A cyclic behavioral modeling aspect to understand the effects of vaccination and treatment on epidemic transmission dynamics

Evolutionary epidemiological models have played an active part in analyzing various contagious diseases and intervention policies in the biological sciences. The design in this effort is the addition of compartments for treatment and vaccination, so the system is designated as susceptible, vaccinated, infected, treated, and recovered (SVITR) epidemic dynamic. The contact of a susceptible individual with a vaccinated or an infected individual makes the individual either immunized or infected. Inventively, the assumption that infected individuals enter the treatment and recover state at different rates after a time interval is also deliberated through the presence of behavioral aspects. The rate of change from susceptible to vaccinated and infected to treatment is studied in a comprehensive evolutionary game theory with a cyclic epidemic model. We theoretically investigate the cyclic SVITR epidemic model framework for disease-free and endemic equilibrium to show stable conditions. Then, the embedded vaccination and treatment strategies are present using extensive evolutionary game theory aspects among the individuals in society through a ridiculous phase diagram. Extensive numerical simulation suggests that effective vaccination and treatment may implicitly reduce the community risk of infection when reliable and cheap. The results exhibited the dilemma and benefitted situation, in which the interplay between vaccination and treatment evolution and coexistence are investigated by the indicators of social efficiency deficit and socially benefited individuals.


Mathematical Analysis
Here, we present some of the actual results related to the theoretical analysis of the model [56][57][58][59][60][61]. The primary goal is to get an asymptotic understanding of how the virus will propagate, ensuring that the model's explanations are accurate by requiring positivity and boundedness. We verify the asymptotic local stability study by finding the model's disease-free equilibrium. We also calculate the reproduction number and the existence of a uniformly stable situation for the exactness of the solution.

Model's positivity and boundedness of the solutions
In this part, we mainly focus on the proposed model's positivity and boundedness, which certifies the exactness of the model's solutions. Thus, for infected class ( ), we may write, Similarly, the treatment class ( ) is expressed as follows, Furthermore, the recovered class ( ) is, Hence, the norm of the domain where ∈ [61], put out in the following way Analogously, utilizing the overhead norm, the vaccinated and susceptible classes are likewise represented as follows, In the same way, ( ) ≥ 0 −( ‖ ‖ ∞ + ) , ∀ ≥ 0.
Finally, we can conclude that the suggested model and its solution are both positive and bound.

Disease-free equilibrium (DFE) point and its stability
The disease-free equilibrium, symbolized by ℰ 0 , is the point at which there is no infection in the population at the equilibrium stage, and all infected classes will have a zero value. To calculate the DFE point of the proposed model, we put = 0 in the system (1-5). Then we get the DFE point of the current model is ℰ 0 = ( 0 , 0 , 0 , 0 , 0 ) = ( * , * , 0,0,0); * + * = (= 1).

Derivation of the basic reproduction ( 0 ) and effective reproduction number ( )
We calculate the basic reproduction number, 0 to show stable equilibrium conditions to analyze the preliminary theoretical investigation. We consider the next-generation matrix [56] technique to evaluate the basic reproduction number as follows: As the basic reproduction number is the most considerable eigenvalue of −1 thus, Determining an epidemiological model's effective reproduction number ( ) is the same as the basic reproduction number [28][29][30]. The effective reproduction number can be estimated by multiplying the basic reproductive number and the proportion of the host population [68][69][70][71]. Therefore, the time-dependent reproduction number is known as the effective reproduction number is, Theorem 1. If 0 < 1, then the disease-free equilibrium 0 is locally asymptotically stable. If 0 > 1, the disease-free equilibrium is unstable.
Proof: Let us compute the proposed model's Jacobian matrix is as Substituting the value of the DFE point 0 , we obtain The characteristics equation | ( 0 ) − | = 0 has five roots, which are, As all eigenvalues are negative or equal to zero, therefore, conferring to Routh-Hurwitz criteria [67], we can easily accomplish that the model is locally asymptotically stable at the disease-free equilibrium point
Theorem 4. The endemic equilibrium point * is locally asymptotically stable and unstable whenever 0 > 1.
Proof: Remember that the system's (1.1-1.6) Jacobian exists at any point of equilibrium ( , , , , , ), and we obtain Hence, at the endemic equilibrium point 1 * , the desired Jacobian matrix is The roots of the characteristic equations | ( 1 * − 6 | satisfies the following equation:  It is simple to demonstrate that all of the roots of equation (A1) will have a negative real portion if 0 > 1 and that the coefficients of equation (A1) will fulfill the Routh-Hurwitz condition [67]. The endemic equilibrium point will thus be locally asymptotically stable for 0 > 1.
We have to prove that ℰ * is globally asymptotically stable for 0 > 1. Where,

Strength number
We use the suitable strength numbers approach to determine the waving tendency in proposed epidemic dynamics [61]. First, we determined the recommended model's strength number under the assumption of a limited population, , by analyzing the partial first derivative of the infected class using next-generation matrix techniques as follows: Therefore, Therefore, as previously, from the spectral radius of ( −1 ) for defining the epidemic wave, the desired strength number is denoted by , When ≤ 0, the disease can only produce one wave, and the infection class would quickly drop below or equal to the equilibrium of disease-free conditions. However, when ≥ 0, multi-waving scenarios are revealed. Here, all parameters of the suggested model are well-defined and positive. More precisely, , , ≥ 0, and 0 ≤ ≤ 1, which shows that the proposed model's strength number ≤ 0 represents only one wave.

Geometrical interpretation of Strength Number
The second-order derivative usually depicts the concavity or curvature of any graph. In epidemic models, a concept like this from fundamental calculus is routinely applied to observe the situation of several layers or waves of epidemic disease cases. To illustrate the second-order time derivative study of our suggested model, we exemplify it below as follows: By using the disease-free equilibrium point, we can demonstrate the concavity of the system of nonlinear ODEs (equation A11). The inflection point occurs when the time derivative of the second order equals zero. Concave up arises if it is more significant than zero, and concaves down if it is less meaningful than zero.
Using the system equation (A11) and the disease-free equilibrium point ℰ 0 , we may conclude that ℰ 0 cannot be concave up or down, ̈= 0, ̈= 0.
Equation (A11) shows that for all second-order time derivatives utilized in the computation of concavity, we only have the case for the inflection or stationary points. In conclusion, the model equation (A11) only provides the infection or the fixed points for the second-order model equation (A10) at the disease-free equilibrium points ℰ 0 instead of the concave up and concave down.