Dynamical analysis and optimal control of the developed information transmission model.

Information transmission significantly impacts social stability and technological advancement. This paper compares the phenomenon of "Super transmission" and "Asymptomatic infection" in COVID-19 transmission to information transmission. The former is similar to authoritative information transmission individuals, whereas the latter is similar to individuals with low acceptance in information transmission. It then constructs an S2EIR model with transmitter authority and individual acceptance levels. Then, it analyzes the asymptotic stability of information-free and information-existence equilibrium on a local and global scale, as well as the model's basic reproduction number, R0. Distinguished with traditional studies, the population density function and Hamiltonian function are constructed by taking proportion of "Super transmitter" and proportion of hesitant group turning into transmitters as optimization control variables. Based on the Pontryagin maximum principle, an optimal control strategy is designed to effectively facilitate information transmission. The numerical simulation corroborates the theoretical analysis results and the system's sensitivity to control parameter changes. The research results indicate that the authoritative "Super transmitter" has a beneficial effect on information transmission. In contrast, the "Asymptomatic infected individual" with poor individual acceptance level negatively affects information transmission.

1 Introduction

Information is a necessary component of human society’s development and has significant impacts on human life. In terms of the impact of information on human society, it can be broadly classified as either positive or negative. For instance, knowledge transmission [1, 2] and innovation capability transmission [3] are beneficial to social development, which constitutes positive information, whereas the spread of rumors [4, 5] and computer viruses [6-8] constitutes negative information. It can be classified into two broad categories of information transmission channels: contact transmission [9, 10] and network transmission [11, 12]. As a result, it is necessary to investigate the mechanisms of information transmission and control.

The information transmission characteristics are strikingly similar to those of infectious disease and rumor transmission [13, 14]. Thus, the information transmission models are frequently enhanced using infectious disease and rumor transmission models. The spread of infectious diseases was the first spreading problem studied by scholars. Infectious disease models that are frequently used include the SI model, the SIS model, and the SIR model [15-17]. Daley and Kendal developed a model of rumor transmission based on the classical infectious disease model. Scholars refer to this model as the DK model [18]. Based on the classical model of rumor transmission, scholars proposed the SEIR model with lurkers [19], the SIVR model with mutants [20], the SIVRS model with restorers [21], the SIHR model with forgetting and memory mechanisms [22], the SHIR model with hesitation mechanisms [23], and the SLIS model with age-structured [24].

In the last five years, scholars have conducted in-depth studies on information transmission from the following perspectives: (1) in terms of information type. Liu et al. developed a new SEIR model for heterogeneous networks to better understand the dynamics of information transmission in microblogs [25]. Wan et al. proposed the SIB model for analyzing the transmission of information about e-commerce discounts on a scale-free network [26]. Hosseini et al. developed the SEIRS-QV model with vaccination and isolation strategies to investigate malware transmission behavior in heterogeneous networks while also accounting for additional influencing factors such as user perception and network delay [27]. He et al. hypothesized that there is competition for various types of information in online social networks. They proposed the CISIR model to shed light on information competition and transmission [28]. Xiao et al. considered the dynamic changes in anti-rumor information from different aspects. They developed an evolutionary game theory-based driving mechanism for information, and ultimately proposed the SKIR model of rumor and anti-rumor competition [29]. (2) in terms of individual transmission, Zhang et al. examined the coexistence of rumors and authoritative information in social networks. They proposed the IS1 S2 C1 C2 R1 R2 model, which includes a super transmitter, a super authoritative information transmitter, a rumor suppressor, and a super authoritative information suppressor [30]. Li et al. believed that education had a significant impact on information transmission in a multi-language and heterogeneous network environment. As a result of this issue, the model was proposed [12]. Sang et al. hypothesized that social network users were heterogeneous and that individual perception behaviors influenced information transmission. As a result of this phenomenon, the SEIRD model was proposed that takes into account individual consciousness, social relationships, and knowledge levels [31]. Additionally, Fu et al. investigated the transmission of e-commerce discount information via social networks, believing that the super transmitter had a significant promotional effect on information and could increase potential users’ degree of acceptance of information. As a result, they proposed the SEIAR model, which takes into account super transmitters and potential users [32]. Yin et al. conducted an in-depth analysis of opinion leaders’ influence on the information transmission process using Weibo data and proposed an OD-SFI model [33]. Zhang et al. developed the SETQR model by considering time delay, trust, and forgetting mechanisms [34]. Additionally, the transmission of COVID-19’s information and the virus has been a hot topic for scholars in recent years [35, 36]. Abdo et al. analyzed and found the solution for the model of nonlinear fractional differential equations describing the deadly and most parlous virus, so-called coronavirus (COVID-19). The study discovered that the susceptibility decreases more rapidly at the lower fractional order of the derivative. Similarly, the increase in infections is also rapid, but in a smaller order [37]. Almalahi et al. used fractal-ABC type fractional differential equations by incorporating population self-protection behavior changes to study the dynamics of 2019-nCoV transmission [38], and investigated sufficient conditions of existence and uniqueness of positive solutions for a finite system of ψ-Hilfer fractional differential equations [39]. Jeelani et al. investigated a fractional-order mathematical model of COVID-19 [40]. These research findings are critical in the prediction of 2019-nCoV.

We have a certain understanding of the characteristics and modes of information transmission after analyzing the information transmission mechanism. Some scholars have conducted additional research on information transmission control to convert information into a controllable variable. Wang et al. developed a model of alcoholism’s transmission dynamics. They used prevention effectiveness, treatment costs, media coverage, and contact ratio as control variables to achieve optimal control of the alcoholism problem [41]. Alzahrani et al. also used a similar approach to research on smoking prevention strategies [42]. Huo discovered that psychological factors have a significant influence on information transmission and that scientific knowledge and official information can be used to guide people’s psychological activities. As a result, the two factors are used as control variables to mitigate the spread of false information [43]. It can generally use network control strategies such as forced silence, information labeling, administrator control, and controlling the relative density of individual nodes to suppress online information transmission [4, 44–46].

The scholars mentioned above have conducted extensive research on transmitting various types of information via various channels, including the transmission of rumors [4], false information [5], competitive information [28] and malware [27]. Scholars who consider information transmission from the perspective of individuals are usually based on the attitude of transmission individuals to an event [22–24, 31, 32]. Few scholars consider the transmission individuals’ attributes. Generally, most scholars primarily focus on controlling and transmitting negative information in networks [44, 46]. On the other hand, confidential information must be transmitted via the contact for information that must be taught in person, such as high technology or practical skills. At the same time, due to the unique characteristics of this type of information, as well as the requirements for the transmitter’s authority and the recipient’s ability to accept it, there is a shortage of literature on the subject. In addition, the spread of COVID-19 is a hot topic in current research. However, few scholars consider the phenomenon of “Asymptomatic infection” in the transmission model. The research on the transmission of the phenomenon of “Asymptomatic infection” is even less, especially in information transmission. Therefore, consider the phenomenon of “Asymptomatic infection” in the information transmission model and investigate the impact of “Asymptomatic infection” on the information transmission system.

As a result of the above considerations, this paper compares the phenomenon of “Super transmission” and “Asymptomatic infection” in COVID-19 transmission to information transmission. It then proposes an S2EIR model that incorporates the phenomena of “Super transmission” and “Asymptomatic infection”. During the transmission of COVID-19, the phenomena of “Super transmission” and “Asymptomatic infection” are common. The term “Super transmission” refers to the phenomenon in which highly contagious individuals are more likely to spread the virus to the majority of patients. The term “Asymptomatic infection” refers to a situation in which a patient has been infected with a virus and has become a carrier of the virus but does not exhibit obvious disease symptoms due to individual resistance and physical quality differences. Both of these phenomena frequently occur during information transmission. “Super transmitters” are analogous to authoritative individuals in information transmission. The information transmission by such individuals is more easily accepted. Whereas “Asymptomatic infected individuals” are analogous to hesitant individuals with a low level of acceptance. However, this group of people has certain infectivity in the transmission of COVID-19. However, individuals who conceal information rarely choose to share it with others during the transmission of information. At the same time, others are unaware that such individuals know the information. This is also the manifestation of “Asymptomatic” in information transmission. The optimal control strategy of information transmission is quantified by scientific methods to test these phenomena.

The remaining sections of this paper are organized as follows. In Section 2, an S2EIR model is constructed that considers the phenomena of “Super transmission” and “Asymptomatic infection”. In Section 3, it will propose the basic reproduction number, R0, and demonstrate the locally and globally asymptotically stability of information-free equilibrium and information-existence equilibrium. Section 4 proposes the optimal control existence and control strategy of information transmission. Section 5, through numerical simulation, analyzes the influence of parameter changes on information transmission, as well as the effect of the optimal control strategy. Section 6 analyzes sensitivity analysis on important control parameter changes in information transmission. The last section provides the conclusion.

2 The model

This paper discusses the concept of an open virtual community. The population size is variable at any point in time t, and the total population size can be represented by N(t). Each group can be classified into one of five categories. (1) The easy group, which is not exposed to information but is receptive to it, is represented by S(t). (2) E1 represents the hesitant group that has a high level of individual acceptance as a result of the influence of the “Super transmitter” E1(t). (3) The hesitant group is represented by E2(t), which exhibits the characteristics of “Asymptomatic infection” in the absence of “Super transmitter” influence. (4) I(t) represents the group that has a high level of individual acceptance and transmits information. (5) R(t) denotes the group without adoption that has a high level of individual acceptance due to external factors’ influence, or that has a low level of individual acceptance as a result of their own factors.

In the constructed model, the authoritative transmitter has a strong ability to interpret information, increasing the degree to which receivers adopt information. However, some individuals transmit information without the assistance of a “Super transmitter”. Due to the difference in an individual’s comprehension ability or learning level, information transmitters also have poor information interpretation. As a result, such individuals are unable to develop an effective understanding of information. Hence, these individuals are unable to disseminate information widely.

This is slightly different from the virus transmission phenomenon referred to as “Asymptomatic infection”. Although the “Asymptomatic infected individuals” of information transmission have received information, they are unable to use it effectively, let alone transmit it to other individuals, due to their insufficient understanding and acceptance of information.

In order to reflect the phenomena in information transmission. An S2EIR model is constructed that considers the phenomena of “Super transmission” and “Asymptomatic infection” in this paper. The model flow diagram is given in Fig 1.

Fig 1

The flow diagram of the model.

The parameters in the S2EIR model can be explained as follows:

The number of individuals in a social system generally varies over time. As a result, B is defined in this paper as the number of immigrants in the social system. Simultaneously, some individuals in the social system may withdraw due to various force majeure factors; thus, μ is used in this paper to define the removal rate;

When information begins to transmit in the system, the easy adopters will make contact with the transmitter with a certain probability, and the contact rate is defined as α. There is a subgroup of information transmitters known as “Super transmitters” that has a proportion of m. Simultaneously, the “Super transmitters” will increase the degree of transmission, converting the easy adopters into a hesitant group of strong recipients with a probability of α(1 + m);

When the hesitant group of strong recipients adopts the information through their own understanding and learning, this group will be transformed into the transmitters with a probability of β. On the contrary, if their learning ability is insufficient, they will become the hesitant people of “Asymptomatic infection” with the probability of ε;

Due to the difference in individual acceptance degrees, the two hesitant groups will become the non-acceptance groups with the probability of γ1 and γ2, respectively. Meanwhile, because the periodicity and timeliness of information constrain the transmitter, it will not transmit such information. Thus, the recovery rate is defined as λ.

The parameters of S2EIR model are summarized in Table 1.

Table 1

The parameter description of S2EIR model.

Parameter	Description
S(t)	The number of easy adopters at the time t.
E1(t)	The number of hesitating individuals that has a high level of individual acceptance as a result of the influence of the “Super transmitter” at the time t.
E2(t)	The number of hesitating individuals that exhibits the characteristic of “Asymptomatic infection” in the absence of “Super transmitter” influence at the time t.
I(t)	The number of transmittable individuals at the time t.
R(t)	The number of not adopters at the time t.
α	Information transmitting rate.
m	The proportion of “Super transmitters”.
β	Transition probability from state E1 to state I.
ε	Transition probability from state E1 to state E2.
γ 1	Transition probability from state E1 to state R.
γ 2	Transition probability from state E2 to state I.
μ	Removal rate per unit time.
B	The number of immigrants in the social system per unit time.

Based on the above analysis, we construct an S2EIR model that considers the phenomena of “Super transmission” and “Asymptomatic infection”. The system dynamics equations are described as follows: Where: and

3 Stability analysis of the model

Firstly, the basic reproduction number R0 of system (1) approach in the next generation matrix [47]. In this paper, R0 represents the number of next generation from a single information transmitter produced.

Let X = (E1, I, R, E2, S), then system (1) can be written as:

Calculate the Jacobian matrices of F(X) and V(X) in system (5) respectively, and then take the sub matrices corresponding to the first two variables (i.e. E1, I) directly related to the number of communicators. The results are as follows: where F and V represent the infection and transition matrices respectively [38]. By simple calculation, the inverse matrix of V can be obtained as:

The next generation matrix [47] is

Hence, the basic reproduction number R0 of system (1) is the spectral radius of the next generation matrix FV−1. Here, the spectral radius is the maximum value of characteristic root of FV−1. Therefore, R0 can be computed as:

Then, the information will disappear if R0 < 1, and the information-free equilibrium point of system (1) can be easily observed as .

While the information will be transmitted if R0 > 1, and the information-existence equilibrium point of system (1) can be obtained as . E* must satisfy the following equations:

Solving the equations of Eq (10), we can get the information-existence equilibrium point :

Theorem 3.1 If R0 < 1, the information-free equilibrium point of system (1) is locally asymptotically stable.

Proof of Theorem 3.1 The Jacobin matrix of system (1) at information-free equilibrium point can be written as:

It is easy to know that J(E0) has three negative eigenvalues Λ01 = Λ02 = −μ < 0, Λ03 = −(γ2 + μ)<0, and the other eigenvalues are the characteristic roots of |hE-J(E0)|, where:

The eigenvalues of Eq (16) can be obviously obtained as: and

If R0 < 1, so Λ04 < 0 and Λ05 < 0. Hence, the information-free equilibrium point of system (1) is locally asymptotically stable if R0 <1 based on the Routh-Hurwitz criterion.

Theorem 3.2 If Bα(2 + m)≤μ2, the information-free equilibrium point of system (1) is globally asymptotically stable.

Proof of Theorem 3.2 It is easy to know that S(t) + E1(t) + E2(t) + I(t) + R(t) = N(t) and satisfy . It illustrates that:

For t ≥ 0, the positive invariant set of system (1) can be written as:

Then, we construct the Lyapunov function L(t) = E1(t) + E2(t) + I(t) + R(t) and L′(t) can be computed as: obviously, L′(t)≤0 if and Bα(2 + m)≤μ2.

In addition, L′(t) = 0 holds if and only if S(t) = S0, E1 = E2 = I = R = 0. From system (1), it is known that E0 is the only solution in T when L′(t) = 0. Therefore, based on the Lyapunov-LaSalle Invariance Principle [48], it is shown that every solution of system (1) approach E0 for t → ∞. Hence, the information-free equilibrium point of system (1) is globally asymptotically stable.

Theorem 3.3 If R0 > 1, the information-existence equilibrium point of system (1) is locally asymptotically stable.

Proof of Theorem 3.3 The Jacobin matrix of system (1) at information-existence equilibrium point can be written as:

It is easy to know that two of the negative eigenvalues of system (22) are Λ11 = −μ and Λ12 = −(γ2 + μ), and the other eigenvalues are the characteristic roots of |hE-J(E*)|, where:

We can obviously obtain the eigenvalues of Eq (23), where:

Then we construct a cubic polynomial and replace the coefficient with a3, a2, a1, a0 to determine the other eigenvalues of system (22). Hence, Eq (24) can be rewritten as: where: and

The condition of information-existence equilibrium point is locally asymptotically stable and the conditions: (i) a3, a2, a1, a0 > 0 and (ii) a2 a1 − a3 a0 > 0 based on the Routh-Hurwitz criterion. It is easy to know that a3, a2 > 0.

If and R0 > 1, then a1, a0 > 0 and a2 a1 − a3 a0 > 0. In this case, the Routh-Hurwitz criterion are satisfied. Hence, the information-existence equilibrium point of system (1) is locally asymptotically stable.

Theorem 3.4 If R0 > 1, the information-existence equilibrium point of system (1) is globally asymptotically stable.

Proof of Theorem 3.4 We construct the Lyapunov function as: and

Because of the existence of , we can know that , i.e., .

Then, Eq (32) can be computed as:

Besides that, W′(t) = 0 holds if and only if . Hence, the information-existence equilibrium point of system (1) is globally asymptotically stable based on Lyapunov-LaSalle Invariance Principle [48].

4 The optimal control model

Two control objectives are proposed to expand the scope of information transmission based on the information transmission model established above. On the one hand, the number of information transmitters grows, and the information transmitted to the maximum. On the other hand, it has an increasing number of hesitant groups capable of comprehending information in depth and choosing to transmit it to expand the group of information transmitters. Hence, the two proportion constants m and β in the model are transformed into control variables m(t) and β(t). The control variable m(t) is used to control the proportion of authoritative “Super transmitters” in the crowd. More individuals in these groups will participate in information transmission through approaches such as policy guidance or macro-control. Thus, groups exposed to information are more receptive to it and can easily become new transmitters. The control variable β(t) is used to control the proportion of hesitant groups turning into transmitters. The acceptance degree of these groups can be improved using media publicity or strengthening education. At the same time, they can be influenced by “Super transmitters” and become information transmitters more easily. In this model, the control variables 0 ≤ m(t) ≤ 1 and 0 ≤ β(t) ≤ 1. While m(t) = 1 and β(t) = 1, it means that the control effect is optimal and the information can be transmitted to the greatest extent. On the contrary, while m(t) = 0 and β(t) = 0, it means that the control measures are completely ineffective.

Hence, we propose an objective function as: and satisfy the state system as:

The initial conditions for system (35) are satisfied: where: while U is the admissible control set. The time interval of control is between 0 and t. c1 and c2 are positive weight coefficients shown the control strength and importance of two control measures.

Theorem 4.1 An optimal control pair (m*, β*)∈U exists so that the function is established below:

Proof of Theorem 4.1 Let X(t) = (S(t), E1(t), E2(t), I(t), R(t)) and

The existence of an optimal pair must satisfy: (i) the set of control variables and state variables is nonempty, (ii) the control set U is convex and closed, (iii) the right-hand side of the state system is bounded by a linear function in the state and control variables, (iv) the integrand of the objective functional is convex on U, (v) there exist constants d1, d2 > 0 and ρ > 1 such that the integrand of the objective functional satisfies:

Conditions (i)-(iii) is clearly established, we just prove the condition (iv) and (v). One can easily obtain inequality:

Hence, condition (iv) is established. Then, for any t ≥ 0, there is a positive constant M which is satisfied |X(t)| ≤ M, therefore

Let and ρ = 2, then condition (v) is established. Hence, the optimal control can be realized.

Theorem 4.2 For the optimal control pair (m*, β*) of state system (35), there exist adjoint variables δ1, δ2, δ3, δ4, δ5 that satisfy:

With boundary conditions:

In addition, the optimal control pair (m*, β*) of state system (35) can be given by:

Proof of Theorem 4.2 Define a Hamiltonian function enlarged with penalty term to obtain the expression of optimal control system and optimal control pair. The Hamiltonian function enlarged can be written as: which the penalty term is ω(t)≥0, and it is satisfied that ω11(t)m(t) = ω12(t)(1 − m(t)) = 0 at optimal control m* and ω21(t)β(t) = ω22(t)(1 − β(t)) = 0 at optimal control β*.

Based on the Pontyragin maximum principle, the adjoint system can be written as: and the boundary conditions of adjoint system are

Let m* as an example to give the optimality conditions. One have and the optimal control formulae can be written as:

To obtain the final optimal control formulae without ω11 and ω12 need to consider the following three situations.

The first situation is that ω11(t) = ω12(t) = 0 in set {t|0

The second situation is that ω11(t) = 0 in set {t|m*(t) = 1}, then the optimal control formulae can be written as:

Due to ω12(t)≥0, it is shown that .

The third situation is that ω12(t) = 0 in set {t|m*(t) = 0}, then the optimal control formulae can be written as:

Based on the above situation, the final optimal control formulae of m*(t) can be written as . Similarly, the final optimal control formulae of β*(t) can be written as .

So far, we get the optimal control system includes state system (35) with the initial conditions S(0), E1(0), E2(0), I(0), R(0) and the adjoint system (43) with boundary conditions with the optimization conditions. The optimal control system can be written as: and

5 Numerical simulations

In this section, some numerical simulations will be given by the Rung-Kutta algorithm. The results of numerical simulation show that the rationality of the theoretical. Due to the range of the parameters has not been explicitly given in previous studies. Therefore, this paper combining with the values of basic regeneration number R0 and stability conditions, and give the numerical values of the parameters in the model.

In order to verify the locally and globally asymptotically stability of information-free equilibrium in Theorem 3.1 and Theorem 3.2. Let B = 1, α = 0.2, m = 0.3, μ = 0.3, γ1 = 0.5, γ2 = 0.7, β = 0.6, ε = 0.2, λ = 0.4. Through calculation, it can be concluded that R0 = 0.681 < 1. Fig 2 verifies the stability of the model and shows that variety groups eventually converge to 0 change over time.

Fig 2

The stability of information-free equilibrium E0 of system 1 with R0 < 1.

In order to verify the locally and globally asymptotically stability of information-existence equilibrium in Theorem 3.3 and Theorem 3.4. Let B = 1, α = 0.5, m = 0.3, μ = 0.3, γ1 = 0.5, γ2 = 0.7, β = 0.6, ε = 0.2, λ = 0.2. Through calculation, it can be concluded that R0 = 1.275 > 1. Fig 3 verifies the stability of the model and shows that variety groups eventually converge to E* change over time.

Fig 3

The stability of information-existence equilibrium E* of system 1 with R0 > 1.

In order to reveal the influence of optimal control pair (m*, β*) on variety groups when we adopt the optimal control strategy. We give the image of “optimal control (m = m*(t), β = β*(t))”, “middle control measure”, “single control measure” and “constant control measure” respectively.

Firstly, let B = 1, α = 0.5, μ = 0.3, γ1 = 0.5, γ2 = 0.7, ε = 0.2, λ = 0.2 and different control strategies are adopted at the same time. Fig 4A–4C illustrate the densities of E1(t), E2(t), I(t) change over time under different control strategies. The densities of E1(t), E2(t), I(t) show that “optimal control” is better than “middle control measure” batter than “single control measure” batter than “constant control measure” in Fig 4. It illustrates that the information is effectively extended when the optimal control strategy is adopted. However, the density of E2(t) should be negative correlation with information extension in the theoretical analysis. It is inconsistent with the phenomenon shown in Fig 4B. Therefore, the further analysis is needed.

Fig 4

The densities of (A) E1(t), (B) E2(t), (C) I(t) change over time under different control strategies, where B = 1, α = 0.5, μ = 0.3, γ1 = 0.5, γ2 = 0.7, ε = 0.2, λ = 0.2.

Then, let B = 5, α = 0.8, μ = 0.2, γ1 = 0.5, γ2 = 0.7, ε = 0.2, λ = 0.2 and different control strategies are adopted at the same time. Fig 5A–5C illustrate the densities of E1(t), E2(t), I(t) change over time under different control strategies. The density of E1(t) can achieve the optimal state when “middle control measure” is adopted shown in Fig 5A. The density of E2(t) is less than the “middle control measure” and “single control measure β” when optimal control strategy is adopted shown in Fig 5B. It illustrates that the density of E2(t) is gradually showed a negative correlation with information extension with the increase of B. And Fig 5C illustrates the density of I(t) can achieve the optimal state when optimal control strategy is adopted.

Fig 5

The densities of (A) E1(t), (B) E2(t), (C) I(t) change over time under different control strategies, where B = 5, α = 0.8, μ = 0.2, γ1 = 0.5, γ2 = 0.7, ε = 0.2, λ = 0.2.

But we further find that the densities of E1(t), E2(t), I(t) are approaching 0 when β = 0. Therefore, let β increases to 0.06 when the values of other parameters remain unchanged. Then, the densities of E2(t), I(t) change over time shown in Fig 6. Finally, the density of E2(t) reaches the minimum and the density of I(t) reaches the maximum when the optimal control strategy is adopted. The information has been effectively transmitted and the density of E2(t) acted the minimum hindrance.

Fig 6

The densities of (A) E2(t), (B) I(t) change over time under different control strategies, where β increases to 0.06 when the values of other parameters remain unchanged.

Based on the above analysis, the density of I(t) can always achieve the optimal state when taking the optimal control strategy in any case. Meanwhile, the influence of the density of E2(t) on information transmission decreases gradually when taking the optimal control strategy with the increase of B. These phenomena demonstrate that the information has been effectively transmitted when the optimal control strategy is adopted for optimal control pair (m*, β*).

6 Sensitivity analysis

To discuss the effect of control variables m and β on the basic reproductive number R0, we need to perform the sensitivity analysis of R0. According to the deduction above, we can figure out , thereby calculating:

As can be seen, R0 increases along with m. This indicates the authoritative “Super transmitters” existing in the social system can promote the transmission of information. In other words, the more authoritative “Super transmitters”, the more expansive the transmitting range of information. Meanwhile, R0 is also positively correlated with β. This demonstrates that when the hesitant individuals affected by the authoritative “Super transmitters” have a higher acceptance level of information, their probability of translating into information transmitters will be higher, and the transmission range of information will be wider.

The sensitivity analysis of R0 is shown in Fig 7.

Fig 7

The sensitivity analysis of the basic reproduction number R0.

7 Conclusions

To investigate the effects of “Super transmission” and “Asymptomatic infection” on information transmission, this paper developed an S2EIR model that included the “Super transmitters” and “Asymptomatic carriers”, which can reflect the transmitters’ authoritativeness and individual acceptance level. After analyzing the equilibrium point and stability of the model and verifying the existence of optimal control, an optimal control strategy was proposed, and its effectiveness was validated through numerical simulation. Ultimately, the sensitivity of the optimal control parameters was analyzed.

Through the research in this paper, we have come to the following results:

The “Super transmission” and “Asymptomatic infection” phenomena that have been valued by the transmission of COVID-19, which still have a significant impact on information transmission. We conduct a quantitative comparison of these two concepts in terms of information transmission. In contrast to traditional studies, our optimal control strategies are based on the optimal value of control variables;

The authoritative “Super transmitters” can promote the transmission of information and increase the acceptance level of information by hesitant individuals, making contact with those “Super transmitters” prone to becoming the transmitters of information. Thus, information transmission can be accelerated by increasing the proportion of “Super transmitters” among the population;

Due to the unique nature of information, it is difficult for individuals who have not come into contact with “Super transmitters” to receive information through self-learning. Therefore, their acceptance level of information is generally low. In this case, these individuals may not be well-informed and thus are not active in information transmission;

The individual’s self-learning and understanding ability guarantee the individuals to receive and transmit information. A greater proportion of hesitant individuals will emerge as information transmitters when their acceptance level of information is improved. To this end, we can strengthen education or popularize knowledge to improve the understanding abilities of hesitant individuals.

The transmission of knowledge, technology, and skill-based information has contributed positively to social development. Expanding the transmission of such information and encouraging people to accept it will have a far-reaching impact on social progress. Our research discovered that information could be effectively disseminated by increasing the proportion of authoritative “Super transmitters” in the social system and improving individuals’ learning abilities through education or knowledge popularization.

We will focus our future research on the following three aspects. First, we will consider the transmission of multi-information in the social system. Second, we will further study the influence of authoritative “Super transmitters” on transmitting various types of information over the same period. Finally, the interference mechanism of multi-information will be introduced to determine an approach for the stable transmission of information within the social system.

14 Mar 2022

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29 Mar 2022

Dear Editors and Reviewers:

Thank you very much for your letter and the reviewers’ comments concerning our manuscript entitled "Dynamical analysis and optimal control for an information transmission model considering the phenomena of 'Super Transmission' and 'Asymptomatic Infection'" (ID: PONE-D-22-05396). These comments are indeed extremely helpful for revising and improving our paper, as well as the important guiding significance to our researches. We have studied these comments with care and have made correction which we hope meet with approval. The main corrections in the paper and the responds to the reviewer’s comments are as flowing:

Responds to the reviewer’s comments:

Reviewer #1:

1. Response to comment: (Make sure all equations are properly cited.)

Response: Thank you for helping us to find the errors and omissions. We are very sorry for not being careful to make this obvious mistake. According to your suggestions and requirements, we have corrected the mistakes cited in the original manuscript. At the same time, according to the whole comments, we have added some equations. Finally, we have checked all equations cited in the manuscript, and then make sure all equations are properly cited in our revised manuscript.

2. Response to comment: (Introductions need to be improved with more details of current work. Please mention which challenges you face to prove the results.)

Response: Thank you very much for your incisive comments and thorough reminders. I'm very sorry for the inaccurate and unclear expression due to our negligence. According to your comments and suggestions, we have improved with more details of current work and mentioned the challenges we face to prove the results in our revised manuscript(Page 3, Line 82-99). The details as following:

“The scholars mentioned above have conducted extensive research on transmitting various types of information via various channels, including the transmission of rumors[4], false information[5], competitive information[28] and malware[27]. Scholars who consider information transmission from the perspective of individuals are usually based on the attitude of transmission individuals to an event[22-24,31,32]. Few scholars consider the transmission individuals' attributes. Generally, most scholars primarily focus on controlling and transmitting negative information in networks[44,46]. On the other hand, confidential information must be transmitted via the contact for information that must be taught in person, such as high technology or practical skills. At the same time, due to the unique characteristics of this type of information, as well as the requirements for the transmitter's authority and the recipient's ability to accept it, there is a shortage of literature on the subject. In addition, the spread of COVID-19 is a hot topic in current research. However, few scholars consider the phenomenon of "Asymptomatic infection" in the transmission model. The research on the transmission of the phenomenon of "Asymptomatic infection" is even less, especially in information transmission. Therefore, consider the phenomenon of "Asymptomatic infection" in the information transmission model and investigate the impact of "Asymptomatic infection" on the information transmission system.”

3. Response to comment: (What is the difference between stability and asymptotic stability?)

Response: Thanks for your sincere comments and reminders, which are indeed to the point. Stability and asymptotic stability are important theories in the field of differential equations. It has a very important application in the study of the dynamic system in infectious disease transmission or information transmission. These theories are also applied in our paper. Therefore, we consult some materials to learn the difference between stability and asymptotic stability.

If for any given \\varepsilon>0 and t_0\\geq0 all exist \\delta=\\delta(\\varepsilon,t_0)>0, so that as long as x0 is satisfied:

x0-x1<δ ,

then we can get

x(t,t0,x0)-φ(t,t0,x1)<ε .

For every t\\geq t_0 establish, then the solution x=\\varphi(t,t_0,x_1) of differential equation \\frac{dx}{dt}=f(t,x) is said to be stability.

Suppose that x=\\varphi(t,t_0,x_1) is stable, and there is \\delta_1(0<\\delta_1<\\delta), so that as long as x0 satisfies:

x0-x1<δ1 ,

then we can get

limt→∞x(t,t0,x0)-φ(t,t0,x1)=0 .

Then the solution x=\\varphi(t,t_0,x_1) of differential equation \\frac{dx}{dt}=f(t,x) is said to be asymptotic stability.

It can be seen that asymptotic stability is a special case of stability. Asymptotic stability is more stringent than stability for the system. Stability requires that the state trajectory converge to a certain range near the equilibrium point. The asymptotic stability requires that the state trajectory converges to the equilibrium point on the premise of stability. The differential equation constructed in this paper has an equilibrium point, and the system finally converges to the equilibrium point. Therefore, the model in this paper meets the requirements of asymptotic stability. We think the model constructed in this paper needs the system to reach an asymptotically stability state. Asymptotic stability can better reflect the state of the model. Your comments have prompted us to deepen our understanding of the concepts of stability and asymptotic stability.

4. Response to comment: (It is very important to know the advantages of the present study? So, the introduction needs to improve by highlighting the main advantages.)

Response: Thanks for your sincere comments and reminders. Your suggestions are of great help to the improvement of the paper. According to your comments and suggestions, we have improved by highlighting the main advantages in our revised manuscript(Page 3, Line 100-Page 4, Line 119). The details as following:

“As a result of the above considerations, this paper compares the phenomena of "Super transmission" and "Asymptomatic infection" in COVID-19 transmission to information transmission. It then proposes an S2EIR model that incorporates the phenomena of "Super transmission" and "Asymptomatic infection". During the transmission of COVID-19, the phenomena of "Super transmission" and "Asymptomatic infection" are common. The term "Super transmission" refers to the phenomenon in which highly contagious individuals are more likely to spread the virus to the majority of patients. The term "Asymptomatic infection" refers to a situation in which a patient has been infected with a virus and has become a carrier of the virus but does not exhibit obvious disease symptoms due to individual resistance and physical quality differences. Both of these phenomena frequently occur during information transmission. "Super transmitters" are analogous to authoritative individuals in information transmission. The information transmission by such individuals is more easily accepted. Whereas "Asymptomatic infected individuals" are analogous to hesitant individuals with a low level of acceptance. However, this group of people has certain infectivity in the transmission of COVID-19. However, individuals who conceal information rarely choose to share it with others during the transmission of information. At the same time, others are unaware that such individuals know the information. This is also the manifestation of "Asymptomatic" in information transmission. The optimal control strategy of information transmission is quantified by scientific methods to test these phenomena.”

5. Response to comment: (What are the means of the matrixes F and V in equation 6, and how do obtain them? please explain it.)

Response: Thank you very much for your comments! In the original manuscript, perhaps our expression is not comprehensive and careful enough. We would like to make a further explanation on this point.

It is explained in the reference (https://doi.org/10.1016/j.rinp.2021.104045) you recommend, F and V represent the infection and transition matrices respectively. Meanwhile, we have modified and added the method of obtaining the matrixed F and V in the revised manuscript (Page 6, Line 190-193). The details as following:

“Calculate the Jacobian matrices of F(X) and V(X) in system (5) respectively, and then take the sub matrices corresponding to the first two variables (i.e. E1, I) directly related to the number of communicators. The results are as follows:

F=\\left(\\begin{matrix}0&\\alpha(1+m)S\\\\\\beta&0\\\\\\end{matrix}\\right)\\ ,\\ V=\\left(\\begin{matrix}\\beta+\\gamma_1+\\varepsilon+\\mu&0\\\\0&\\lambda+\\mu\\\\\\end{matrix}\\right)\\ , (6)

where F and V represent the infection and transition matrices respectively[38].”

6. Response to comment: (How obtain equation 7? add some information.)

Response: Thank you very much for the attentive and earnest comments! We do regret for the lack of smooth logical narration due to the omission of necessary steps and instructions in the original manuscript, which has brought you a dissatisfactory reading experience.

We have modified and added relevant parts in the revised manuscript (Page 6, Line 193-198). The details as following:

(Tip: Due to the addition of equation, equation 7 in the original manuscript has changed to equation 9 in the revised manuscript.)

“By simple calculation, the inverse matrix of V can be obtained as:

V^{-1}=\\left(\\begin{matrix}\\frac{1}{\\beta+\\gamma_1+\\varepsilon+\\mu}&o\\\\o&\\frac{1}{\\lambda+\\mu}\\\\\\end{matrix}\\right)\\ . (7)

The next generation matrix[47] is

{FV}^{-1}=\\left(\\begin{matrix}0&\\frac{\\alpha(1+m)S}{\\lambda+\\mu}\\\\\\frac{\\beta}{\\beta+\\gamma_1+\\varepsilon+\\mu}&0\\\\\\end{matrix}\\right)\\ . (8)

Hence, the basic reproduction number R0 of system (1) is the spectral radius of the next generation matrix FV-1. Here, the spectral radius is the maximum value of characteristic root of FV-1. Therefore, R0 can be computed as:

R_0=\\rho({FV}^{-1})=\\sqrt{\\frac{B\\alpha\\beta(1+m)}{\\mu(\\beta+\\gamma_1+\\varepsilon+\\mu)(\\lambda+\\mu)}}\\ . (9)’’

7. Response to comment: (In system 8, replace S; E1; E2; I; R with S*; E_1^\\ast; E_2^\\ast; I*; R*.)

Response: Thank you for helping us to find the omissions. We are very sorry for not being careful to make this obvious mistake. According to your suggestions and requirements, we have replaced S; E1; E2; I; R with S*; \\mathbit{E}_\\mathbf{1}^\\ast; \\mathbit{E}_\\mathbf{2}^\\ast; I*; R* in system 8 in our revised manuscript (Page 6, Line 203).

(Tip: Due to the addition of equation, system 8 in the original manuscript has changed to system 10 in the revised manuscript.)

8. Response to comment: (How was equation 29 chosen in this form? which conditions followed for it?)

Response: Thanks for your sincere comments and reminders, which are indeed to the point. According to our understanding，there is no unified standard and rule for constructing Lyapunov function. According to the conditions to be satisfied by Lyapunov function, we have made a lot of attempts. The method we choose just satisfy the conditions of Lyapunov function. Therefore we construct the Lyapunov function as:

(Tip: equation 29 has been changed to equation 31 due to the change of equation number. We apologize for the inconvenience.).

Wt=[St-S*+E1t-E1*+E2t-E2*+It-I*⬚+Rt-R*]2, (31)

and

\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ W\\prime\\left(t\\right)=2\\left[\\left(S\\left(t\\right)-S^\\ast\\right)+\\left(E_1\\left(t\\right)-E_1^\\ast\\right)+\\left(E_2\\left(t\\right)-E_2^\\ast\\right)+\\left(I\\left(t\\right)-I^\\ast\\right)\\ \\ \\ \\ \\ \\ \\ \\ \\ +\\left(R\\left(t\\right)-R^\\ast\\right)\\right]\\left[S\\prime\\left(t\\right)+E_1\\prime\\left(t\\right)+E_2\\prime\\left(t\\right)+I\\prime\\left(t\\right)+R\\prime\\left(t\\right)\\right]=2\\left[\\left(S\\left(t\\right)-S^\\ast\\right)+\\left(E_1\\left(t\\right)-E_1^\\ast\\right)+\\left(E_2\\left(t\\right)-E_2^\\ast\\right)+\\left(I\\left(t\\right)-I^\\ast\\right)+\\left(R\\left(t\\right)-R^\\ast\\right)\\right]\\left[B-\\mu S-\\mu E_1-\\mu E_2-\\mu I-\\mu R\\right]\\ .\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (32)

Because of the existence of E^\\ast\\left(S^\\ast,E_1^\\ast,E_2^\\ast,I^\\ast,R^\\ast\\right), we can know that B-\\mu S^\\ast-\\mu{E_1}^\\ast-\\mu{E_2}^\\ast-\\mu I^\\ast-\\mu R^\\ast=0, so that B=\\mu S^\\ast+\\mu{E_1}^\\ast+\\mu{E_2}^\\ast+\\mu I^\\ast+\\mu R^\\ast.

Then, Eq.(32) can be written as:

W\\prime\\left(t\\right)=2\\left[\\left(S\\left(t\\right)-S^\\ast\\right)+\\left(E_1\\left(t\\right)-E_1^\\ast\\right)+\\left(E_2\\left(t\\right)-E_2^\\ast\\right)+\\left(I\\left(t\\right)-I^\\ast\\right)\\ \\ \\ \\ \\ \\ \\ \\ \\ +\\left(R\\left(t\\right)-R^\\ast\\right)\\right]\\left[\\mu S^\\ast+\\mu{E_1}^\\ast+\\mu{E_2}^\\ast+\\mu I^\\ast+\\mu R^\\ast-\\mu S-\\mu E_1-\\mu E_2-\\mu I-\\mu R\\right]=-2\\left[\\left(S-S^\\ast\\right)+\\left(E_1-E_1^\\ast\\right)+\\left(E_2-E_2^\\ast\\right)+\\left(I-I^\\ast\\right)+\\left(R-R^\\ast\\right)\\right]^2\\le0\\ .\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (33)

Besides that, W\\prime\\left(t\\right)=0 holds if and only if S\\left(t\\right)=S^\\ast,E_1\\left(t\\right)=E_1^\\ast,E_2\\left(t\\right)=E_2^\\ast,I\\left(t\\right)=I^\\ast,R\\left(t\\right)=R^\\ast. Hence, the information-existence equilibrium point E^\\ast\\left(S^\\ast,E_1^\\ast,E_2^\\ast,I^\\ast,R^\\ast\\right) of system (1) is globally asymptotically stable based on Lyapunov-LaSalle Invariance Principle[48].

We choose a simple and effective way to construct Lyapunov function in this paper. We will try our best to study the construction of Lyapunov function by other methods in the future.

9. Response to comment: (I suggest to improve the introduction section by studying some useful recent papers/books, for example: https://doi.org/10.1016/j.chaos.2020.109867, https://doi.org/10.1016/j.chaos.2021.110931, https://doi.org/10.1016/j.rinp.2021.104045.)

Response: Thanks for your comments and suggestions. We have downloaded and read these references. These references are closely related to the research of this paper, and they deepen our understanding. It is more helpful for us to clearly understand the characteristics of virus transmission and the choice of methods. Therefore, we have added refers to these useful recent papers in the revised manuscript (Page 2, Line 56-Page 3, Line 67). The details are as follows:

“Abdo et al. analyzed and found the solution for the model of nonlinear fractional differential equations describing the deadly and most parlous virus, so-called coronavirus (COVID-19). The study discovered that the susceptibility decreases more rapidly at the lower fractional order of the derivative. Similarly, the increase in infections is also rapid, but in a smaller order[37]. Almalahi et al. used fractal-ABC type fractional differential equations by incorporating population self-protection behavior changes to study the dynamics of 2019-nCoV transmission[38], and investigated sufficient conditions of existence and uniqueness of positive solutions for a finite system of \\varphi-Hilfer fractional differential equations[39]. Jeelani et al. investigated a fractional-order mathematical model of COVID-19[40]. These research findings are critical in the prediction of 2019-nCoV.”

10. Response to comment: (Update references according to the style of journal.)

Response: Thanks very much for your comments. According to your request, we have checked upon and updated references according to the style of journal. And we have already put them right in the revised version. As these minor errors are more, we have not marked each and every of them corresponding page and line numbers in details.

Special thanks to you for your good comments!

Reviewer #2:

1. Response to comment: (There are some typos and grammatical errors in some parts of this text, especially in the introduction section. Please double-check all sentences and correct all sentences that need to be corrected grammatically.)

Response: Thanks very much for your comments. According to your request, we have checked upon a number of grammatical and typo mistakes once again. And we have already put them right in the revised version. As these minor errors are more, we have not marked each and every of them corresponding page and line numbers in details.

2. Response to comment: (Please pay attention to all punctuation marks in the text.)

Response: Thanks very much for your comments. Your comments are conducive us to be more careful. It is more helpful for us to improve our good scientific research literacy. According to your request, we have checked upon all punctuation once again. And we have already put them right in the revised version. As these minor errors are more, we have not marked each and every of them corresponding page and line numbers in details.

3. Response to comment: (Update the recent references related to this work; Chaos, Solitons & Fractals, 135, 109867.‏ https://doi.org/10.1016/j.chaos.2020.109867; Axioms 2021, 10(3), 228; https://doi.org/10.3390/axioms10030228.)

Response: Thanks for your comments and suggestions. We have downloaded and read these references. These references are closely related to the research of this paper, and they deepen our understanding. It is more helpful for us to clearly understand the characteristics of virus transmission and the choice of methods. Therefore, we have added refers to these useful recent references in the revised manuscript (Page 2, Line 56-Page 3, Line 67). The details are as follows:

“Abdo et al. analyzed and found the solution for the model of nonlinear fractional differential equations describing the deadly and most parlous virus, so-called coronavirus (COVID-19). The study discovered that the susceptibility decreases more rapidly at the lower fractional order of the derivative. Similarly, the increase in infections is also rapid, but in a smaller order[37].

Jeelani et al. investigated a fractional-order mathematical model of COVID-19[40]. These research findings are critical in the prediction of 2019-nCoV.”

4. Response to comment: (I suggest that the authors amend the article title as follows:

"Dynamical analysis and optimal control of the developed information transmission model".)

Response: Thanks for your sincere comments and reminders. Compared with our original title, the title you proposed is more concise and more widely applicable. According to your suggestions and requirements, we have amended the article title as "Dynamical analysis and optimal control of the developed information transmission model" in our revised manuscript.

Special thanks to you for your good comments!

We tried our best to improve the manuscript and made some changes in the manuscript. These changes will not influence the content and framework of the paper.

We appreciate for Editors/Reviewers’ warm work earnestly, and hope that the correction will meet with approval.

Once again, thank you very much for your comments and suggestions.

Submitted filename: Response to Reviewers.pdf

Click here for additional data file.

27 Apr 2022

Dynamical analysis and optimal control of the developed information transmission model

PONE-D-22-05396R1

Dear Dr. Hou,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

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Kind regards,

Mohammed S. Abdo

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

The authors did their best to improve the manuscript and made some changes to the manuscript according to the reviewers' reports. These changes did not affect the content and frame of the paper.

Therefore, I decided to accept the manuscript provided that the authors delete some of the references suggested by the reviewers and keep the references related to the topic of the paper.

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

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Reviewer #1: All comments have been addressed

Reviewer #2: All comments have been addressed

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The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #2: Yes

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Reviewer #2: N/A

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Reviewer #1: The authors compared the phenomenon of "Super transmission" and "Asymp-

tomatic infection" in COVID-19 transmission to information transmission. The

former is similar to authoritative information transmission individuals, whereas

the latter is similar to individuals with low acceptance in information trans-

mission. Also, they constructed an S2EIR model with transmitter authority

and individual acceptance levels. Then, they analyzed the asymptotic stability

of information-free and information-existence equilibrium on a local and global

scale, as well as the model’s basic reproduction number, R0. Based on the Pon-

tryagin maximum principle, an optimal control strategy is designed to e¤ectively

facilitate information transmission. The numerical simulation corroborates the

theoretical analysis results and the system’s sensitivity to control parameter

changes. This work is well written and the results are new

The authors take into account the suggestions. All comments have been addressed. I recommend the publication of the paper.

Reviewer #2: Dear Authors, I carefully reviewed the revised manuscript PONE-D-22-05396R1 entitled “Dynamical analysis and optimal control of the developed information transmission model” and I found the following:

This paper is very interesting to read. The analysis in this paper is very good. The results are original and present a good degree of novelty. The techniques in this paper present are well-employed to obtain the intended results, and the proofs are correct.

So, the revised manuscript is suitable for publication.

**********

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Reviewer #1: No

Reviewer #2: No

12 May 2022

PONE-D-22-05396R1

Dynamical analysis and optimal control of the developed information transmission model

Dear Dr. Hou:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

If we can help with anything else, please email us at plosone@plos.org.

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on behalf of

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