Public opinion and opinion dynamics can have a strong effect on the transmission rate of an infectious disease for which there is no vaccine. The coupling of disease and opinion dynamics however, creates a dynamical system that is complex and poorly understood. We present a simple model in which susceptible groups adopt or give up prophylactic behaviour in accordance with the influence related to pro- and con-prophylactic communication. This influence varies with disease prevalence. We observe how the speed of the opinion dynamics affects the total size and peak size of the epidemic. We find that more reactive populations will experience a lower peak epidemic size, but possibly a larger final size and more epidemic waves, and that an increase in polarization results in a larger epidemic.
Citation: Rebecca C. Tyson, Noah D. Marshall, Bert O. Baumgaertner. Transient prophylaxis and multiple epidemic waves[J]. AIMS Mathematics, 2022, 7(4): 5616-5633. doi: 10.3934/math.2022311
Related Papers:
[1]
Mona Hosny .
Rough sets theory via new topological notions based on ideals and applications. AIMS Mathematics, 2022, 7(1): 869-902.
doi: 10.3934/math.2022052
[2]
Rehab Alharbi, S. E. Abbas, E. El-Sanowsy, H. M. Khiamy, K. A. Aldwoah, Ismail Ibedou .
New soft rough approximations via ideals and its applications. AIMS Mathematics, 2024, 9(4): 9884-9910.
doi: 10.3934/math.2024484
[3]
Rukchart Prasertpong .
Roughness of soft sets and fuzzy sets in semigroups based on set-valued picture hesitant fuzzy relations. AIMS Mathematics, 2022, 7(2): 2891-2928.
doi: 10.3934/math.2022160
[4]
Mona Hosny, Tareq M. Al-shami .
Rough set models in a more general manner with applications. AIMS Mathematics, 2022, 7(10): 18971-19017.
doi: 10.3934/math.20221044
[5]
Mostafa A. El-Gayar, Radwan Abu-Gdairi .
Extension of topological structures using lattices and rough sets. AIMS Mathematics, 2024, 9(3): 7552-7569.
doi: 10.3934/math.2024366
[6]
Rabia Mazhar, Shahida Bashir, Muhammad Shabir, Mohammed Al-Shamiri .
A soft relation approach to approximate the spherical fuzzy ideals of semigroups. AIMS Mathematics, 2025, 10(2): 3734-3758.
doi: 10.3934/math.2025173
[7]
Nurettin Bağırmaz .
A topological approach for rough semigroups. AIMS Mathematics, 2024, 9(10): 29633-29644.
doi: 10.3934/math.20241435
[8]
D. I. Taher, R. Abu-Gdairi, M. K. El-Bably, M. A. El-Gayar .
Decision-making in diagnosing heart failure problems using basic rough sets. AIMS Mathematics, 2024, 9(8): 21816-21847.
doi: 10.3934/math.20241061
[9]
Jamalud Din, Muhammad Shabir, Nasser Aedh Alreshidi, Elsayed Tag-eldin .
Optimistic multigranulation roughness of a fuzzy set based on soft binary relations over dual universes and its application. AIMS Mathematics, 2023, 8(5): 10303-10328.
doi: 10.3934/math.2023522
[10]
Saqib Mazher Qurashi, Ferdous Tawfiq, Qin Xin, Rani Sumaira Kanwal, Khushboo Zahra Gilani .
Different characterization of soft substructures in quantale modules dependent on soft relations and their approximations. AIMS Mathematics, 2023, 8(5): 11684-11708.
doi: 10.3934/math.2023592
Abstract
Public opinion and opinion dynamics can have a strong effect on the transmission rate of an infectious disease for which there is no vaccine. The coupling of disease and opinion dynamics however, creates a dynamical system that is complex and poorly understood. We present a simple model in which susceptible groups adopt or give up prophylactic behaviour in accordance with the influence related to pro- and con-prophylactic communication. This influence varies with disease prevalence. We observe how the speed of the opinion dynamics affects the total size and peak size of the epidemic. We find that more reactive populations will experience a lower peak epidemic size, but possibly a larger final size and more epidemic waves, and that an increase in polarization results in a larger epidemic.
1.
Introduction
Rough set theory, introduced by Pawlak [17], is a powerful mathematical tool for effectively transacting with imprecise and uncertain information. A key advantage of rough set theory is its ability to represent data using granular computing inspired by an equivalence relation. As we know, the granular computing represented by equivalence classes in the original model of Pawlak has been updated using some neighborhood systems inspired by relations weaker than equivalence relation or arbitrary relations; for more details about these neighborhood systems, we refer the readers to [6,14,22] and references mentioned therein. This development assists in canceling a strict condition of an equivalence relation and expanding the scope of its applications in diverse disciplines.
The interconnection of topological and rough set theory was first put forward by Wiweger [21], who explored the topological aspects of rough sets. This led to a fusion of rough set theory and topological structures becoming a central focus of numerous studies [19,25]. Then, some techniques to institute a topology using rough neighborhoods were proposed to represent rough approximation operators and analyze information systems; see [1,13]. One of the important topological tools to reduce the vagueness of knowledge is nearly open sets, so many authors applied to describe rough set models, such as δβ-open sets [2], somewhat open sets [3], and somewhere dense sets [4]. Quite recently, this interaction has been developed to involve generalizations of topology, such as supra topology [5], infra topology [7], minimal structures [9], and bitopology [18]. It is worth noting that several manuscripts investigated rough set models from different views as [8,20].
In 2013, Kandi et al. [12] integrated the abstract principle so-called ideal I with rough neighborhoods to provide a new framework of rough sets paradigms called ideal approximation spaces. As illustrated in published literature like [10,15,24], this framework proves its capability in terms of enlarging the domain of confirmed knowledge and thereby maximizing the value of accuracy.
In 2022, Nawar et al. [16] proposed two new rough set models, the first one generated by one of nearly open sets, namely, θβσ-open sets, and the second generated by ideals and I-θβσ-open sets. However, we note that they provided some incorrect results and relationships that cannot be overlooked and require correction, particularly those that compare the superiority of their approach over the one introduced by Hosny [11]. In this regard, we construct a counterexample to show that Theorem 4.1, Corollary 4.1, and items (2) and (4) of Corollary 4.2 displayed in [16] are false. Moreover, we prove that their approach and Hosny's approach [11] are incomparable. Ultimately, we evidence that three observations on page 2494 of [11] about the given application are incorrect, in general.
2.
Preliminaries
Here, we recall some definitions and results that are required to understand this work.
Remember that a relation λ on a nonempty set X is a subset of X×X. We write aλb when (a,b)∈λ.
Definition 2.1[17] Let λ be an equivalence relation on X. The lower approximation and upper approximation of Z⊆X are, respectively, given by:
λ_(Z)=∪{V∈X/λ∣V⊆Z}.
¯λ(Z)=∪{V∈X/λ∣V∩Z≠∅},
where X/λ denotes the family of equivalence classes induced by λ.
The triple (X,λ_,¯λ) is called Pawlak rough set models; it is known as the original (standard) model. The core features of this model are enumerated in the subsequent proposition.
Proposition 2.2.[17] Consider an equivalence relation λ defined on X. For sets V,W, the next characteristics hold:
In many positions, the equivalence relations are not attainable. Consequently, the classical approach has been extended by employing weaker relations than full equivalence. This led to the proposal of different types of neighborhoods as granular computing alternatives for the equivalence classes.
Definition 2.3[1,22,23] Consider an arbitrary relation λ on X. If σ∈{r,⟨r⟩,l,⟨l⟩,i,⟨i⟩,u,⟨u⟩}, then the σ-neighborhoods of a∈X, symbolized by Nσ(a), are identified as:
(i) Nr(a)={b∈X:aλb}.
(ii) Nl(a)={b∈X:bλa}.
(iii)
N⟨r⟩(a)={⋂a∈Nr(b)Nr(b):∃Nr(b)involvinga∅:Elsewise
(iv)
N⟨l⟩(a)={⋂a∈Nl(b)Nl(b):∃Nl(b)involvinga∅:Elsewise
(v) Ni(a)=Nr(a)⋂Nl(a).
(vi) Nu(a)=Nr(a)⋃Nl(a).
(vii) N⟨i⟩(a)=N⟨r⟩(a)⋂N⟨l⟩(a).
(viii) N⟨u⟩(a)=N⟨r⟩(a)⋃N⟨l⟩(a).
Henceforward, unless otherwise specified, we will consider σ to belong to the set {r,⟨r⟩,l,⟨l⟩,i,⟨i⟩,u,⟨u⟩}.
Remark 2.4.The authors of [16] incorrectly mentioned the definitions of N⟨r⟩(a) and N⟨l⟩(a). They overlooked the cases that do not exist Nr(b) containing a and Nl(b) containing a, which leads to incorrect computations for some cases, especially when the given binary relation is not serial or inverse serial. Therefore, we should consider these cases when we define N⟨r⟩(a) and N⟨l⟩(a) as given in (iii) and (iv) of Definition 2.3.
Definition 2.5.[1] Consider a relation λ on X and let ζσ denote a mapping from X to 2X, associating each member a∈X with its σ-neighborhood in 2X. Consequently, the triple (X,λ,ζσ) is termed a σ-neighborhood space, abbreviated as σ-NS.
Theorem 2.6.[1] It may generate a topology ϑσ on X using Nσ-neighborhoods by the next formula
ϑσ={V⊆X:Nσ(a)⊆Vfor eacha∈V}
The main concepts of rough set paradigms inspired by a topology given in Theorem 2.6 are mentioned in the next two definitions.
Definition 2.7.[1] Let (X,λ,ζσ) be a σ-NS and ϑσ be a topology described in Theorem 2.6. For each σ, the σ-lower and σ-upper, σ-boundary, and σ-accuracy of a subset Z of X are, respectively, defined by the following formulas.
(ⅰ)H_σ(Z)=∪{V∈ϑσ:V⊆Z}=intσ(Z), where intσ(Z) is the interior points of Z in (X,ϑσ).
(ⅱ)¯Hσ(Z)=∩{W:Z⊆W and ∈Wc∈ϑσ}=clσ(Z), where clσ(Z) is the closure points of Z in (X,ϑσ).
(ⅲ)Bσ(Z)=¯Hσ(Z)∖H_σ(Z)
(ⅳ)Aσ(Z)=∣H_σ(Z)∣∣¯Hσ(Z)∣, where Z is a nonempty set.
Definition 2.8.[1] A subset Z of (X,λ,ζσ) is called an σ-exact (resp., σ-rough) set if H_σ(Z)=¯Hσ(Z) (resp., H_σ(Z)≠¯Hσ(Z))
Definition 2.9.A nonempty subclass I of 2X is called an ideal on X provided that the next conditions are satisfied.
(ⅰ)The union of any two members in I is a member of I.
(ii)If V∈I, then any subset of V is a member of I.
Theorem 2.10.[11,12] It may generate a topology ϑIσ on X using Nσ-neighborhoods and an ideal I by the next formula
ϑIσ={V⊆X:Nσ(a)∖V∈Ifor eacha∈V}
The main concepts of rough set paradigms inspired by a topology given in Theorem 2.10 are mentioned in the next two definitions.
Definition 2.11.[11] Let (X,λ,ζσ) be a σ-NS, I be an ideal on X, and ϑIσ be a topology described in Theorem 2.10. For each σ, the Iσ-lower and Iσ-upper, Iσ-boundary, and Iσ-accuracy of a subset Z of X are, respectively, defined by the following formulas.
(ⅰ)H_Iσ(Z)=∪{V∈ϑIσ:V⊆Z}=intIσ(Z), where intIσ(Z) is the interior points of Z in (X,ϑIσ).
(ⅱ)¯HIσ(Z)=∩{W:Z⊆W and ∈Wc∈ϑIσ}=clIσ(Z), where clIσ(Z) is the closure points of Z in (X,ϑIσ).
(ⅲ)BIσ(Z)=¯HIσ(Z)∖H_Iσ(Z)
(ⅳ)AIσ(Z)=∣H_Iσ(Z)∣∣¯HIσ(Z)∣, where Z is a nonempty set.
Definition 2.12.[11] Let (X,λ,ζσ) be a σ-NS and I be an ideal on X. A subset Z of X is called an Iσ-exact (resp., Iσ-rough) set if H_Iσ(Z)=¯HIσ(Z) (resp., H_Iσ(Z)≠¯HIσ(Z)).
3.
Main results
This section rectifies some invalid claims and relationships presented in [16]. An elucidative example is provided to support the amendments that were made.
To begin, we recall the definition of I-θβσ-open sets as introduced in [16].
Definition 3.1.[16] Let (X,λ,ζσ) be a σ-NS and I be an ideal on X. Then, a subset of V of X is called I-θβσ-open if V⊆clσ(intσ(cl⋆θσ(V))) such that
The complement of an I-θβσ-open set is named I-θβσ-closed. The classes of I-θβσ-open subsets and I-θβσ-closed subsets are respectively symbolized by I-θβσO(X) and I-θβσC(X).
Then, they introduced the ideas of lower and upper approximations, boundary regions, and accuracy measures in relation to the classes of I-θβσO(X) and I-θβσC(X) as follows.
Definition 3.2.[16] Let (X,λ,ζσ) be a σ-NS and I be an ideal on X. For each σ, the I-θβσ-lower and I-θβσ-upper, I-θβσ-boundary, and I-θβσ-accuracy of a subset Z of X are, respectively, defined by the following formulas.
(ⅰ)H_I−θβσ(Z)=∪{V∈I-θβσO(X):V⊆Z}=intI−θβσ(Z), where intI−θβσ(Z) is the interior points of Z in (X,I-θβσO(X)).
(ⅱ)¯HI−θβσ(Z)=∩{W∈I-θβσC(X):Z⊆W}=clI−θβσ(Z), where clI−θβσ(Z) is the closure points of Z in (X,I-θβσO(X)).
(ⅲ)BI−θβσ(H)=¯HI−θβσσ(Z)∖H_I−θβσσ(Z)
(ⅳ)AI−θβσ(Z)=∣H_I−θβσσ(Z)∣∣¯HI−θβσσ(Z)∣, where Z is a nonempty set.
Remark 3.3.One can prove that the structure (X,I-θβσO(X)) is closed under arbitrary unions. In contrast, the intersection of two I-θβσ-open sets fails to be an I-θβσ-open set, in general. Therefore, (X,I-θβσO(X)) forms a supra topology on X.
Definition 3.4.[16] Let (X,λ,ζσ) be a σ-NS and I be an ideal on X. A subset Z of X is called an I-θβσ-exact (resp., I-θβσ-rough) set if H_I−θβσ(Z)=¯HI−θβσ(Z) (resp., H_I−θβσ(Z)≠¯HI−θβσ(Z))
The authors of [16] claimed the following theorem and two corollaries; they were presented in [16] in the following order: Theorem 4.1, Corollary 4.1, and items (2) and (4) of Corollary 4.2.
Theorem 3.5.[16] Let (X,λ,ζσ) be a σ-NS and I be an ideal on X. Then, we have the following properties for any subset Z of X.
(ⅰ)H_σ(Z)⊆H_Iσ(Z)⊆H_I−θβσ(Z).
(ⅱ)¯HI−θβσ(Z)⊆¯HIσ(Z)⊆¯Hσ(Z).
Corollary 3.6.[16] Let (X,λ,ζσ) be a σ-NS and I be an ideal on X. Then, we have the following properties for any subset Z of X.
(ⅰ)BI−θβσ(Z)⊆BIσ(Z)⊆Bσ(Z).
(ⅱ)Aσ(Z)≤AIσ(Z)≤AI−θβσ(Z).
Corollary 3.7.[16] Let (X,λ,ζσ) be a σ-NS and I be an ideal on X. Then,
(ⅰ)Every I-σ-exact set is I-θβσ-exact.
(ⅱ)Every I-θβσ-rough set is I-σ-rough.
We give the subsequent counterexample to show that Theorem 3.5 is incorrect in general.
Example 3.8.Let λ={(a,a),(b,b),(c,c),(a,b),(b,a),(x,a),(x,b),(x,c)} be a binary relation on X={a,b,c,x} and I={∅,{a},{b},{a,b}} be an ideal on X. Then, the r-neighborhoods of elements of X are Nr(a)=Nr(b)={a,b}, Nr(c)={c}, and Nr(x)={a,b,c}. Accordingly, we compute the following classes:
(ⅰ)ϑr={∅,X,{c},{a,b},{a,b,c}},
(ⅱ)ϑIr=ϑr∪{{a},{b},{a,c},{b,c},{c,x},{a,c,x},{b,c,x}}, and
On the one hand, one can see from Table 1 that H_I−θβr({a,c})={c}⊆H_Ir({a,c})={a,c} and ¯HIr({a,c,x})={a,c,x}⊆¯HI−θβr({a,c,x})=X. Therefore, there exist subsets V,W such that H_Ir(V)⊈H_I−θβr(V) and ¯HI−θβr(W)⊈¯HIr(W), which revokes the claims of Theorem 3.5. On the other hand, H_Ir({b,x})={b}⊆H_I−θβr({b,x})={b,x} and ¯HI−θβr({a,c})={a,c}⊆¯HIr({a,c})={a,c,x}.
This implies that the rough approximation operators generated by the methods introduced in [11] and [16] are independent of each other. Hence, the claims given in Theorem 3.5 are false.
Accordingly, one can note that Corollary 4.2 of [16] (mentioned here as Corollary 3.7) is also wrong. To confirm this matter, take a subset V={a}. By Table 1, we find that H_Ir(V)=¯HIr(V)=V, so V is I-r-exact. However, H_I−θβr(V)=∅≠H_I−θβr(V)={a}, so V is not I-θβr-exact. Equivalently, V is I-θβr-rough but not I-r-rough.
Consequently, Corollary 4.1 of [16] (mentioned here as Corollary 3.6) is false. To illustrate this conclusion, we provide Table 2 which is based on the computations of Table 1.
Table 2.
Boundary region and accuracy in relation to the methods of Hosny [11] and Nawar et al. [16].
On the one hand, one can see from Table 2 that BIr({a})=∅⊆BI−θβr({a})={a} and AI−θβr({a})=0<AIr({a})=1. Therefore, there exist subsets V,W such that BI−θβr(V)⊈BIr(V) and AIr(W)≮AI−θβr(W), which revokes the claims of Corollary 3.6. On the other hand, BI−θβr({c})=∅⊆BIr({c})={x} and AIr({c})=0<AI−θβr({c})=12.
This implies that the boundary regions and accuracy induced by the methods introduced in [11] and [16] are independent of each other. Hence, the claims given in Corollary 3.6 need not be true, in general.
Now, we put forward the correct relationships between the notions presented in Theorem 3.5 and Corollaries 3.6 and 3.7 in the following remark.
Remark 3.9.Let (X,λ,ζσ) be a σ-NS and I be an ideal on X. Then, there is no relationship between the following concepts inspired by the approaches of [11] and [16].
(ⅰ)The lower approximations H_Iσ and H_I−θβσ.
(ⅱ)The upper approximations ¯HIσ and ¯HI−θβσ.
(ⅲ)The boundary regions BIσ and BI−θβσ.
(ⅳ)The accuracy measures AIσ and AI−θβσ.
(ⅴ)I-σ-exact and I-θβσ-exact (I-σ-rough and I-θβσ-rough) sets.
In Figure 1, three observations were mentioned on page 2494 of [16] concerning the suggested application.
Figure 1.
Observations given on page 2494 of [16].
In what follows, we show that these observations are incorrect or inaccurate.
(ⅰ) The first and second items are not true since the approach of [16] is not the finest one. It is not stronger than the approach proposed by Hosny [11] as we illustrated in the aforementioned discussion. The appropriate description is that the methods of [11] and [16] are incomparable.
(ⅱ) The third item is inaccurate since the approach of [16] does not preserve all properties of the standard model of Pawlak (we mentioned these properties in Proposition 2.2) since it can be noted that the properties L5 and U6 are not satisfied. To confirm this point, take subsets V={a,b}, W={a,x}, Y={a}, and Z={x}. Then, H_I−θβσ(V)=V and H_I−θβσ(W)=W, whereas H_I−θβσ(V∩W)=∅ is a proper subset of H_I−θβσ(V)∩H_I−θβσ(W). Also, ¯HI−θβσ(Y∪Z)={a,b,x}, whereas ¯HI−θβσ(Y)∪¯HI−θβσ(Z)={a,x} is a proper subset of ¯HI−θβσ(Y∪Z). This means that rough set models provided in [16] violate some properties of the standard model of Pawlak, which disproves the third observation of Figure 1.
4.
Conclusions
In this note, we have showed invalid results and relationships introduced in [16]. With the help of an illustrative example, we have demonstrated that Theorem 4.1, Corollary 4.1, and items (2) and (4) of Corollary 4.2 given in [16] are incorrect. Also, we have concluded that the rough set paradigms proposed by Hosny [11] and Nawar et al. [16] are independent of each other; that is, they are incomparable. Then, we pointed out the concrete relationships between these rough set models. Finally, we have elucidated that three observations on page 2494 of [16] about the given application are false, as well as emphasized that there is no preponderance for Nawar et al.'s approach [16] over Hosny's approach [11] and vice versa in terms of improving the approximation operators and reducing the size of uncertainty.
Author contributions
Tareq M. Al-shami: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Writing-original draft, Writing-review and editing, Supervision. Mohammed M. Ali Al-Shamiri: Validation, Formal analysis, Investigation, Funding acquisition. Murad Arar: Validation, Formal analysis, Investigation, Funding acquisition. All authors have read and agreed to the published version of the manuscript.
Acknowledgments
The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP2/302/45.
Conflict of interest
The authors declare that they have no competing interests.
References
[1]
S. Ghosh, J. Heffernan, Influenza Pandemic Waves under Various Mitigation Strategies with 2009 H1N1 as a Case Study, PLoS One, 5 (2010), e14307. http://dx.doi.org/10.1371/journal.pone.0014307 doi: 10.1371/journal.pone.0014307
[2]
A. R. Tuite, D. N. Fisman, A. L. Greer, Mathematical modelling of COVID-19 transmission and mitigation strategies in the population of Ontario, Canada, Cmaj, 192 (2020), E497–E505.
[3]
B. M. Althouse, S. V. Scarpino, L. A. Meyers, J. W. Ayers, M. Bargsten, J. Baumbach, et al., Enhancing disease surveillance with novel data streams: challenges and opportunities, EPJ Data Sci., 4 (2015), 1–8. http://dx.doi.org/10.1140/epjds/s13688-015-0054-0 doi: 10.1140/epjds/s13688-015-0054-0
[4]
M. Roser, H. Ritchie, E. Ortiz-Ospina, J. Hasell, Coronavirus Pandemic (COVID-19), Our World in Data, 2020. Available from: https://ourworldindata.org/covid-cases
[5]
T. Oraby, O. Vasilyeva, D. Krewski, F. Lutscher, Modeling seasonal behavior changes and disease transmission with application to chronic wasting disease, J. Theor. Biol., 340 (2014), 50–59. http://dx.doi.org/10.1016/j.jtbi.2013.09.003 doi: 10.1016/j.jtbi.2013.09.003
[6]
A. Mummert, H. Weiss, L.-P. Long, J. M. Amigo, X.-F. Wan, A Perspective on Multiple Waves of Influenza Pandemics, PLoS One, 8 (2013), e60343. http://dx.doi.org/10.1371/journal.pone.0060343 doi: 10.1371/journal.pone.0060343
[7]
N. C. Grassly, C. Fraser, Seasonal infectious disease epidemiology, P. Roy. Soc. B - Biol. Sci., 273 (2006), 2541–2550. http://dx.doi.org/10.1098/rspb.2006.3604 doi: 10.1098/rspb.2006.3604
[8]
W. P. London, J. A. Yorke, Recurrent outbreaks of measles, chickenpox, and mumps.1. Seasonal-variation in contact rates, Am. J. Epidemiol., 98 (1973), 453–468. http://dx.doi.org/10.1093/oxfordjournals.aje.a121575 doi: 10.1093/oxfordjournals.aje.a121575
[9]
D. Glabska, D. Skolmowska, D. Guzek, Population-based study of the influence of the COVID-19 pandemic on hand hygiene behaviors - Polish adolescents' COVID-19 experience (PLACE-19) study, Sustainability, 12 (2020), 4930. https://doi.org/10.3390/su12124930 doi: 10.3390/su12124930
[10]
C. O'Connor, J. O. Weatherall, The misinformation age: How false beliefs spread, Yale University Press, 2019. https://doi.org/10.2307/j.ctv8jp0hk
[11]
E. Dubois, G. Blank, The echo chamber is overstated: the moderating effect of political interest and diverse media, Inf., commun. soc., 21 (2018), 729–745. https://doi.org/10.1080/1369118X.2018.1428656 doi: 10.1080/1369118X.2018.1428656
[12]
D. M. Kahan, A risky science communication environment for vaccines, Science, 342 (2013), 53–54. https://doi.org/10.1126/science.1245724 doi: 10.1126/science.1245724
[13]
B. Baumgaertner, B. J. Ridenhour, F. Justwan, J. E. Carlisle, C. R. Miller, Risk of disease and willingness to vaccinate in the United States: A population-based survey, PLoS med., 17 (2020), e1003354. https://doi.org/10.1371/journal.pmed.1003354 doi: 10.1371/journal.pmed.1003354
[14]
I. C.-H. Fung, S. Cairncross, How often do you wash your hands? A review of studies of hand-washing practices in the community during and after the SARS outbreak in 2003, Int. J. Environ. Heal. R., 17 (2007), 161–183. https://doi.org/10.1080/09603120701254276 doi: 10.1080/09603120701254276
[15]
S. Jamshidi, M. Baniasad, D. Niyogi, Global to USA country scale analysis of weather, urban density, mobility, homestay, and mask use on COVID-19, Int. J. Env. Res. Pub. Heal., 17 (2020), 7847. http://dx.doi.org/10.3390/ijerph17217847 doi: 10.3390/ijerph17217847
[16]
A. Matakos, E. Terzi, P. Tsaparas, Measuring and moderating opinion polarization in social networks, Data Min. Knowl. Disc., 31 (2017), 1480–1505. https://doi.org/10.1007/s10618-017-0527-9 doi: 10.1007/s10618-017-0527-9
[17]
J.-M. Esteban, D. Ray, On the mesaurement of polarization, econometrica, 62 (1994), 819–851. https://doi.org/10.2307/2951734 doi: 10.2307/2951734
[18]
P. Manfredi, A. d'Onofrio, Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, Springer, New York, 2017. http://dx.doi.org/10.1007/978-1-4614-5474-8
[19]
N. Perra, D. Balcan, B. Gonçalves, A. Vespignani, Towards a characterization of behavior-disease models, PloS one, 6 (2011), e23084. https://doi.org/10.1371/journal.pone.0023084 doi: 10.1371/journal.pone.0023084
[20]
L. Mao, Predicting self-initiated preventive behavior against epidemics with an agent-based relative agreement model, J. Artif. Soc. Social S., 18 (2015), 6. https://doi.org/10.18564/jasss.2892 doi: 10.18564/jasss.2892
[21]
A. Mummert, H. Weiss, Get the news out loudly and quickly: the influence of the media on limiting emerging infectious disease outbreaks, PloS one, 8 (2013), e71692. https://doi.org/10.1371/journal.pone.0071692 doi: 10.1371/journal.pone.0071692
[22]
P. Poletti, B. Caprile, M. Ajelli, A. Pugliese, S. Merler, Spontaneous behavioural changes in response to epidemics, J. Theor. Biol., 260 (2009), 31–40. https://doi.org/10.1016/j.jtbi.2009.04.029 doi: 10.1016/j.jtbi.2009.04.029
[23]
P. Poletti, M. Ajelli, S. Merler, Risk perception and effectiveness of uncoordinated behavioral responses in an emerging epidemic, Math. Biosci., 238 (2012), 80–89. https://doi.org/10.1016/j.mbs.2012.04.003 doi: 10.1016/j.mbs.2012.04.003
[24]
S. Maharaj, A. Kleczkowski, Controlling epidemic spread by social distancing: Do it well or not at all, BMC Public Health, 12 (2012), 1–16. https://doi.org/10.1186/1471-2458-12-679 doi: 10.1186/1471-2458-12-679
S. C. Anderson, A. M. Edwards, M. Yerlanov, N. Mulberry, J. E. Stockdale, S. A. Iyaniwura, et al., Estimating the impact of COVID-19 control measures using a Bayesian model of physical distancing, medRxiv, 2020. https://doi.org/10.1101/2020.04.17.20070086 doi: 10.1101/2020.04.17.20070086
[27]
N. M. Ferguson, D. Laydon, G. Nedjati-Gilani, N. Imai, K. Ainslie, M. Baguelin, et al., Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand, Imperial College London, (2020). https://doi.org/10.25561/77482 doi: 10.25561/77482
[28]
L. G. Nardin, C. R. Miller, B. J. Ridenhour, S. M. Krone, P. Joyce, B. O. Baumgaertner, Planning horizon affects prophylactic decision-making and epidemic dynamics, PeerJ, 4 (2016), e2678. https://doi.org/10.7717/peerj.2678 doi: 10.7717/peerj.2678
[29]
D. Weston, K. Hauck, R. Amlôt, Infection prevention behaviour and infectious disease modelling: a review of the literature and recommendations for the future, BMC Public Health, 18 (2018), 336–351. https://doi.org/10.1186/s12889-018-5223-1 doi: 10.1186/s12889-018-5223-1
[30]
R. C. Tyson, S. D. Hamilton, A. S. Lo, B. O. Baumgaertner, S. M. Krone, The Timing and Nature of Behavioural Responses Affect the Course of an Epidemic, B. Math. Biol., 82 (2020), 1–28. https://doi.org/10.1007/s11538-019-00684-z doi: 10.1007/s11538-019-00684-z
[31]
A. K. Misra, R. K. Rai, Y. Takeuchi, Modeling the control of infectious diseases: Effects of TV and social media advertisements, Math. Biosci. Eng., 15 (2018), 1315–1343. https://doi.org/10.3934/mbe.2018061 doi: 10.3934/mbe.2018061
[32]
MATLAB, 9.8.0.1323502 (R2020a), The MathWorks Inc., Natick, Massachusetts, 2020.
[33]
M. J. Keeling, L. Danon, Mathematical modelling of infectious diseases, Brit. Med. Bull., 92 (2009), 33–42. https://doi.org/10.1093/bmb/ldp038 doi: 10.1093/bmb/ldp038
[34]
A. Huppert, G. Katriel, Mathematical modelling and prediction in infectious disease epidemiology, Clin. Microbiol. Infec., 19 (2013), 999–1005. https://doi.org/10.1111/1469-0691.12308 doi: 10.1111/1469-0691.12308
[35]
M. W. Fong, H. Gao, J. Y. Wong, J. Xiao, E. Y. C. Shiu, S. Ryu, et al., Nonpharmaceutical Measures for Pandemic Influenza in Nonhealthcare Settings—Social Distancing Measures, Emerg. Infect. Dis., 26 (2020), 976–984. https://doi.org/10.3201/eid2605.190995 doi: 10.3201/eid2605.190995
[36]
A. E. Aiello, R. M. Coulborn, V. Perez, E. L. Larson, Effect of Hand Hygiene on Infectious Disease Risk in the Community Setting: A Meta-Analysis, Am. J. Public Health, 98 (2008), 1372–1381. https://doi.org/10.2105/ajph.2007.124610 doi: 10.2105/ajph.2007.124610
[37]
T. Jefferson, R. Foxlee, C. Del Mar, L. Dooley, E. Ferroni, B. Hewak, et al., Physical interventions to interrupt or reduce the spread of respiratory viruses: systematic review, BMJ, 336 (2007), 77–80. https://doi.org/10.1136/bmj.39393.510347.be doi: 10.1136/bmj.39393.510347.be
[38]
F. Chen, A. Griffith, A. Cottrell, Y.-L. Wong, Behavioral Responses to Epidemics in an Online Experiment: Using Virtual Diseases to Study Human Behavior, PLoS ONE, 8 (2013), e52814. https://doi.org/10.1371/journal.pone.0052814
Tareq M. Al-shami, M. Hosny, Murad Arar, Rodyna A. Hosny,
Cardinality rough neighborhoods via ideals with medical applications,
2025,
44,
2238-3603,
10.1007/s40314-024-03069-8
2.
Rani Sumaira Kanwal, Saqib Mazher Qurashi, Rizwan Gul, Alaa M. Abd El-latif, Tareq M. Al-shami, Faiza Tufail,
New insights into rough approximations of a fuzzy set inspired by soft relations with decision making applications,
2025,
10,
2473-6988,
9637,
10.3934/math.2025444
3.
Mona Hosny,
Idealizing Rough Topological Structures Generated by Several Types of Maximal Neighborhoods and Exploring Their Applications,
2025,
14,
2075-1680,
333,
10.3390/axioms14050333
Figure 1. Compartmental diagram for the coupled opinion and disease dynamics model. Variables appear inside boxes; rates appear above arrows. The Sn and Sp populations are the susceptibles exhibiting non-prophylactic and prophylactic behaviour, respectively. The I and R populations are the populations of infective and removed (dead, or recovered and immune). Note that the transition rates between the Si groups is affected by the size of the I group through the influence functions ωn(I) and ωp(I)
Figure 2. The prophylactic influence ωp(I) (solid black) and non-prophylactic influence ωn(I) (dashed black) plotted as functions of the proportion of the population that is infected. Susceptibles become prophylactic at an effective rate of (ωp−ωn) shown in blue (colour online). The net shift of susceptibles is toward prophylaxis once I>Icross, which is the point at which the dashed and solid black curves intersect and the effective rate curve (blue) is equal to zero
Figure 3. Intermediate disease transmission rate with fast opinion dynamics leads to multiple epidemic waves. Number of epidemic waves as a function of the baseline speed of the opinion dynamics, ω0, and the non-prophylactic infection rate, β0. Results were obtained with a grid spacing of Δβ0=Δω0=0.03. Dark blue: disease dies out; Light blue: one wave; Turquoise: two waves; Orange: three waves; Yellow: four waves
Figure 4. Second waves may or may not trigger opinion dynamics. Multiple waves are obtained when the speed of the opinion dynamics ω0 is increased relative to the speed of the infection dynamics β0. The populations are: Sn; blue, Sp; green, I; red (right axis), R; black. The red dashed line indicates Icross and the black dashed line indicates herd immunity. All parameters except ω0 are at their default values. When ω0=0.1033 (top plot) the rate of switching to prophylaxis results in a single epidemic wave with herd immunity achieved exactly. When ω0 is increased, the first wave does not confer herd immunity (R plateaus at a value less than the black dashed line), and a secondary wave of disease breaks out once a sufficient proportion of the population has reverted to non-prophylactic behaviour. The second wave can be small enough (middle plot), so that I<Icross throughout, or larger (bottom plot). On all three plots, the infectious population and the threshold Icross are plotted on a different scale for visibility (right axis)
Figure 5. Faster opinion dynamics always decrease peak size but can increase final size. Final epidemic size (top) and peak epidemic size (bottom) as functions of ω0, with c≤ω0≤0.5. Increasing ω0 increases the speed of the prophylactic response. Final size decreases with increasing ω0 until a new wave appears, at which point final size first increases with increasing ω0 before decreasing again. Peak size is monotonically decreasing with respect to ω0
Figure 6. Peak and final size both increase with faster disease dynamics, but only peak size always decreases with faster opinion dynamics. Final epidemic size (left) and peak size (right) as functions of the baseline rate of the opinion dynamics ω0, and the non-prophylactic infection rate β0. Other parameter values are as listed in Table 1
Figure 7. Polarization increases both peak and final size Contour plots of the peak and final size as a function of increasing polarization due to either an increase in the cost of prophylactic behaviour c (vertical axis) or a decrease in the responsiveness to disease prevalence through increasing k (horizontal axis). The results depend on the initial proportion of the population already prophylactic, Sp(0) (columns). The remaining parameter values were fixed at β0=0.2, a=2, and ω0=0.04