of generality, that I 1 becomes zero at some time t 1 before another I p becomes zero, p ∈ P \ {1}. Then from (2.2b), is a non-decreasing function of t at t 1 . Hence, I 1 stays non-negative. Similar reasoning holds for all other I p , p ∈ P. Again, without loss of generality, now suppose that at some time t 2 , S 1 (t 2 ) = 0 before any other S p goes to zero, p ∈ P \ {1}. Then at t 2 , from (2.2b), Hence, S 1 stays positive for all t > 0 when the initial condition S p (0) > 0. By the similar argument, we obtain the positivity of all S p , p ∈ P.
Let N p (t) = S p (t) + I p (t) denote the total population of patch p ∈ P at time t and N (t) = p∈P N p (t) be the total population in the system. Adding up the equations in (2.2) is invariant under (2.2) and the total population is bounded, the individual components S p and I p are also bounded.

Moreover, the closed subset of
is also positively invariant under the flow of (2.2).

Disease-free equilibrium
System (2.2) always has a unique disease-free equilibrium (DFE). To compute it, set I p = 0 for all p ∈ P (which we write I = 0) and rewrite the remaining equations (2.2a) in vector form: Consider equilibria of (3.1). We have additionally, M is irreducible, in other words, if the digraph of patches is strongly connected, then it is not required for the death rates d p to all be positive in order for (d − M) −1 0, i.e., be entry-wise positive. Irreducibility of the movement digraph is a reasonable assumption to make, and we assume from now on it holds. We therefore have the DFE with S 0.

Basic reproduction number
The stability of the disease-free equilibrium E is considered in terms of the general basic reproduction number R (2.2) 0 , i.e., the basic reproduction number for the whole system. We follow the next generation matrix method for deterministic compartmental models [42]. Let be, respectively, the vectors of new infections and all other flows within the infected compartments (the latter with a negative sign). Taking partial derivatives of F and V with respect to I and evaluating at E where F is non-negative and V is an M -matrix. Then F V −1 is non-negative. From [42, Theorem 2], we have the following result.
Note that neither α q nor σ q appear in R . This is to be expected: computation of R (2.2) 0 takes place where I = 0; when disease is absent, there is no change in travel rates and system (2.2) is reduced to a regular SIS metapopulation model. This is similar to what was observed, for example, in [40].

Mathematical analysis of the two-patch model
We now consider the case of only two patches, i.e, where P = {1, 2}, which we assume from now on. This allows us to investigate global properties of the model more in detail. It also provides a much easier framework for performing numerical simulations. In this case, model (2.2) takes the form (4.1a) (4.1b) (4.1c) with non-negative initial conditions for p ∈ P. All results established in Section 3 hold for system (4.1), of course. In particular, we have that, at the disease-free equilibrium E Also, when computing the basic reproduction number, F = diag(β 1 , β 2 ) and that give the basic reproduction number Note that parameters σ P and α P do not appear in R , so the introduction of media effect on travel does not change the basic reproduction number. When media coverage does not affect travel rates, i.e., σ P = 0 and or α P → ∞, the disease-free equilibrium E Are conditions in Theorem 4.1 stronger than R  Proof. One may compute By the Gershgorin circle theorem, and where λ 1 , λ 2 are the eigenvalues of F V −1 . Since Similarly, one finds that |λ 2 | < 1. Therefore, R (4.1) 0 = ρ(F V −1 ) < 1. Next, we discuss for system (4.1) the questions of uniform strong persistence and existence of endemic equilibrium. A system being persistent means that no components within the system approach zero. For example, there is no extinction for any populations that make up a biological system. The definition of uniform strong persistence below comes from [1]. x 2 (t), . . . , x n (t)) T , is said to be uniformly strongly persistent if for any positive initial condition X 0 > 0, there exists δ > 0 such that lim inf t→∞ x i (t) > δ for i = 1, 2, . . . , n.
An infectious disease is endemic if system (4.1) is uniformly strongly persistent. In other words, the numbers of susceptible and infectious individuals remain above a certain positive level. The following lemma is summarized from [50].
Using Definition 4.1 and Lemma 4.1, we obtain the following result.
for all non-negative solutions of (4.1) with initial conditions (4.2).
Obviously, uniform strong persistence implies uniform weak persistence.

Mobility between patches with large and small populations
In this section, we consider the interconnection through travel of two patches with very different population sizes, large and small. When the population size is small, it is assumed that each infectious individual may potentially meet almost all susceptible individuals in the population, so that the incidence function for disease transmission is of mass action type. In the case of a large population, each infectious individual can only contact a proportion of the susceptible individuals, leading to the use of a proportional incidence function. Without loss of generality, we assume that patch 1 is populous and patch 2 is underpopulated. Then system (2.2) becomes the following system of ODE. (5.1a) (5.1d)

DFE and basic reproduction number
System (5.1) has the same disease-free equilibrium as system (4.1). However, the basic reproduction number R  can be obtained by the next generation matrix method [42] as well, and it presents as follows after a straightforward calculation: where S 2 is the number of susceptibles in patch 2 at disease-free equilibrium E (5.1) 0 , and v ij (i, j = 1, 2) are listed in Section 4.
The parameters α 1 , α 2 , σ 1 and σ 2 do not appear in R . This implies that media coverage cannot determine disease elimination and persistence although it modifies individuals' travel rates.
Lemma 5.1. For any non-negative solutions (S 1 (t), I 1 (t), S 2 (t), I 2 (t)) of system (5.1), there exists some constant δ such that lim sup t→∞ N (t) ≤ δ and The proof of Lemma 5.1 is straightforward and is omitted.

Uniform persistence and existence of endemic equilibrium
Theorem 5.1. The disease in system (5.1) is uniformly weakly persistent if R Proof. We introduce the notation S := (S 1 , S 2 ) T , I := (I 1 , I 2 ) T , S(t) := S 1 (t)+ S 2 (t) and I(t) := I 1 (t) + I 2 (t). The proof is completed in three steps. Suppose the statement is false.
Step 1. There exists some c > 0 such that lim sup t→∞ I(t) ≤ c.
To prove the statement above, the following differential inequality derived from (5.1a), (5.1c) and (5.2) is required.
Step 3. Find the contradiction.
Shifting forward in time and from discussions above, we may assume that S i (t) ≥ δ 1 (δ 1 ≤ δ 1 ), I i (t) ≤c(c ≥ c) and I i (t)/N i (t) ≤¯ (¯ < ), i = 1, 2 for all t ≥ 0, wherē δ,c,¯ > 0. Then S i (t)/N i (t) ≥ 1 −¯ . From (5.1b) and (5.1d), we deduce that (5.4a) (5.4b) By Step 2 and Lemma 5.1, and again shifting forward in time, we may assumē Since the functionsm 21 (I 2 ) andm 12 (I 1 ) are continuous, thenm 21 (I 2 ) →m 21 (0) = m 21 andm 12 (I 1 ) →m 12 (0) = m 12 as I(t) → 0. Therefore, for any η ∈ (0, 1), we may choose > 0 small enough such that for all t ≥ 0 (bear in mind that I(t) < δ =: c for all t ≥ 0), From (5.4) we have I ≥ A I where the linear operator A is associated with an irreducible quasi-positive matrix. Moreover, A → A as → 0, and A is an operator in the form where the positive operator A 1 is represented by a diagonal matrix with all diagonal entries being the capita infection rate β i > 0, and the operator A 2 is represented by a matrix with −m ij (0) of off-diagonal entries and γ i + d i +m ij (0) of diagonal entries for i, j = 1, 2 and i = j. Let s(A) denote the spectral bound of a linear bounded operator A, i.e., the largest real part of its eigenvalues. Since R We next study the uniform strong disease persistence (Definition 4.1) and the existence of an endemic equilibrium of system (5.1).
Theorem 5.2. System (5.1) with initial conditions (4.2) is uniformly strongly persistent, and there exists an endemic equilibrium E Proof. Let the state space X be as in Theorem 4.3. Define ρ : X → R + by ρ(S 1 , I 1 , S 2 , , I 2 ) = Ii Ni where N i = S i +I i , andρ : X → R + byρ(S 1 , I 1 , S 2 , , I 2 ) = I i , i = 1, 2. By Lemma 5.1 and Lemma 5.2, the compactness condition of [41,Theorem A.34] is satisfied. Notice that every total orbit ω : R → X of φ t , where φ t is the semiflow induced by the non-negative solutions of (5.1), is associated with a solution of (5.1) that is defined for all t ∈ R and takes value in X. By the fact that the matrix below is irreducible, it follows thatρ(ω(0)) > 0 whenever ρ(ω(t)) > 0 for all t ∈ R. The claim for lim inf t→∞ I i (t) ≥δ, a given constant, i = 1, 2 now follows from [41,Theorem A.34] which requires R . The greater the basic reproduction number is, the harder the epidemic is to control. This implies that the form of incidence function does play an important role in determining disease persistence and extinction.

Numerical results
To complement the mathematical analysis performed in previous sections, we now carry out numerical simulations of system (4.1). Choose parameters characteristic of influenza, as detailed in Table 1. The initial total population in each patch is assumed to be N 1 (0) = 80, 000 and N 2 (0) = 100, 000 with initial infectives I 1 (0) = 8000 and I 2 (0) = 6000. The effects of media coverage on the prevalence of an epidemic and on the time the epidemic takes to go extinct are illustrated in Fig. 3. The corresponding system, with media alert present in patch 1 and absent in patch 2, can be derived by setting σ 1 = 0 and σ 2 = 0 in system (4.1). It appears in Fig. 3(a) that the presence of media coverage in patch 2 leads to a respective reduction and increase in the time to extinction of disease in patch 1 and patch 2. And meanwhile, it is shown in Fig.  3(b) that it lowers the magnitude of an epidemic in patch 1 and enhances that in patch 2.   Table 1 such that R In Fig. 4, we show the trend of I 1 and I 2 at the endemic equilibrium as the fraction of known cases σ 1 and σ 2 , characterizing the strength of media coverage in patch 2 and patch 1 respectively, are varied in a simulation R     Table 1 such that R below unity. It is observed that travel-related media coverage in patch i may respectively exacerbate and reduce the disease burden of patch i and patch j (i, j = 1, 2 and i = j). Fig. 5 illustrates the endemic equilibrium values for I 1 and I 2 as the insensitivity of response to knowledge of cases covered by media, α 1 and α 2 , are varied. And it shows that the insensitivity of travel-related media coverage in patch i may impose positive and negative effects on disease propagation in patch i and patch j, respectively. The results here can be applied to models with border screening. Fig. 6 presents a sensitivity analysis of the value of R   Table 1, and each of b i , β i , m ij , d i , γ i , α i and σ i (i, j = 1, 2, i = j) is successively made to vary 10,000 times in the ranges indicated above. Obviously, R Finally, we study in Fig. 7 the effect of population size on the prevalence of an existing epidemic by comparing the disease dynamics numerically between (4.1) and (5.1) when R  /N 2 (0), to be equal in (4.1) and (5.1) with N 1 (0) = 80, 000, N 2 (0) = 5000, I 1 (0) = 8000, I 2 (0) = 1000. It shows that small population incurring frequent contacts of individuals may increase disease prevalence within the patch, but have little effect on that of the destination patch.

Discussion
This paper introduces an SIS metapopulation model accounting for media-induced modification (reduction) of travel rates. After brief consideration of local properties of solutions to the general model, the model is specialised to two patches and two scenarios are taken into consideration.
The systems under consideration, (2.2), (4.1) and (5.1), are in essence different from previous models of media effect on disease transmission ( [9, 14-16, 21-24, 30-32, 37, 38, 40, 43-45, 47, 48] and references therein). Global stability of system equilibria is still an open problem. Note that neither the standard comparison principle [39,Theorem B.1] in proving the global asymptotic stability of diseasefree equilibrium when the basic reproduction number is less than unity, nor the application of LaSalle Invariance Principle by Li and Shuai [29,Theorem 4.1] in proving the uniqueness and global asymptotic stability of endemic equilibrium when the basic reproduction number is greater than unity, are applicable for our model systems. Indeed, the travel rates between patches in our models are variable. Consider the uncoupled system without travel and denote R Obviously, R and that these quantities do not involve parameters of media coverage. This implies that the differentiation in population size does change the dynamics and potentially accelerate disease transmission. It is then inferred that disease can be persistent in both patches in system (5.1) while it dies down in both patches for system (4.1).
The models presented and studied here are toy models. The first limitation is that they are based on a simple SIS model, whereas a lot of diseases follow an SLIRS-type progression, where L and R stand for latently infected and recovered individuals, respectively. The second limitation is that intervention strategies such as vaccination and treatment are not included in the model. Moreover, the waning effect of media alert at individual level is not taken into account. Indeed, one may become less and less concerned about the epidemic after initially experiencing high anxiety of getting infected with the disease [26]. Work is in progress to formulate a model addressing such limitations.