Research on Nonlinear Infectious Disease Models Inﬂuenced by Media Factors and Optimal Control

In this article, a mathematical model is developed to specify a speciﬁc function to describe the degree of disease control by mediating factors. The Lambert W function is used to convert the system deﬁned by implicit functions into explicit functions. We analyze the dynamics of the deﬁned segmented smooth system and verify the correctness of the theoretical analysis through numerical simulation. Through research, it is found that media inﬂuences can delay the peak of the epidemic and lead to a reduction in the scale of the epidemic. Then, we ﬁnd that diﬀerent control have a certain eﬀect on the scale of the epidemic, our analysis shows that implementing dual is the most eﬀective approach in limiting the spread of diseases.


Introduction
2019 coronavirus disease (COVID-19) originates from a new type of severe acute respiratory syndrome coronavirus [1].According to existing case data, some patients may experience respiratory and digestive symptoms such as nasal congestion, runny nose, diarrhea [2,3].In severe cases, they may rapidly progress to acute respiratory distress syndrome, septic shock, difficult to correct metabolic acidosis, coagulation disorders, and multiple organ failure [3].It should be noted that elderly people and those with chronic underlying diseases have a poorer prognosis, and the symptoms of youngster cases are relatively mild [4,5].During the sudden outbreak of the epidemic, for avoiding cross infection of patients with multiple strains [6], it is important to practice self-protection, wash hands frequently, maintain hygiene habits, raise health awareness, correctly wear disposable masks, avoid contact with infected individuals, and avoid going to dangerous crowded and enclosed places [7].
Media factors play a crucial role in the outbreak of an epidemic [8,9].They can detect and report on the epidemic early on, provide a warning, and use public opinion to take effective measures to eliminate potential crises before they escalate [10,11].By disseminating information about the epidemic, the media can help the public understand its severity and better control its development [12].During the initial phases of the SARS outbreak in 2003, speculations circulated amongst the general public, causing widespread social unrest due to media coverage and insufficient dissemination of information regarding SARS prevention and treatment [13,14].In the wake of COVID-19, the media disseminated information to assist medical personnel and warn the public to prioritise personal protection, contributing significantly to guiding public perception.This statement highlights the need to explore how media coverage impacts the transmission and management of infectious diseases.
Mathematical models have the ability to predict disease development trends through dynamic analysis [15,16,17,18,19].Currently, numerous studies have established mathematical models to simulate the dynamics of COVID-19 [20,21,22].Jia  and numerical simulation reveal that social distance has a significant impact on reducing the spread of COVID-19 [22].Mathematical models enable the simulation of the impact of various factors on diseases and offer optimal control strategies for infectious diseases, thus providing valuable ideas for disease control.
The remaining structure of this article is as follows: in the second section, a SIR model is proposed, which includes media related factors.Using the properties of the Lambert W function to convert the system into an explicitly defined system through an implicit function.This system is a piecewise smooth system that can analyze the dynamic of the system.In the third section, we study the dynamics of piecewise smooth system, as well as the existence and local stability of endemic equilibrium points.Finally, the impact of various control methods on disease prevalence is modeled using optimal control theory.The numerical simulation results indicates that verifying the theoretical analysis results, and discovering that the vector function caused by infected individuals and infection rates can affect the scale of epidemic outbreaks.And our analysis shows that implementing dual measures is the most effective way to limit the spread of diseases, this may provide clues for disease control.

Model description
We consider the dynamics in susceptible population S(t), infected population I(t) and convalescent population R(t).The exponential decreasing factor of I(t) is used to express the reducing effect of media influence on the infection rate of infectious diseases, similar to reference [12], the function is defined as , where N (I, dI dt ) is shown as follows Among them, m 1 and m 2 are non-negative parameters, which are used to represent the impact of media factors on media coverage cases and change rates.
The N (t) = N (I, dI dt ) function is guaranteed to be nonnegative.For the R(t) population after recovery, they will no longer impose risk on susceptible individuals.In most studies, the model is established by assuming that the total population is constant or satisfies exponential growth [23,24].A new epidemic dynamic model is obtained as follows In model ( 1), all parameters are positive based on theoretical facts, and the meaning of each parameter is as follows: a is the inherent growth rate of population, k is the population carrying capacity of a given region.We hypothesis b is the basic propagation coefficient, f (I) = be −M (I, dI dt ) is the term of contact and transmission, which measures the spread of the virus from an infected person to a susceptible individual.α is the mortality rate associate with the disease, β is the natural death rate, or death caused by the sequelae of disease recovery, γ is the rate of recovery from infection.To simplify, let n = α + β + γ, and It can be observed that the form of equation ( 2) is quite complex, then we introduce the definition and properties of Lambert W function, which is Definition 1. [25] The Lambert W function is the inverse of the function f (z) = ze z and satisfies the following conditions By definition, we have .
Using the definition of Lambert W function, we can obtain Therefore N (t) reads Then, we study that N 1 (t) is greater than zero, for this we consider is N 1 (t) we use the properties of Lambert W function, then we obtain Because N 1 (t) > 0 is strictly monotone for S, which yields, In order to study the properties of the epidemic model, we remove the dynamics of the individual R(t), and system ( 1) is transformed into system ( 8) with Equations ( 8) and ( 9) indicate that the system has a susceptibility threshold, and there is no influence of media factors below the threshold.Above the threshold, the media has a certain role in reducing the spread of disease, so the media has an impact on the disease to a certain extent.The susceptibility threshold is analyzed in the following content.
We set P (Z) = S − S q with X = (S, I) T , then the equations ( 8) and ( 9) become a non-smooth system with and the system (1) invariant set is > 0 holds, we have S q > 0; if S q < 0, the set A 1 becomes an empty set.Then the non-smooth system (10) becomes smooth system Ẋ(t) = P A2 (X).

The dynamics of piecewise smooth system
In this part, we analyze the different areas S A1 and S A2 of the system to study the global dynamics of the system (10).

The dynamics of S A1
The system dynamics of S A1 has a disease-free equilibrium point E * 0 , which is (k, 0), then 0 is locally stable in the system of S A1 .The interior equilibrium point Ẽ1 = ( S1 , Ĩ1 ) of S A1 , which is exists only if R 0 > 1, and note that S1 > S q holds, which means that equilibrium Ẽ1 is located in region A 2 , so it is a virtual equilibrium.
The dynamics of S A2 120 For the smooth system S A2 , the equation is as follows Due to N 1 (t) contains the Lambert W function, which is difficult to study its dynamics through theoretical analysis.The disease-free equilibrium of system ( 11) is E * 0 = (k, 0), which is consistent with the corresponding of system S A1 .Existence of endemic equilibrium Proof: In the function N 1 (t), we define , by using the properties of Lambert W function, we can get Substituting equation (12) into model (11), we have If the second of the above formulas is equal to zero, then there is W(G 1 (S, I)) = m 2 nI, utilizing the properties of Lambert W function, we can obtain Substituting ( 14) into the first formula in ( 13) and combining with W(G 1 (S, I)) = m 2 nI, we get 130 We consider I * > 0, and the parameters are all positive, that is which is equivalent to If R 0 > 1, the interior equilibria Ẽ2 = (S * , I * ) is satisfies condition S * > S q , which means that Ẽ2 is located in the area of A 2 and it is a regular equilibrium.
Local stability of the endemic equilibrium Ẽ2 Lemma 2. If the parameters satisfy the following relationship then the system (13) has the point Ẽ2 is locally asymptotically stable.
Proof : We use the Jacobian matrix to analyze the stability of equilibrium point the Jacobian matrix is as follow Where T 1 and T 2 are defined as with ∂G1 ∂S = m 2 bIe cI , ∂G1 ∂I = m 2 bIe cI (1+cI).In order to simplify the calculation at the equilibrium point Ẽ2 , we can get Thus, the characteristic equation at point Ẽ2 is Then if the characteristic equation satisfies the following conditions it can be obtained that the characteristic equation has two negative roots in 140 the region A 2 , which means that the point Ẽ2 is locally asymptotically stable.

Optimal control strategies
In this section, we investigate the dynamic behaviors of the system under control variables u 1 (t) and u 2 (t).The first control equation u 1 (t) is to enhance prevention strategies, reduce the number of patients, vaccinate and wear masks, or increase social distance.This control can reduce the probability of illness among susceptible populations, represented by (1 − u 1 (t)).The second control equation u 2 (t) represents accelerating the recovery time of patients, enhancing medical conditions, developing specific drugs, and enhancing human immune capacity.We focus on the impact of media factor m 1 on the infected individuals, while ignoring the impact of m 2 on infection rates, we set m 1 = 0, m 2 = 0, the system become When u i = 1, (1 = 1, 2) indicate complete control, and u i = 0 indicate that control is ineffective.We consider the following optimal control problem to minimize the objective functional is given by The weight constant ω 1 represents the infected population, while ω 2 and ω 3 represent the weight constants of personal protection, and improvement of medical conditions, respectively.The terms 1 2 ω 2 u 2 1 (t) and 1 2 ω 3 u 2 2 (t) describe the costs associated with the corresponding intervention measures over the time interval [0, T f ].Assuming that the cost is proportional to the square of the corresponding control function.
Where, U is defined by Therefore, in order to determine the necessary conditions that the optimal control (u1 * , u2 * ) must satisfy, Pontryagin's maximum principle is used [29] and the Hamiltonian H for the control problem is defined by: the adjoint variables λ i (i = 1, 2, 3) are associated with the state variables of the 160 model (23).The expression satisfies the following: with the terminal (transversality) conditions Further, the optimal control double (u * 1 , u * 2 ) are given as follows:

Numerical Simulations
In this section, we use numerical simulations to verify the rationality of the theory, observe and study the dynamic phenomena of the system.
In Fig. 1 and Fig. 2 display the relationship between the system and R 0 .
165 When R 0 < 1, the disease-free equilibrium point (k, 0) of the system is located in region A 1 or A 2 , depending on the values of m 1 and m 2 .This indicates that the region where the equilibrium point is located is closely related to the degree of media influence.When the equilibrium point is located in A 1 , it is shown in Fig. 1 (a), and it can be observed that changing different initial values of system will cross the line S = S q and stabilize at the equilibrium point; when the equilibrium point is located at A 2 , it is shown in Fig. 1 (b).It can be found that the trajectory starting from A 2 will stabilize at the equilibrium point, while the trajectory with an initial value at A 1 will directly pass through S = S q to reach the equilibrium point at A 2 .
On the other hand, when R 0 > 1, the system dynamics are shown in Fig. 2.
The trajectory with an initial value at A 1 will directly pass through S = S q to reach the equilibrium point at A 2 .If the trajectory starting from region A 2 will cross the line S = S q and then return to the A 2 , because its asymptotically stable equilibrium point E * 0 is virtual and located in A 2 , finally, the trajectory stabilizes at point Ẽ2 , which is regular and is globally asymptotically stable in Next, we continue to analyze the impact of R 0 and media factors on the disease, it can be seen from Fig. 3 (a) and Fig. 3 (b) that whether R 0 is greater than one has obvious impact on the disease.In Fig. 3 (a), we consider the case of m 1 = 0, m 2 = 0.When R 0 > 1, the number of diseases will first increase and then decrease, and eventually asymptotically stabilize at a steady-state value.
When R 0 < 1, the number of patients shows a downward trend, which is much lower than the number of diseases in the system when R 0 > 1.In Fig. 3 (b), we consider media influencing factors, which is m 1 = 0.2, m 2 = 0.8.The situation is similar to Fig. 3 (a), but when R 0 < 1, we change media related parameters m 1 and m 2 has no effect on the number of diseases.This shows that when the basic reproduction number R 0 < 1, the influence of media factors on the disease condition is less.Therefore, in the event of a disease outbreak such as COVID-19, we should use the media to have an impact on the disease at an appropriate time.
In order to explore the influence of media factors on diseases, we first select   significant.
Sensitivity analysis can analyze and understand the impact of different parameters on specific variables, which may help us control disease transmission or provide guidance.Referring to the definition in [27], the sensitivity index of R 0 for each parameter c is defined as follows In Fig. 5, it can be observed that among all positive correlation factors, the population carrying capacity k and basic transmission coefficient b are the highest.This indicates that the value of k or b is positively correlated to R 0 , with a degree of 100 %.However, among all the negative correlation factors, the rate of recovery from infection γ is the most sensitive parameter, and the value of γ is negatively correlated to R 0 , with a degree of 78.95 %.
The numerical simulation of the optimal control system is implemented using MATLAB, and the equation is solved using the forward backward of the fourth order Runge-Kutta method in [28].The weight constants are chosen as ω 1 = Using sensitivity analysis, it can be found that the population carrying capacity k and basic transmission coefficient b are positively correlated with R 0 , while the rate of recovery from infection γ is negatively correlated with R 0 .This may provide a control strategy for disease control.Finally, we used the optimal control strategy to analyze the impact of different controls on the system and found that dual-control have the best effect on disease control, and a single strategy that improved personal protection or vaccination is also have effective in disease control.
and his colleagues proposed a dynamic model of COVID-19 based on official data to analyze the impact of non-pharmaceutical interventions on transmission dynamics during the COVID-19 pandemic [20].Huang et al. established a COVID-19 mathematical model to analyze how spontaneous social distance and public social distance can increase the outbreak threshold of asymptomatic infections [21].Nyaberi et al. through theoretical analysis

Lemma 1 .
The interior equilibrium point Ẽ2 = (S * , I * ) is located in the area of A 2 and it is a regular equilibrium, where

Figure 2 :
Figure 2: The phase diagram of the SIR model under the case of R 0 = 1.9737 > 1.The dotted line indicates S = Sq, the left side of the dotted line is the A 1 area, and the right side is the A 2 area.The black points are the local equilibrium points E * 0 and Ẽ2 , and the parameters are b = 1.5, r = 1.5, β = 0.2, α = 0.2, k = 2.5, a = 0.1, µ = 0.1, m 1 = 0.2, m 2 = 0.8.