1 Introduction

Psoriasis is a common, chronic inflammatory skin disease with a complex etiology involving immune disorder and environmental triggers. According to World Health Organization (WHO), it affects about 2% to 3% of the world population, although it is more prevalent in American, Canadian, and European populations [35]. This disease is characterized by scaly red lesions on skin surface due to a huge proliferation of epidermal keratinocytes. Although, it is believed that psoriasis is a treatable but not curable disease, enlargement of treatment procedure is still in progress.

Biologic and biosimilar medications for psoriasis is the best current therapy proposed by many clinical and experimental researchers [20, 21]. These drugs fight with the causes rather than just eases the symptoms, and they target a specific part of body immune system. Specifically, biologics are used to suppress the action of a specific type of immune cells or they block some particular cytokines signaling. Tumor necrosis factor alpha (), Interleukin 17 or Interleukin 12 Interleukin 23 etc. Over the past decades, several biological therapies have been approved based upon numerous parameters: FDA- and EU-approved indications, therapeutic efficacy and impact on quality of life, cost-effectiveness, and safety profile [4]. Though some biologics (Infliximab, Etanercept, Adalimumab, Efalizumab, etc.) have been already approved as therapeutic agent of psoriasis, yet researchers are still testing the efficacy of new biologics (Fezakinumab, Alefacept, etc.) [4,11]. Etanercept, inhibitor, is accepted for the treatment of psoriasis at a dose of 25 mg or 50 mg twice weekly for 3 months followed by a maintenance dosage of 50 mg weekly thereafter [30,31]. This biologic binds with mediated cytokine that can bind to receptor 1 or receptor 2 and is involved in keratinocytes hyper-proliferation) to inhibit the inflammatory responses in skin which is characteristic of psoriasis [14, 17]. On the other hand, dendritic cells and keratinocytes mediated cytokine Interleukin stimulates cells within dermis to create , which drives keratinocytes hyper-proliferation in psoriasis [6]. Many clinical trials suggested that Fezakinumab, a well-tolerated monoclonal antibody, which is a promising therapeutic agent to treat epidermal hyperplasia and abnormal keratinocytes differentiation [9,19].

Recent evidence indicates that the activated cells release a large amount of , which lead the activation and hyper-proliferation of keratinocytes by the process of biochemical requirements [15, 33]. Activated cells mediated , has been shown to be a highly effective promoter of cell trafficking to the skin in psoriasis [32]. Th. participate in promoting cell activation with enhanced cytotoxicity through Interleukin 21 signaling cascade [28]. a new subpopulation of suppressor cells, are typically known as inhibitors of autoimmune responses. For the psoriatic case, can differentiate into inflammation-associated cells (paradigm shift) [2]. Due to the pro-inflammatory cytokine milieu in the psoriasis lesion, especially high levels of Interleukin 6 secreted from endothelial cells and cells, which inhibit activity [3]. The effects of two biologics (Etanercept and Fezakinumab) have been aggregated in this model by taking the mathematical approaches from [29] including some clinical data from [19,30,31].

Earlier, many researchers have discussed the dynamics of psoriasis to explore the etiology and possible treatment policies of psoriasis using mathematical models [12,16, 27]. Grigorieva and coworkers have studied the possible types of the control strategies associated with the first, second, and third orders singular arcs to enrich the control approach for the treatment of psoriasis [7,10]. Recently, Roy et al. have studied the impacts of T helper cells along with the pro-inflammatory and anti-inflammatory cytokines on psoriasis progression and also discussed about the dominating roles of few biologics inhibitor, inhibitor) in the treatment of psoriasis [23–26].

In this research, a mathematical model has been formulated with an aim to study the etiology of psoriasis development including the dynamics of regulatory cell () and activated natural killer cells ( cells). Natural killer cells (type of lymphocyte and a component of innate immune system) become activated in presence of macrophage secreted cytokines, and that activated cells can produce various inflammatory cytokines [18, 34]. Here we have focused on the crucial role of regulatory T cell and activated cell along with the other helper cells (naïve cells, ) for capturing the excessive rapid proliferation of keratinocytes. We have observed the effect of Etanercept ( inhibitor) and Fezakinumab ( monoclonal antibody) therapy by considering impulse differential equation. We have determined the change in drug concentration, which occurs when a new dose is administered by using the impulsive model. We have found that the minimum dosing interval is required to maintain the keratinocyte density under a certain threshold. Furthermore, we have determined the efficacy of the combined drug doses as well as the individual drug doses through numerical analysis. It is also revealed that the Etanercept alone could theoretically maintain the keratinocytes level, but Fezakinumab alone may not be able to control the hyperproliferation of keratinocytes.

This manuscript is organized as follows. In the afterward Section 2, we have formulated the full mathematical model based on the above discussion. In Section 3, we have theoretically analyzed some basic properties of the system when no drugs is applied. In Section 4, we have investigated the impulse therapeutic approach with four extreme cases. The numerical outcomes of our formulated model (without and with therapy) have been demonstrated in Section 5. Finally, in Section 6, we have discussed the main results including the limitations, future scope, and novelty for our formulated mathematical system.

2 The model

In this section, we have formulated a mathematical system to describe the psoriasis dynamics by considering naive cell, four different types of helper cells and ), activated natural killer cells ( cells), and keratinocytes (skin cells). Here the density of naive cell, activated natural killer cells ( cells), and keratinocytes are represented by respectively, at any time .. The concentration of our considered two drugs (Etanercept and Fezakinumab) are represented by and at time where

In order to develop the growth equation of naive cell, we assume that is the constant accumulation of naive cells from bone marrow. Here we have considered that are the production rate of respectively, from naive cell. represent the negative regulatory effect of cell on naive cell, cell, respectively, and and denote the negative regulatory effect of cell on and cell, respectively. The inhibitory effect of cell on and are denoted by and , these inhibitory effects are associated with cytokines signaling, respectively. We have assumed that and are the proliferation rate of and , respectively, due to self released cytokines. Due to paradigm shift, differentiate into , this rate is denoted by and, at a rate activated cells promote through signaling cascade.

Let be the constant accumulation of activated natural killer cells ( cells) from bone marrow. Also, let and are the proliferation rates of activated cells by the effect of self released and mediated cytokines, respectively.

We assume that is the constant growth of keratinocytes due to the constant migration of cells from dermal layer to epidermal layer. The effect of the vast released by activated cells, which help to proliferate the keratinocytes at a rate Furthermore, the positive regulatory effect of and cells on the proliferation of keratinocytes is given by the rate and , respectively, and at the same time, cells inhibit the proliferation of keratinocytes at the rate of . Natural death rates of naive cell, activated natural killer cells, and keratinocytes are denoted by respectively.

Based on the above assumptions and discussion, our formulated mathematical model to describe psoriasis dynamics is as follows:

In the seventh equation of system (1), the Hill functions represent the degree to which Etanercept and Fezakinumab inhibit the positive regulatory effects of and , respectively, on keratinocytes. Value of and decrease due to the inhibitory effects of these two biologics. and represent the concentration of Etanercept and Fezakinumab, which required to inhibit hyper-proliferation of keratinocytes by 50%.

The dynamics of two drugs are described by the following:

In system (2), and denote the rate at which the drugs are cleared. Here and are used to denote the concentration of Etanercept and Fezakinumab, respectively, at just after the th number of doses are taken. The concentration of these biologics at just before the intake of th number doses are represented by and Here and are the dosages of Etanercept and Fezakinumab, respectively, those drugs are taken at each impulse time where In general, so that the two drugs are taken at different times.

3 The system without drugs

When drugs are absent, , the system becomes:

subject to the following initial conditions:

In this section, we have studied the boundedness property of the solutions of system (3). We have also discussed the existence of the interior equilibria and stability criteria.

3.1 Boundedness

It is very important to establish that all the model variables are bounded for all time . This will ensure that the model is well-posed and plausible to represent the cell populations. The following theorem describes the boundedness of the solutions of system (3).

Theorem 1. All the solutions of system (3) enter into the region and are ultimately bounded, where Ω is defined as

if the following conditions hold:

and

Here the symbol “T” denotes the transpose, and are defined by the following formulas:

Proof. Let us consider the first equation of system (3) and neglecting the term we have the relationship:

By applying the well-known comparison principle described by [1] to (4) and considering we obtain the inequality

from which it follows that By using similar arguments (one can see [25] for detail analysis) we can easily show that all solutions , of system (3) that start in remain in this set for all t > 0. It means that . is invariant set of this system. Moreover, the region . is bounded, and therefore, all mentioned solutions ultimately bounded.

3.2 Equilibrium analysis

Here we have considered interior equilibrium as a steady state solution where the disease persists.

For the existence of interior equilibrium its coordinates should satisfy the conditions: In order to obtain the existence condition of the interior equilibrium, we set the first, second, third, fifth, sixth, and seventh equations of system (3) to zero. Solving state variables in terms of we obtain the following:

Now using the values expressed in system (5), we have obtained the tenth degree polynomial of from the fourth equation of system (3). Now, we compare with the standard form

Since it is hard to evaluate all coefficients, viz. so we can construct the following lemma to conclude the existence criteria by using the fact that if then it has at least one positive root in

Lemma 1. holds, then there exists an interior equilibrium point I. of system (1).

Analytically, it is also very difficult to determine the steady state values of system populations. We have determined numerically by performing the numerical stability analysis. For the stability analysis, we need the Jacobian at any steady point at

The matrix has characteristic equation

where

Let elements of the above matrix are in the form of where Now, det gives

where are the coefficient of the above polynomial. The coefficients are defined as

Here followed a rule that , then can go to 6 or 5. Following same rule for Similarly, 2 can go 5 or 6, 3 can go 5, 4 can go 5 or 3, 5 can go 2 or 3 or 4 or 6, and 6 can go 1 or 2 or 5. In let then can go 5 or 6, but ._15.51 does not exist because 5 cannot go to 1. For example, applying the above rule, we get the expression of as follows:

Now, we can conclude the stability criteria by the following remark.

Remark 1. All the roots of the characteristic equation are negative or have negative real part if the determinants of all the Hurwitz matrices are positive, Thus, according to the Routh–Hurwitz criterion [8, 22], the system is asymptotically stable if where the Hurwitz matrices are given by

4 The system with drugs

In this section, we have analyzed the drug-induced system using modified impulsive method to evaluate the dosing interval. The consideration of impulsive differential equations to measure the drug adherence will obviously perturb the interior steady state. Due to these impulse effect, the solutions of our formulated model exhibit periodic orbits with discontinuities. It is clear from models (1), (2) that only the drugs will exhibit discontinuities directly, and the other parameters may have discontinuities in their derivatives, but will have continuous solutions.

Here we have considered that the drugs (Etanercept and Fezakinumab) are given at fixed intervals. Let be the period of Etanercept and let be the period of Fezakinumab ( monoclonal antibody) For satisfying we have

The recursion relation at the moments of impulse is given by

From the first equation of system (7) along with the recursion relation (8), we have obtained

as Similarly, as we have

In order to obtain the ends of a positive impulsive periodic orbit of Etanercept, we have considered

Furthermore,

Hence, the ends of a positive impulsive periodic orbit of Etanercept has been defined by the impulse points to which the endpoints of each cycle monotonically increase.

Similarly, the ends of a positive impulsive periodic orbit of Fezakinumab has been defined by the impulse points to which the endpoints of each cycle monotonically increase.

4.1 Extreme cases analysis

Here we have analyzed four extreme cases to manifest the different outcomes that can occur depending on the dosing intervals. Before analysing the case, we cite the following lemma from [29].

Lemma 2. Suppose x is a variable satisfying

where c is a constant, and is independent of and . Then iit follows that for all t.

We have considered that a small dosing interval corresponds to frequent drug administration. It is obvious that small dosing intervals should provide the most effective therapy, whereas large dosing intervals should have little effect on over proliferated keratinocytes. We have also considered , the normal keratinocytes density. The value of has been taken 200 mm⁻³. Our target is to keep the keratinocytes density below by administrating drug with suitable dosing interval. Furthermore, only the keratinocytes population has been considered in describing the following four extreme cases depending on the dosing intervals. In order to avoid mathematical complexity, we have ignored the effect of drugs on the other immune cells.

The initial conditions on the drug concentrations and the monotonicity of the impulsive trajectories imply that

for all .. Since the impulsive drug orbits are asymptotically stable, it follows that for any there exists such that

Case 1. Absence of both drugs

If no drugs have been applied, then we can consider the situation where and activated cells dominate, while is present at the low density level. By using Theorem 1 we can consider the keratinocytes density as follows:

where have been defined in Theorem 1. Here we have concluded by the following remark.

Remark 2. Drugs will be applied when the keratinocytes density satisfies the condition for any time Note that keratinocytes density at interior equilibrium also satisfies the condition

Case 2. The absence of Fezakinumab but frequent use of Etanercept. Here, for a fixed value of the dynamics of keratinocyte is described as follows:

Suppose that Etanercept is applied at dosing interval, for which keratinocytes density will be less than or equal to Hence, the following inequality holds:

where

From (9) we get the upper value of dosing interval of the frequent dose of Etanercept. By the following remark we can conclude this case.

Remark 3. In absence of Fezakinumab, if only Etanercept is applied at the dosing interval described in equation (9) the keratinocytes density will be less than or equal to For a fixed value of we can determine the value of which have been demonstrated in numerical section.

Case 3. The absence of Etanercept but frequently use of Fezakinumab. Here, for a fixed value of the dynamics of keratinocyte is described as follows:

Again, suppose that Fezakinumab is applied at dosing interval, for which keratinocytes density will be less than or equal to . Hence, by using Lemma 2 the following inequality can be established:

where

From (10) we get the upper bound of dosing interval of the frequent dose of Fezakinumab. By the following remark we make a conclusion on Case 3.

Remark 4. In absence of Etanercept, if only Fezakinumab is applied at the dosing interval , described in equation (10) the keratinocytes density will be less than or equal to For a fixed value of we can determine the value of , which have been demonstrated in numerical section.

Case 4. Frequent dosing of both drugs.

Here we have recalled inequalities (9) and (10) to describe the case when Etanercept and Fezakinumab are taken frequently. and are the fixed dosing intervals of Etanercept and Fezakinumab. Hence, for a fixed value of the dynamics of keratinocyte is described as follows:

Hence, the following inequality holds:

Where

We write our conclusion on Case 4 in the following remark.

Remark 5. If Etanercept and Fezakinumab are taken frequently at the dosing interval and satisfying inequalities (9) and (10), then for a fixed value of we can determine the value of and by using inequality (11).

5 Numerical simulations and discussion

In this section, we have performed the numerical simulations of the mathematical systems (1), (2), and (3) to understand the analytic results. First, we have shown the solution of different cells and their effects on the psoriatic dynamics that is reflected in the model (without drug). Then the dynamical behavior of keratinocytes have been numerically evaluated under impulse therapeutic approach. The corresponding drugs (Etanercept and

Fezakinumab) dynamics also have been demonstrated. Parameter’s value used for numerical simulations are listed in Table 1.

In Fig. 1, we have plotted the numerical solution of considered immune cells (naive cell, cell, cell, cell, regulatory cell, activated cell) and keratinocytes to investigate the qualitative behavior of model populations for psoriatic state. This figure manifests that the psoriatic situation is dominated by T, and activated cells as well as , and regulatory cell are in suppressed condition. In presence of various pro-inflammatory cytokines (released by and activated cells) the

keratinocytes population undergone a hyperproliferative nature and reach to a high density level (350 mm⁻³).

Figure 2 shows the comparison of actual keratinocytes (seventh equation of system (3)) with the overestimated keratinocytes (described by Theorem 1). From this figure it is clear that the overestimated keratinocytes concentration reaches to stable condition after 50 days at density level about 350 mm⁻³, this is almost same as the dynamical behavior of actual keratinocytes. In order to simulate the impulsive effect on keratinocytes, we have considered the keratinocytes dynamics (with no drug situation) as described by the seventh equation of system (3). In impulsive approach, it is mandatory that the population must be in equilibrium state; thus, for this approximation, the initiation of treatment policy will not be hampered [27].

In Fig. 3, we have demonstrated the qualitative behavior of keratinocytes for the cases: with and without impulse effect by considering different doses of Etanercept when Fezakinumab is absent. This figure manifests that for the low dose of Etanercept, keratinocytes density is oscillating with small magnitude and chronologically decreasing towards the normal condition Furthermore, for the high dose of Etanercept ), keratinocytes density rapidly decreases and reaches to a density level below the desired threshold The corresponding time-course of drug concentration also has been illustrated in Fig. 3. In absence of Fezakinumab, the dynamical behavior of keratinocytes and corresponding drug dynamics have been scrutinized for different dosing intervals of Etanercept by considering the fixed dose (see Fig. 4). From these two Figs. 3 and 4 it is clear that Etanercept (dose regime: = 6 days) alone is able to control the hyper-proliferation of keratinocytes after 20 days of treatment.

In Fig. 5, we have exposed the dynamical nature of keratinocytes for with and without impulse therapy by considering different dosages of Fezakinumab (= 0.02, 0.002, and 0.0002) when Etanercept is absent. This figure exhibits that for any dose of Fezakinumab (low to high), keratinocytes density shows oscillating nature with very small magnitude. Though keratinocytes density is chronologically decreasing due to this inductive phase, but it not achieve the predefined healthy condition The time-course of drug

concentration by considering dosing interval . = 2 days has been illustrated in Fig. 3. For the different dosing intervals of Fezakinumab (. = 3 and 6 days), by considering the fixed dose F. = 0.002 the dynamical behavior of keratinocytes and corresponding drug dynamics have been plotted in Fig. 6. These two Figs. 5 and 6 show that Fezakinumab alone can only partially control the hyper-proliferation of keratinocytes.

The effect of combined biologic (Etanercept, Fezakinumab) on keratinocytes, taking dosing intervals 6, 4 days and doses 0.0002, F. = 0.0002, = 0.0002has been illustrated in Fig. 7. This figure depicts that within 20 days of this combined impulse therapy keratinocytes decrease to the preferred density level Single drug and the combined drug’s levels have been provided in the corresponding drug dynamics.

6 Conclusion

In this paper, formulating a mathematical model, we have studied the roles of different immune cells viz. and activated cells to encounter the hyperproliferation of keratinocytes during the disease progression. In absence of drugs, the proposed model system exhibits a unique steady state, namely, the coexistence equilibrium, which represents the psoriatic state. By using modified impulse theory we have also measured the efficacy and safety of Etanercept and Fezakinumab, which are applied as combined biologics. We have successfully demonstrated the effects of taking single drug and the combined drug by considering different dosing intervals for altered doses. The numerical simulations revel that high dose ) of Etanercept is able to clear the psoriatic lesions within 3 weeks. We have not found any suitable dose regime for Fezakinumab to maintain the normal keratinocytes level. On the other hand, our study shows that treatment with the low doses of that combined biologics (Etanercept: 6 days interval and Fezakinumab: 4 days interval) may be an effective dosing schedule for the treatment of psoriasis. This study finally leads to the conclusion for short time treatment policy, instead of taking Etanercept with high dose for the entire length of the induction period, it would be better if the patient takes the drug combination (Etanercept and Fezakinumab) with low doses.

Due to the lack of sufficient primary data, we have chosen our parameter values to see the dynamical behaviors, namely, asymptotic stability, periodic oscillations, etc. Furthermore, the proposed treatment regime using Etanercept and Fezakinumab is based on hypothetical value of model parameters, but the result can be proposed for future clinical trials. It is also possible to determine the therapy schedule for the similar combined drugs using the analytical and numerical techniques. Moreover, if proper data are obtained, a particular situation can be modeled, then we will be able to make these proposed results more applicable and biologically reasonable.

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Notes

Notas de autor