The influence of awareness on epidemic spreading on random networks

The influence of awareness on epidemic spreading on random networks

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The influence of awareness on epidemic spreading on random networks Meili Li, Manting Wang, Shuyang Xue, Junling Ma PII: DOI: Reference:

S0022-5193(19)30459-X https://doi.org/10.1016/j.jtbi.2019.110090 YJTBI 110090

To appear in:

Journal of Theoretical Biology

Received date: Revised date: Accepted date:

10 May 2019 18 November 2019 19 November 2019

Please cite this article as: Meili Li, Manting Wang, Shuyang Xue, Junling Ma, The influence of awareness on epidemic spreading on random networks, Journal of Theoretical Biology (2019), doi: https://doi.org/10.1016/j.jtbi.2019.110090

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Highlights • Awareness acquired during an epidemic does not affect the basic reproduction number. Awareness acquired from outbreaks circulated/circulating in other populations before the local epidemic reduces the basic reproduction number. • Awareness can significantly reduce the epidemic final size. Breaking infectious edges causes a larger reduction than reducing the infection rate. • The reduction in final size may not increase monotonically with awareness. • The reduction in final size may not depend monotonically on the infection rate. • Whether local or global awareness has a larger reduction on the final size depends on the network and the infection rate.

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The influence of awareness on epidemic spreading on random networks Meili Li1 , Manting Wang1 , Shuyang Xue2 , Junling Ma3,∗ 1. School of Science, Donghua University, Shanghai 201620, China 2. School of Information Science and Technology, Donghua University, Shanghai 201620, China 3. Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 2Y2, Canada *email: [email protected]

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November 17, 2019

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Abstract

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During an outbreak, the perceived infection risk of an individual affects his/her behavior during an epidemic to lower the risk. We incorporate the awareness of infection risk into the Volz-Miller SIR epidemic model, to study the effect of awareness on disease dynamics. We consider two levels of awareness, the local one represented by the prevalence among the contacts of an individual, and the global one represented by the prevalence in the population. We also consider two possible effects of awareness: reducing infection rate or breaking infectious edges. We use the next generation matrix method to obtain the basic reproduction number of our models, and show that awareness acquired during an epidemic does not affect the basic reproduction number. However, awareness acquired from outbreaks in other regions before the start of the local epidemic reduces the basic reproduction number. Awareness always reduces the final size of an epidemic. Breaking infectious edges causes a larger reduction than reducing the infection rate. If awareness reduces the infection rate, the reduction increases with both local and global awareness. However, if it breaks infectious edges, the reduction may not be monotonic. For the same awareness, the reduction may reach a maximum on some intermediate infection rates. Whether local or global awareness has a larger effect on reducing the final size depends on the network degree distribution and the infection rate.

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Keywords awareness; contact network; disease dynamics; basic reproduction

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number; final epidemic size

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Introduction

During a disease outbreak, individuals adapt their behavior to lower their perceived risk of infection [2,6,12,20,21]. This risk can be estimated by surveying the infections among local contacts (the local awareness of the risk), or the whole population (the global awareness). With the improvement of modern communication technology, the spread of information and the change of travel patterns, global awareness becomes increasingly influential in human behavior, which in turn affects the disease dynamics [1, 3, 5, 13, 16]. Thus behavior changes become more and more influential to disease dynamics. It is thus important to understand how much influence the global and local awareness has on disease dynamics, and which type of awareness is more influential. Mathematical models are powerful tools to study the effect of awareness on disease dynamics. However, classical models assume homogeneous mixing, i.e., all pairs of individuals have equal probability of contact. Thus, such models can only study global awareness, because they cannot easily model local contact structure of the population. For example, Funk et al. [7] studied the spread of awareness and its effect on disease outbreaks. They proved that the spread of awareness not only reduces the incidence of disease, but also prevents the disease from developing into an epidemic in some cases. Wu et al. [25] considered the SIS model when studying the influence of consciousness on disease transmission. They considered the awareness has three kind of forms: contact awareness, local awareness and global awareness. The results showed that the contact awareness and the local awareness have an effect on the transmission threshold, while the global awareness has no effect on it. To study local awareness, it is more appropriate to use contact networks to model individual contacts [4, 8, 10, 18]. In a contact network model, the population is represented by a graph, in which individuals are represented by nodes, and the contacts between individuals are represented by edges connecting the corresponding nodes. Rizzo [22] proposed that self-protection and quarantine behavior are effective means to control the spread of the epidemic in the SIS dynamic model, which leads to the increase of epidemic threshold and the decrease of steady-state prevalence. Massaro and Bagnoli [15] proposed a two-layer UAU-SIR model. The first layer represents the physical contact network for epidemic transmission, while the other layer is the awareness network. However, this model is huge and difficult to analyze. Many of the recent outbreaks of emerging infectious diseases, such as SARS, pandemic influenza, Ebola, and Zika, are viral diseases, for which infected individuals develop immunity after recovery. For these diseases, an Susceptible-Infectious-Recoverd (SIR) model is more appropriate than an Susceptible-Infectious-Susceptible (SIS) model, where the former assumes that an infectious individual recovers with acquired life time immunity, while the latter assume that an infectious individual recovers with no acquired immunity.

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Diseases with or without immunity have different disease thresholds on the same contact network [14]. So, in this paper, we concentrate on SIR epidemics. Because it is difficult to predict the emergence of a new disease, when a disease first appears, there is typically little awareness. Note that the disease threshold is determined near the disease free equilibrium, i.e., when a disease first appears, awareness should not affect the disease threshold, and thus the emergence, of the disease. However, with the spread of the disease, awareness reduces the transmission, and thus lowers the epidemic final size. On the other hand, if the disease has been noticeably circulating in other populations, then this prior knowledge of an epidemic can affect individual behavior, resulting in the decrease of the basic reproduction number (i.e., making the disease harder to invade), before the introduction of the disease into the population. To verify these effects of awareness on emerging diseases, in this paper, we incorporate the initial awareness, the local awareness and the global awareness into a well established SIR epidemic model on a random contact network, namely, the Volz-Miller model [17, 24]. We develop our models in Section 2. In Section 3, we calculate the basic reproduction number of our models. In Section 4, we use numerical simulations to study the impact of awareness on the final size of the disease. Concluding remarks are given in Section 5.

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Disease dynamics

In this section, we incorporate awareness into the Miller model [17], which is equivalent to an earlier Volz model [24]. To do so, we first give a quick introduction to these models. Both models assume that the underlying contact network is a configuration network, which is determined by its degree distribution. A network with n nodes can be constructed as follows. The degree k of each node in the network is given by the degree distribution pk . We assign a degree ki from the distribution for node i (i = 1, 2, · · · , n), P and assign ki stubs the node. If the total degree L = ni=1 ki is odd, then the degrees ki should be reassigned to make L even, because L is twice the total number of edges. Then we randomly select a pair of stubs to form an edge [19]. Large configuration networks have negligible self-loops, multiply edges, clustering and degree correlation. Like the pair approximation approach [9], both models keep track of new infections through the (infection) state change of the neighbors of a susceptible node. The state of neighbors in turn depends on the states of their neighbors, and thus, to study the state change of a node, we need to keep track of an infinite chain of state changes of neighbors. Unlike the pair approximation approach, the Volz-Miller approach assumes that the states of the neighbors of a susceptible node are independent, i.e., the number of infectious neighbors of a susceptible node is, on average, its degree multiplied by the probability that a neighbor is infectious. In addition, this probability is the same for all susceptible nodes. Kiss et al. proved that this additional assumption is true for all time as long as it is true initially [11]. Both models depend on a key quantity, the probability that a susceptible node with

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degree 1 has not been infected by time t, denoted as θ(t). With the assumptions above, θ(t) is also the probability that a random neighbor of a susceptible node has not transmitted by time t. A degree-k node is susceptible because none of its neighbors has k transmitted, which has a probability P k θ . Thus, a random node is susceptible at time t with a probability g(θ) = k θ pk . Note that the function g(θ) is the probability generation function of the degree distribution pk . Let λ be the infection rate along a random edge, and γ be the recovery rate of an infectious node. A random neighbor of a susceptible node is infectious or susceptible with probabilities PI and PS , respectively. Then, the Volz model [24] can be written as S = g(θ), I˙ = λPI θg 0 (θ) − γI, θ˙ = −λPI θ,

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g (θ) P˙I = −λPI (1 − PI ) + λPI PS θ 0 − PI γ, g (θ) 00 g (θ) ˙ PS = λPI PS (1 − θ 0 ). g (θ) 111 112 113

Miller [17] showed that, with the variable change φ = PI θ where φ is the fraction of edges that has not transmitted infection (θ edges) but can cause infection (i.e., connected to an infectious node), the Volz model can be simplified as: S = g(θ), I˙ = λφg 0 (θ) − γI, θ˙ = −λφ,

(1a) (1b) (1c) 00

g (θ) φ˙ = −(λ + γ)φ + λφ 0 . g (1)

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(1d)

Now we incorporate awareness into the Miller model (1). Here we assume that awareness is represented by the perception of a susceptible node on the prevalence of the disease. The local awareness means that an individual surveys his/her neighbors for the prevalence. On random contact networks, the local prevalence is indeed the fraction of infectious neighbors, which is equal to the probability PI = φ/θ. Global awareness means that he/she surveys the whole population for the prevalence I. The population may also be alerted by an outbreak circulating (or circulated) in another region before the epidemic starts locally, and thus may have an initial awareness before the disease breaks out in this population. This is illustrated by recent outbreaks of emerging diseases. For example, populations around the world have been alerted by the Chinese and Hong Kong outbreaks of SARS during the 2002 epidemic, by the Mexican outbreak of 2009 influenza A/H1N1 during the 2009 pandemic, by the western African outbreaks of Ebola in 2013, and by the Brazil Zika outbreak in 2015.

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In the following subsections, we consider two situations in which awareness affects the disease dynamics. In the first one, we assume that awareness reduces the infection rate. In the second, we assume that awareness causes an susceptible individual to avoid the infectious neighbors by breaking their edges.

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2.1

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Awareness reduces the infection rate

Following the tradition of Wu et al. [25], the effect of local awareness is modeled as a reduction factor r` on the infection rate λ, where r` ∈ (0, 1] decreases with the local prevalence PI about disease transmission. We assume that r` = e−αPI = e−αφ/θ , where α ≥ 0 is the local awareness coefficient. The global awareness is modeled as an additional reduction factor rg on the infection rate, where rg ∈ (0, 1] decreases with the prevalence I in the population. Similarly, we choose rg = e−βI , where β ≥ 0 is the global awareness coefficient. The initial awareness is modeled as a constant b ≥ 0 where b = 0 means that there is no initial awareness. It reduces the infection rate further by a factor e−b . Thus, the infection rate under the influence of awareness, denoted as q, is φ

q = λe−b r` rg = λe−b−α θ −βI . 141

(2)

We incorporate the effect of awareness into (1c) and (1d) as: θ˙ = −qφ,

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g (θ) φ˙ = −(q + γ)φ + qφ 0 . g (1) 142

Thus, our SIR model with awareness can be written as: S = g(θ), φ I˙ = λe−b−α θ −βI φg 0 (θ) − γI, φ θ˙ = −λe−b−α θ −βI φ,

(3a) (3b) (3c)

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φ g (θ) φ˙ = λe−b−α θ −βI ( 0 − 1)φ − γφ, g (1)

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with the initial conditions θ(0) = 1, φ(0) = I(0)  1 and S(0) = 1.

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2.2

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(3d)

Awareness breaks infectious edges

Here we consider that awareness does not reduce the infection rate λ, but causes the susceptible individuals to break their edges with the infectious neighbors. Such edges are in class φ. Assume that the φ edges break with a rate δ. Thus, (1d) becomes 00

g (θ) − δφ. φ˙ = −(λ + γ)φ + λφ 0 g (1)

(4)

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These broken edges can be regarded as edges that lost the ability to infect, thus effectively keeping the network unchanged. These not-infectious edges do not affect disease dynamics. Thus, we do not keep track of them. The infectious node of such an edge still recovers at the rate γ. Hence, the dynamics of I remains the same as (1b). We can model δ as an increasing function of the awareness coefficients α and β, for example, φ

δ = δ0 eb+α θ +βI , 153 154 155 156

where δ0 is the maximum breaking rate of the edges. However, this method introduces an extra parameter δ0 , which makes the comparison with model (3) difficult. To avoid this problem, we combine γ and δ together, and assume that the φ edges leave the class from recovery and edge breaking at a rate γ e−b r` rg

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φ

= γeb+α θ +βI .

Note that infectious nodes still recover with the rate γ. Thus, the model is S = g(θ), I˙ = λφg 0 (θ) − γI, θ˙ = −λφ, 00 φ g (θ) ˙ − 1)φ − γeb+α θ +βI φ. φ = λ( 0 g (1)

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(5a) (5b) (5c) (5d)

The basic reproduction number

We use the next generation matrix method to compute the basic reproduction number of our models. The basic reproduction number R0 is the spectral radius of the next generation matrix (see [23]). This approach needs to rewrite the Jacobian of the infectious classes at the disease free equilibrium as F − V , where the (i, j) element of the newinfection matrix F is the rate of new infections in class i caused by a individual in class j, and the (i, j) element of the transition matrix V is the rate that individuals in class j transfer to class i (i 6= j) or leave i (i = j). Then, R0 = ρ(F V −1 ),

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where ρ is the spectral radius of a matrix, and the matrix F V −1 is the next generation matrix of the model. The threshold condition for disease outbreaks in population is R0 = 1. The disease will invade the population if R0 > 1, and die out if R0 < 1. We consider the “infected classes” I and φ in both models (3) and (5).

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3.1

Awareness reduces the infection rate

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Let us linearize model (3) about the disease-free equilibrium (S = 1, θ = 1, I = 0, φ = 0):

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I˙ = −γI + λe−b g 0 (1)φ,

(6a) 00

g (1) φ˙ = −(λe−b + γ)φ + λe−b 0 φ. g (1) 173 174

Note that the dynamics of I is determined by φ, so we thus only need to study equation (6b). This is a one-dimensional system. The new infection matrix is F = λe−b

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(6b)

g 00 (1) , g 0 (1)

and the transition matrix is V = λe−b + γ .

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Thus, R0 =

λe−b g 00 (1) F = −b . V λe + γ g 0 (1)

(7)

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3.2

Awareness breaks infectious edges

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We linearize the model (5) about the disease-free equilibrium (S = 1, θ = 1, I = 0, φ = 0):

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I˙ = −γI + λg 0 (1)φ,

(8a) 00

g (1) φ˙ = −(λ + γeb )φ + λ 0 φ. g (1) 180 181

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(8b)

Similar to (6), the dynamics of I is determined by φ, and thus we only need to study (8b). Here g 00 (1) F =λ 0 , V = λ + γeb . g (1) Thus, the basic reproduction number of model (5) is also given by (7). In summary, the basic reproduction number does not depend on whether awareness reduces infection rate or breaks infectious edges. It is also independent of the awareness coefficients α and β, and it decreases with the initial awareness b. When there is no λ g 00 (1) initial awareness (i.e., b = 0), R0 = λ+γ g0 (1) , which is equal to the basic reproduction number of the Miller model.

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4

Final epidemic size

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In this section, we consider the impact of awareness on the final epidemic size Z(α, β) = 1 − S(∞) = 1 − g(θ(∞)).

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Note that θ(∞), and thus S(∞), depends on α and β. Deriving a formula for the final size as in [17] is difficult. We thus rely on numerical simulations to reveal the effect of awareness on final size. We compute the percentage reduction in final epidemic size, defined as Ψ(α, β) = 1 −

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(9)

Z(α, β) , Z(0, 0)

for models (3) and (5). Here we consider two contact networks: a Poisson random network and a scalefree network, to study the effect of network degree distribution on the effectiveness of awareness. For comparison, we set the recovery rate γ = 1 to keep all simulations on the same time scale. We also set the initial awareness b = 0, because the factor e−b can be absorbed into the infection rate λ. We also choose a range of values for the infection rate λ for each network (the values are listed in figure captions) to study how the reduction in final size depends on the infection rate. Then we choose different values for the awareness coefficients α and β, and numerically integrate our model to the equilibrium state. Figures 1 and 2 show the percentage reduction in final size of models (3) and (5), respectively, on the scale-free network. For model (3), the percentage reduction Ψ monotonically increases with both awareness coefficients α and β. However, for model (5), Ψ may not be monotonic in the awareness coefficients. Though not shown in figures, Ψ shows similar behavior on the Poisson network. To compare the effect of the awareness coefficients α and β on the reduction, we keep one coefficient 0, and vary the other. For the Poisson network, the comparisons are shown in Figures 3 and 4 for models (3) and (5), respectively. For the scale-free network, the comparisons are shown in Figures 5 and 6, respectively. Whether the local awareness or the global awareness has a larger impact on the reduction depends on the network degree distribution and the infection rate. On the Poisson network, the global awareness always has a larger effect. On the scale-free network, the local awareness has a larger effect with a smaller infection rate, and as the infection rate increases, the global awareness has an increasingly larger effect, and may surpass that of the local awareness, especially for model (5). For each fixed α and β, while we increase the infection rate λ, the reduction may reach a maximum at an intermediate λ. To investigate which model has a larger reduction, we plot in Figure 7 the difference between the percentage reductions of the two models (5) and (3) (the percentage reduction from breaking edges minus that from reducing infection rate), on the Poisson network. The positive difference implies that breaking edges always has a larger percentage reduction than reducing infection rate. This difference increases as the infection rate

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λ increases, but is not monotonic with α and β. Though not shown in a figure, the same is true for the scale-free network.

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Concluding remarks

We studied the effect of awareness on the dynamics of an SIR epidemic on a random networks, by incorporating both the local awareness and the global awareness into the Miller SIR model (1). The local awareness is represented by the fraction of infectious neighbors, and the global awareness is represented by the prevalence of the disease in the population. We additionally considered an initial awareness from outbreaks in other regions before the epidemic started locally. We considered two possible effects of awareness, a) the reduction of the infection rate and b) the breaking of infectious edges. Both scenarios have the same basic reproduction number. The initial awareness reduces the basic reproduction number. However, neither the local nor the global awareness affects the basic reproduction number. We believe that this is because the basic reproduction number is calculated when almost everyone is susceptible (i.e., when the population is close to the disease free equilibrium). At that time, there is little prevalence, thus both the local awareness and global awareness are negligible. Thus, for an emerging infectious disease with no initial awareness, awareness gained during the epidemic cannot prevent an epidemic. Awareness always reduces the final epidemic size. It causes a larger percentage reduction in final size in the second scenario than in the first. The percentage reduction increases with both the local and global awareness coefficients in the first scenario. However, it may not increase monotonically with the awareness coefficients in the second scenario when the infection rate is high. This may be caused by the complex interactions between the local and global prevalence. Specifically, it causes local prevalence to drop quickly while the global prevalence is still increasing. Whether the local or the global awareness has a larger effect in reducing the final size depends on the network degree distribution and the infection rate. For example, on a Poisson random network, the global awareness has a larger effect. But this is not true on a scale-free network. Interestingly, on both types of networks, the awareness may is more effective on intermediate infection rates, and less effective on very large or very small infection rates. For small infection rates, this may be because the prevalence is low, resulting in a low awareness. For large infection rates, this may be because the disease spread is so fast that the increase in the prevalence (and thus the awareness) is not as efficient to slow down the epidemic as in a slower epidemic. It is possible to use other measures for global awareness, such as cumulative cases. It is not clear which measure is more realistic. Using another measure will require modifications to our models, and will be investigated in future studies.

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λ = 0.3

λ = 0.6

λ = 1.2

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0.45

0.4

0.4

5

0.5

0.5

0.5

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0.4 5

10 5

0.35

0 0.005.10.150.2 0.25 .3

α

0

0 0.00.01.10.02.250.3 0.35 .4 5 5

λ = 2.4

0.000. .100. .2 0.30.35 0.4 51 5 2 5

λ = 4.8

λ = 9.6

5

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0.

5 0.

45 0.

55

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10

0.

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5

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0.

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5 0

0. 0 0. 0.0.2 0.30.35 05.1 15 2 5

0.000. .10.0.2 0.30.35 0.4 51 5 2 5 0

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0. 0 0. 0. 0.2 0.3 05.1 15 2 5 15

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Figure 1: The contour plot of the percentage reductions Ψ(α, β) for model (3), on a scale-free network with a degree distribution pk ∝ k −2 for k = 1, 2, . . . , 66. We consider 6 infection rates, λ = 0.3, 0.6, 1.2, 2.4, 4.8, 9.6. The initial awareness b = 0; the recovery rate γ = 1. The initial conditions are S(0) = 1, θ(0) = 1, and I(0) = φ(0) = 10−6 .

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Acknowledgments We thank the anonymous reviewers for their constructive comments. This research was supported by National Natural Science Foundation of China (No.11771075) (ML) and Natural Sciences and Engineering Research Council Canada Discovery Grant (JM).

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References

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[1] Apolloni, A., Poletto, C., Colizza, V., 2013. Age-specific contacts and travel patterns in the spatial spread of 2009 H1N1 influenza pandemic. BMC Infectious Diseases 13, 176. [2] Bagnoli, F., Li´o, P., Sguanci, L., 2007. Risk perception in epidemic modeling. Phys Rev E Stat Nonlin Soft Matter Phys 76, 061904.

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λ = 0.3

λ = 0.6

λ = 1.2

20

45

0.2

5

55

5

α

0

0.

0.3 5 0.2 0.050.10.1

5 0.4 0.40.35

0.

0. 0.3 4 0 5 0.2.3 00.25 00...1015 5

10 0. 35

λ = 2.4

0.6

00..115 05

5

0.

10

15

20

0.6 0.55 0.5 0.45 0.4 .35 .3 25.2 15.1 05 0 0 0. 0 0. 0 0.

5 0 0..2

0

0.6 λ = 9.6

0.6

0.55 0.5

0.45 0.40.350.3.205.2.15.1 0 00

0.55 0.5 0.45 .4 35 3 5 0 0. 0.0.2

15

0

0 0..3 0 0 .225 00...1015 5

λ = 4.8

20

10

0.55

5

0.4

0.5

0.5

0.4

15

05 0

5

10

15

20

0

5

0.

65

10

15

20

β

Figure 2: The contour plot of the percentage reductions Ψ(α, β) for model (5) on a scalefree network. The network, initial awareness, disease parameters and initial conditions are the same as in Figure 1.

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λ = 0.6

λ = 1.2

λ = 2.4

0.3

0.2

Ψ(α, β)

0.1

line

0.0 λ = 3.6

λ = 4.8

λ = 6.0

α=0 β=0

0.3

0.2

0.1

0.0 0.0

2.5

5.0

7.5 10.0 12.50.0

2.5

5.0

7.5 10.0 12.50.0

2.5

5.0

7.5 10.0 12.5

α, β

Figure 3: The dependence of the percentage reduction in final size Ψ(α, β) for model (3) on either α or β on a Poisson network with an average degree 3, while keeping the other awareness coefficient 0. The infection rates considered here are λ = 0.6, 1.2, 2.4, 3.6, 4.8, 6.0. The recovery rate, initial awareness, and initial conditions are the same as in Figure 1.

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λ = 0.6

λ = 1.2

λ = 2.4

0.3 0.2

Ψ(α, β)

0.1

line

0.0 λ = 3.6

λ = 4.8

λ = 6.0

α=0 β=0

0.3 0.2 0.1 0.0 0.0

2.5

5.0

7.5 10.0 12.50.0

2.5

5.0

7.5 10.0 12.50.0

2.5

5.0

7.5 10.0 12.5

α, β

Figure 4: The dependence of the percentage reduction in final size Ψ(α, β) for model (5) on either α or β on a Poisson network, while keeping the other awareness coefficient 0. The network, initial awareness, disease parameters, and initial conditions are the same as in Figure 3.

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λ = 0.3

λ = 0.6

λ = 1.2

0.5 0.4 0.3 0.2

Ψ(α, β)

0.1

line

0.0 λ = 2.4

λ = 4.8

α=0

λ = 9.6

β=0

0.5 0.4 0.3 0.2 0.1 0.0 0

5

10

15

20 0

5

10

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20 0

5

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α, β

Figure 5: The dependence of the percentage reduction in final size Ψ(α, β) for model (3) on either α or β on a scale-free network, while keeping the other awareness coefficient 0. The network, initial awareness, disease parameters, and initial conditions are the same as in Figure 1.

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λ = 0.3

λ = 0.6

λ = 1.2

0.6 0.4

Ψ(α, β)

0.2

line

0.0 λ = 2.4

λ = 4.8

α=0

λ = 9.6

β=0 0.6 0.4 0.2 0.0 0

5

10

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20 0

5

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15

20 0

5

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Figure 6: The dependence of the percentage reduction in final size Ψ(α, β) for model (5) on either α or β on a scale-free network, while keeping the other awareness coefficient 0. The network, initial awareness, disease parameters, and initial conditions are the same as in Figure 1.

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α

0.0

0.

02

0.0 4 5 .0 0 0.06

0. 03

7

0.01

λ = 3.6

0.0 15

0.0 98 0.0.1 0

0.01 0

2.5

0.0 2 0.025 0.03 0.035 0.045 4 0.0 05 0.

λ = 2.4

0.0

5.0

0

0.003 0.0035 0.004 0.00 45 0 0.0 0.00 .005 0.0 55 065 06

0

7.5

0.005

10.0

λ = 1.2

0.0025 0.002 0.0015 0.001 5e-04

λ = 0.6

12.5

λ = 4.8

λ=6

0.02

5.0

7.5

10.0 12.50.0

2.5

5.0

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10.0 12.50.0

0. 04

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Figure 7: The contour plot of the difference in the percentage reductions Ψ(α, β) between model (5) and (3), on a Poisson random network. Positive difference means that Ψ of model (5) is larger. The network, initial awareness, disease parameters, and initial conditions are the same as in Figure 3.

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