A study of qualitative correlations between crucial bio-markers and the optimal drug regimen of Type I lepra reaction: A deterministic approach

Dinesh Nayak; Anamalamudi Vilvanathan Sangeetha; Dasu Krishna Kiran Vamsi

doi:10.1515/cmb-2023-0117

Article Open Access

A study of qualitative correlations between crucial bio-markers and the optimal drug regimen of Type I lepra reaction: A deterministic approach

Dinesh Nayak , Anamalamudi Vilvanathan Sangeetha and Dasu Krishna Kiran Vamsi

Published/Copyright: December 31, 2023

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From the journal Computational and Mathematical Biophysics Volume 11 Issue 1

Abstract

Mycobacterium leprae is a bacterium that causes the disease leprosy (Hansen’s disease), which is a neglected tropical disease. More than 2,00,000 cases are being reported per year worldwide. This disease leads to a chronic stage known as lepra reaction that majorly causes nerve damage of the peripheral nervous system leading to loss of organs. The early detection of this lepra reaction through the level of bio-markers can prevent this reaction occurring and the further disabilities. Motivated by this, we frame a mathematical model considering the pathogenesis of leprosy and the chemical pathways involved in lepra reactions. The model incorporates the dynamics of the susceptible Schwann cells, infected Schwann cells, and the bacterial load and the concentration levels of the bio-markers interferon- γ , tumor necrosis factor- α , IL (interleukin)- 10 , IL- 12 , IL- 15 , and IL- 17 . We consider a nine-compartment optimal control problem considering the drugs used in multi drug therapy (MDT) as controls. We validate the model using 2D heat plots. We study the correlation between the bio-markers levels and drugs in MDT and propose an optimal drug regimen through these optimal control studies. We use the Newton’s gradient method for the optimal control studies.

Keywords: Hansen’s disease; Type I lepra reaction; bio-markers; multidrug therapy; Newton’s gradient method; optimal drug regimen

MSC 2010: 37Nxx; 92BXX; 92Cxx

1 Introduction

Leprosy, the oldest disease known to human civilization, is one of the highly neglected tropical diseases caused by a slow-growing bacterium called Mycobacterium leprae (M. leprae). Mainly, this bacterium causes damage to the Schwann cells and hence the skin; thereby, the peripheral nervous system of host body gets impacted. It also has bad impact on eyes and mucosa of the upper respiratory tract. The report of World Health Organization (WHO) states that about 120 countries are still reporting new cases of leprosy which accounts to more than 2,00,000 per year [39]. In the year 2021, India alone spotted about 75,395 new cases [38]. Leprosy disease can lead to a chronic phase of lepra reaction that causes permanent disabilities and loss of organs. Early detection of the disease by observing the key changes in the bio-marker level will play a vital role to prevent the losses.

The chemical and metabolic properties of the cytosol environment of host cell that gets altered in the presence of the M. leprae were first explained by Rudolf Virchow (1821–1902) in the late nineteenth century [37]. Furthermore, different clinical studies explained about the pathway of cytokine responses based upon which there are mainly two types of lepra reactions. The Type-1 lepra reactions are associated with cellular immune response, whereas Type 2 reactions are associated with humoral immune response [3,24]. Both these pathways involve the crucial bio-markers/cytokines such as interferon- γ , tumor necrosis factor- α , IL (interleukin)- 10 , IL- 12 , IL- 15 , and IL- 17 [30].

There are quite a number of biochemical studies that deal with the pathogenesis of lepra reaction [29] and some on growth of the bacteria and chemical consequences [30]. On the mathematical modeling front, it can be seen that modeling of leprosy began in the 1970s with simple compartmental models, such as the susceptible-infectious-recovered framework [18]. Later, various complex models were framed to evaluate the effectiveness of long-term control and elimination strategies, such as mass drug administration, contact tracing, and vaccination [19]. The very first stochastic leprosy model to investigate uncertainties in leprosy epidemiology was developed in 1999 [27]. This modeling framework is known as simulation model for leprosy transmission and control (SIMLEP) model. This SIMLEP model was further developed to include the disease dynamics in households and heterogeneity in susceptibility, and a new individual-based model stochastic individual-based model on control of leprosy was proposed [10].

In addition to these individual-based models, a mathematical simulation model was studied with a delay time taken into account during the development of the disease [15]. Furthermore, some population studies that combine a simple leprosy model with a tuberculosis model for exploring the contribution of the immunity acquired from tuberculosis to the disappearance of leprosy in Western Europe can be found in [23]. The work [4] deals with a comprehensive review of population models in leprosy. The mathematical models developed for Leprosy are also used for understanding the 65 cancer immunosurveillance and T cell energy [34].

From the literature, we see that limited studies explore the cellular dynamics of leprosy. The very first within-host model for leprosy predicts the actual role of M. leprae bacteria for the infection on human Schwann cells and dissemination of the disease leprosy [12]. The work [13] formulates a three-compartment within-host leprosy model and studies the effects of the density-dependent parameter on M. leprae. The work [14] deals with a four-compartment within-host leprosy model and concludes that the process of demyelination, nerve regeneration, and infection of the healthy Schwann cells are the three most crucial factors in the pathogenesis of leprosy. The authors in previous work [28] presented a comprehensive and detailed within-host modeling study involving crucial biomarkers and optimal drug regimen for Type I lepra reaction. These models help researchers better understand the nuances of leprosy epidemiology and its implications for control and treatment strategies.

To our knowledge, there is no work performed yet dealing with the dynamics of the bio-markers involved in lepra reactions. Also, there seems to be neither any clinical work that clearly deals with the dynamics of bio-markers during lepra reactions. Hence, it is extremely important to study the dynamics at the bio-markers levels and their correlation with the multidrug therapy (MDT) drugs that can help the clinicians for control of occurrence of lepra reactions.

Motivated by these observations in this work, we propose to study the dynamics of the bio-markers through chemical reactions. In the next section, we discuss and detail the proposed mathematical model. We use the drugs in MDT as control variables. We frame an optimal control problem along with a cost function J min . In Section 3, we validate this model using the 2D heat plots. Furthermore, in Section 4, we establish the existence of an optimal solution for the proposed optimal control problem. Next, in Section 5, we do the numerical studies. Initially, we discuss the numerical scheme used, namely, the Newton’s gradient method for the optimal control studies. Later, we discuss the inferences from the numerical simulations. Finally, Section 6, we do the discussions and conclusions for this work.

2 Mathematical model formulation

Based on the clinical literature, we consider a model that consists of susceptible Schwann cells S ( t ) , infected Schwann cells I ( t ) , bacterial load B ( t ) along with five cytokines that play a crucial role in Type I lepra reaction. We have considered the cytokines IFN- γ , TNF- α , IL- 10 , IL- 12 , IL- 15 , and IL- 17 by capturing their concentration dynamics in Type I lepra reaction. In the following, we discuss briefly each of the compartments in the model.

S ( t ) compartment: The first term of equation (1) deals with the natural birth rate of the susceptible Schwann cell. The second term describes the decrease in the number of susceptible cells S ( t ) at a rate β due to infection by the bacteria (followed by the law of mass action). γ represents the death of S due to the cytokines response and μ 1 represents the natural death rate of S . The rest of the terms account for the controls owing to MDT interventions.

I ( t ) compartment: The increase of the infected cells is accounted by the term β B S in equation (2). Decrease of these cells due to the cytokine response is at a rate δ and the natural death rate is μ 1 . The rest of the terms are associated with controls owing to MDT interventions.

B ( t ) compartment: The bacterial load increases indirectly due to an increase in I ( t ) as the burst of more cells with bacteria increases their replication. This rate α is accounted in the first term of (3). y denotes the rate of cleaning of B ( t ) due to cytokine responses and μ 2 is the natural death rate of bacteria. The rest of the terms are associated with controls owing to MDT interventions.

I γ ( t ) compartment: This compartment deals with the concentration level of IFN- γ through equation (4). The first term represents the production of IFN- γ in the presence of the infection. The second term deals with the inhibition of the concentration level due to other cytokines [24,33], and the last term accounts for the natural decay of the concentration.

T α ( t ) compartment: The concentration of TNF- α is increased by the inter action between IFN- γ and the infected cells [24,33]. This is being captured in the first term of equation (5). The second term represents the natural decay.

Similar formulations are used for studying the concentration levels of IL-12, IL-15, and IL-17 compartments by equations (7)–(9), respectively.

I 10 ( t ) compartment: Equation (6) deals with the concentration levels of IL- 10 . The first term accounts for the production and the last term for the decay. The middle term considers the inhibition of IL- 10 due to IFN- γ [24].

2.1 Description of the control variables dealing with the MDT drugs

WHO guidelines 2018 recommends an MDT for leprosy that consist of three drugs rifampin, dapsone, and clofazimine [25,36]. The impact of each of these drugs and their mathematical articulation as control variables is as follows:

Rifampin ( D 11 , D 12 , D 13 ): Rifampin is known for rapid bacillary killing. Due to this, there is an indirect decrease in the amount of cells obtaining infected [7]. Hence, we incorporate this with the control variable D 12 ( t ) in the compartment of infected cells of (1)–(9) with a negative sign, and in the bacterial load compartment, this control is introduced as D 13 2 ( t ) . Here, the square on D 13 ( t ) is used to capture the extent of intense action on bacterial load. This drug also reduces the susceptible cells; hence, the control D 11 ( t ) is introduced with a negative sign.

Dapsone ( D 21 , D 22 , D 23 ): The drug dapsone is bactericidal and bacteriostatic against M. leprae [32]. In a similar way to capture the drug action, we incorporate D 21 ( t ) and D 22 ( t ) in compartment S (equation (1)) and compartment I (equation (2)), respectively, and D 23 2 ( t ) in the B (equation (3)) compartment.

Clofazimine ( D 31 , D 33 ): Clofazimine, the third drug in MDT for leprosy acts as an immuno-suppressive and also causes the static level of bacteria (bacteriostatic) against M. leprae by binding with DNA of the bacteria and hence causing the inhibition of template function of DNA [11]. Therefore, to incorporate this action, we include the control variable D 31 ( t ) in the S ( t ) compartment in equation (1), resulting in increase of these cells. D 33 ( t ) is negatively incorporated to account for the inhibition of bacterial replication.

We next propose the mathematical model dealing with the mechanism of drug action on each compartment of susceptible cells, infected cells, and bacterial load along with concentration level of the cytokines.

(1) d S d t = ω − β S B − γ S − μ 1 S − D 11 ( t ) S − D 21 ( t ) S + D 31 ( t ) S ,

(2) d I d t = β S B − δ I − μ 1 I − D 12 ( t ) I − D 22 ( t ) I ,

(3) d B d t = ( α − D 23 2 ( t ) − D 33 ( t ) ) I − y B − μ 2 B − D 13 2 ( t ) B ,

(4) d I γ d t = α I γ I − [ δ T α I γ T α + δ I 12 I γ I 12 + δ I 15 I γ I 15 + δ I 17 I γ I 17 ] I − μ I γ ( I γ − Q I γ ) ,

(5) d T α d t = β T α I γ I − μ T α ( T α − Q T α ) ,

(6) d I 10 d t = α I 10 I − δ I γ I 10 I γ − μ I 10 ( I 10 − Q I 10 ) ,

(7) d I 12 d t = β I 12 I γ I − μ I 12 ( I 12 − Q I 12 ) ,

(8) d I 15 d t = β I 15 I γ I − μ I 15 ( I 15 − Q I 15 ) ,

(9) d I 17 d t = β I 17 I γ I − μ I 17 ( I 17 − Q I 17 ) .

Symbols	Biological meaning
S	Susceptible Schwann cells
I	Infected Schwann cells
B	Bacterial load
I γ	Concentration of IFN- γ
T α	Concentration of TNF- α
I 12	Concentration of IL-12
I 15	Concentration of IL-15
I 17	Concentration of IL-17
ω	Natural birth rate of the susceptible cells
β	Rate at which Schwann cells are infected
γ	Death rate of the susceptible cells due to cytokines
μ 1	Natural death rate of Schwann cells and infected Schwann cells
δ	Death rate of infected Schwann cells due to cytokines
α	Burst rate of infected Schwann cells realising the bacteria
y	Rates at which M. leprae is removed by cytokines
μ 2	Natural death rate of M. leprae
α I γ	Production rate of IFN- γ
δ T α I γ	Inhibition of IFN- γ due to TNF- α
δ I 12 I γ	Inhibition of IFN- γ due to IL-12
δ I 15 I γ	Inhibition of IFN- γ due to IL-15
δ I 17 I γ	Inhibition of IFN- γ due to IL-17
μ I γ	Decay rate of IFN- γ
β T α	Production rate of TNF- α
μ T α	Decay rate of TNF- α
α I 10	Production rate of IL- 10
δ I γ I 10	Inhibition IL- 10 of due to IFN- γ
μ I 10	Decay rate of IL- 10
β I 12	Production rate of IL- 12
μ I 12	Decay rate of IL- 12
β I 15	Production rate of IL- 15
μ I 15	Decay rate of IL- 15
β I 17	Production rate of IL- 17
μ I 17	Decay rate of IL- 17
Q I γ	Quantity of IFN- γ before infection
Q T α	Quantity of TNF- α before infection
Q I 10	Quantity of I 10 before infection
Q I 12	Quantity of I 12 before infection
Q I 15	Quantity of I 15 before infection
Q I 17	Quantity of I 17 before infection

Now, mathematically, we define the set of all control variables as follows:

U = { D i j ( t ) , D i j ( t ) ∈ [ 0 , D i j max ] , 1 ≤ i , j ≤ 3 , i j ≠ 32 , t ∈ [ 0 , T ] } ,

where D i j max represents the upper limit of the corresponding control variable that depends on the availability and limit of the drugs recommended for patients, and T is the final time of observation.

Since the drugs in MDT can lead to some hazards, we consider a cost functional that minimizes the drug concentrations along with the infected cell count and the bacterial load.

Based on this, we define the following cost functional:

(10) J min ( I , B , D 1 , D 2 , D 3 ) = ∫ 0 T ( I ( t ) + B ( t ) + P [ D 11 2 ( t ) + D 12 2 ( t ) + D 13 3 ( t ) ] + Q [ D 21 2 ( t ) + D 22 2 ( t ) + D 23 3 ( t ) ] + R [ D 31 2 ( t ) + D 33 2 ( t ) ] ) d t ,

where the integrand of the cost functional J min is denoted by:

(11) L ( I , B , D 1 , D 2 , D 3 ) = ( I ( t ) + B ( t ) + P [ D 11 2 ( t ) + D 12 2 ( t ) + D 13 3 ( t ) ] + Q [ D 21 2 ( t ) + D 22 2 ( t ) + D 23 3 ( t ) ] + R [ D 31 2 ( t ) + D 33 2 ( t ) ] ) ,

which denotes the running cost and is commonly known as Lagrangian of the optimal control problem.

Now, the admissible set of solution of the aforementioned optimal control problems (1)–(10) is given by:

Ω = { ( I , B , D 1 , D 2 , D 3 ) : I , B satisfying ( 1 ) – ( 9 ) ∀ ( D 1 , D 2 , D 3 ) ∈ U } ,

where D 1 = ( D 11 , D 12 , D 13 ) , D 2 = ( D 21 , D 22 , D 23 ) , and D 3 = ( D 31 , D 33 ) .

We next validate the model using 2D heat plots with control variables as zero.

3 Model validation through 2D heat plots

We validate the above-framed model based on the clinical characteristics of leprosy. From the clinical studies, it can be seen that the doubling rate the bacteria (M. leprae) is approximately 14 days [20]. We use this clinical characteristic to validate our model.

To generate these heat plots, we use two pairs of parameters and their range of variation. We consider the pairs of parameters α − γ and α − y . The range of values for α was from 0.0563 to 0.0763, the range for γ is from 0.15 to 0.2090, and the range for y is from 0.0002 to 0.5003. All the other parameter values are taken from Table 1, and a feasible initial condition S ( 0 ) = 5,200 , I ( 0 ) = 0 , B ( 0 ) = 40 , I γ ( 0 ) = 5 , T α ( 0 ) = 5 , I 10 ( 0 ) = 15 , I 12 ( 0 ) = 12 , I 15 ( 0 ) = 12 , and I 17 ( 0 ) = 10 was chosen. Using the aforementioned values, we simulate the model through MATLAB taking 50 pairs of different values for each of the parameters within the range provided, and we consider the value of the control variables as zero at that time. Then, we record the value of B ( t ) at 14th day in a 50 × 50 matrix. We used the function imagesc() to create these heat plots in MATLAB.

Table 1

Values of the parameters complied from clinical literature

Symbols	Values	Units
ω	0.0220 [17]	pg mL ‒ 1 day ‒ 1
β	3.4400 [16]	pg mL ‒ 1 day ‒ 1
γ	0.1795 [30]	day ‒ 1
μ 1	0.0018 [30]	day ‒ 1
δ	0.2681 [30]	day ‒ 1
α	0.0630 [20]	pg mL ‒ 1 day ‒ 1
y	0.0003 [12]	day ‒ 1
μ 2	0.5700 [1]	day ‒ 1
α I γ	0.0003 [31]	pg mL ‒ 1 day ‒ 1
δ T α I γ	0.005540*	pg mL ‒ 1
δ I 12 I γ	0.009030*	pg mL ‒ 1
δ I 15 I γ	0.006250*	pg mL ‒ 1
δ I 17 I γ	0.004990*	pg mL ‒ 1
μ I γ	2.1600 [31]	day ‒ 1
β T α	0.0040 [31]	pg mL ‒ 1 day ‒ 1
μ T α	1.1120 [31]	day ‒ 1
α I 10	0.0440 [21]	pg mL ‒ 1 day ‒ 1
δ I γ I 10	0.001460*	pg mL ‒ 1
μ I 10	16.000 [21]	day ‒ 1
β I 12	0.0110 [21]	pg mL ‒ 1 day ‒ 1
μ I 12	1.8800 [31]	day ‒ 1
β I 15	0.0250 [34]	pg mL ‒ 1 day ‒ 1
μ I 15	2.1600 [34]	day ‒ 1
β I 17	0.0290 [34]	pg mL ‒ 1 day ‒ 1
μ I 17	2.3400 [34]	day ‒ 1
Q I γ	0.1000 [35]	Relative concentration
Q T α	0.1400 [6]	Relative concentration
Q I 10	0.1500 [35]	Relative concentration
Q I 12	1.1100 [6]	Relative concentration
Q I 15	0.2000 [6]	Relative concentration
Q I 17	0.3170 [6]	Relative concentration

The (*) marked values of the parameters are assumed.

In Figure 1(a), the value of γ is taken on ordinate and α on abscissa. From the color bar of this figure, we see that the model is able to reproduce the characteristic that the initial count of bacteria doubles in 14 days (in this case, it is 80 with B ( 0 ) = 40 ). The same inference can be made for Figure 1(b), where y is taken on ordinate and α is taken on abscissa.

$Figure 1 Heat plots: (a) taking 50 × 50 50\times 50 pairs of values of the parameters α \alpha and γ \gamma and (b) taking 50 × 50 50\times 50 pairs of values of the parameters α \alpha and y y .$

Figure 1

Heat plots: (a) taking 50 × 50 pairs of values of the parameters α and γ and (b) taking 50 × 50 pairs of values of the parameters α and y .

4 Existence of an optimal solution

In this section, we establish the existence of solution for the optimal control Problems (1)–(10) using the existence Theorem 2.2 of [5].

Theorem 1

There exists an 8-tuple of optimal controls ( D 1 * ( t ) , D 2 * ( t ) , D 3 * ( t ) ) in the set of admissible controls U, and hence, the optimal state variables, ( S * ( t ) , I * ( t ) , and B * ( t ) ) such that the cost function is minimized, i.e.,

J min ( I * ( t ) , B * ( t ) , D 1 * ( t ) , D 2 * ( t ) , D 3 * ( t ) ) = min D 1 , D 2 , D 3 ∈ U J min ( I , B , D 1 , D 2 , D 3 ) ,

corresponding to the control Systems (1)–(10), where D 1 = ( D 11 , D 12 , D 13 ) , D 2 = ( D 21 , D 22 , D 23 ) , and D 3 = ( D 31 , D 33 ) .

Proof

We consider d s d t = f 1 ( t , x , D ) , d I d t = f 2 ( t , x , D ) , d B d t = f 3 ( t , x , D ) , d I γ d t = f 4 ( t , x , D ) , d T α d t = f 5 ( t , x , D ) , d I 10 d t = f 6 ( t , x , D ) , d I 12 d t = f 7 ( t , x , D ) , d I 15 d t = f 8 ( t , x , D ) , and d I 17 d t = f 9 ( t , x , D ) of the control Systems (1)–(10). Here, x ∈ X denotes the state variables and D ∈ U denotes the control variables. With f = ( f 1 , f 2 , f 3 , f 4 , f 5 , f 6 , f 7 , f 8 , f 9 ) , we see that X ⊆ R 9 and

f : [ 0 , T ] × X × U → R 9

is a continuous function of t and x for each D i j ∈ U .

Now, we intend to show that ( F 1 ) to ( F 3 ) of Theorem 2.2 of [5] holds true for all f i ’s.

F1: Here, each of the f i ’s has a continuous and bounded partial derivative, implying that f is Lipschitz’s continuous.

F2: We consider g 1 ( D 11 , D 21 , D 31 ) = − D 11 − D 12 + D 31 , which is bounded on U .

Thus,

f 1 ( t , x , D ( 1 ) ) − f 1 ( t , x , D ( 2 ) ) [ g 1 ( D ( 1 ) ) − g 1 ( D ( 2 ) ) ] = [ D 11 ( 2 ) + D 12 ( 2 ) − D 31 ( 2 ) − D 11 ( 1 ) − D 12 ( 1 ) + D 31 ( 1 ) ] S [ D 11 ( 2 ) + D 12 ( 2 ) − D 31 ( 2 ) − D 11 ( 1 ) − D 12 ( 1 ) + D 31 ( 1 ) ] ≤ η S = F 1 ( t , x ) ∴ f 1 ( t , x , D ( 1 ) ) − f 1 ( t , x , D ( 2 ) ) ≤ F 1 ( t , x ) ⋅ [ g 1 ( D ( 1 ) ) − g 1 ( D ( 2 ) ) ] .

Here, η > 1 is a real number. Moreover, since U is compact and g 1 is continuous, we have g 1 ( U ) to be compact. Also, since the function g 1 ( U ) is linear so the range of g 1 , i.e., g 1 ( U ) will be convex. Since U is non-negative so g 1 − 1 is non-negative.

Similarly, for f 2 ( t , x , D ) , we choose g 2 ( D 12 , D 22 ) = − D 12 − D 22 and F 2 ( t , x ) = I and F2 can be proven in a similar way.

Now, for f 3 ( t , x , D ) , we choose g 3 ( D 23 , D 33 ) = − D 23 2 − D 33 .

Hence,

f 3 ( t , x , D ( 1 ) ) − f 3 ( t , x , D ( 2 ) ) [ g 3 ( D ( 1 ) ) − g 3 ( D ( 2 ) ) ] = [ D 23 2 ( 2 ) + D 33 ( 2 ) − D 23 2 ( 1 ) − D 33 ( 1 ) ] I − [ D 13 2 ( 1 ) − D 13 2 ( 2 ) ] B [ D 23 2 ( 2 ) + D 33 ( 2 ) − D 23 2 ( 1 ) − D 33 ( 1 ) ] ≤ [ D 23 2 ( 2 ) + D 33 ( 2 ) − D 23 2 ( 1 ) − D 33 ( 1 ) ] I [ D 23 2 ( 2 ) + D 33 ( 2 ) − D 23 2 ( 1 ) − D 33 ( 1 ) ] = I = F 3 ( t , x ) ( provided D 13 2 ( 1 ) ≥ D 13 2 ( 2 ) ) ∴ f 3 ( t , x , D ( 1 ) ) − f 3 ( t , x , D ( 2 ) ) ≤ F 3 ( t , x ) ⋅ [ g 3 ( D ( 1 ) ) − g 3 ( D ( 2 ) ) ] .

But to satisfy this hypothesis for remaining f i ’s, we take the help of Corollary 2.1 of [5], i.e., F4 must be satisfied in the place of F2. Hence, considering g 4 ( D 1 , D 2 , D 3 ) = 0 , which is bounded measurable function and F 4 ( t , x ) = 1 , we establish the relation:

f 3 ( t , x , D ( 1 ) ) − f 3 ( t , x , D ( 2 ) ) = 0 ≤ 1 = 1 ⋅ 0 = F 4 ( t , x ) ⋅ [ g 4 ( D ( 1 ) − D ( 2 ) ) ] .

Similarly, taking F i ( t , x ) = i and g i ( D 1 , D 2 , D 3 ) = 0 for i = 5 , 6 , 7 , 8 , 9 , we obtain the relations:

f i ( t , x , D ( 1 ) ) − f i ( t , x , D ( 2 ) ) ≤ F i ( t , x ) ⋅ [ g i ( D ( 1 ) − D ( 2 ) ) ] .

Hence, we are done in satisfying the hypothesis F2.

F3: Since S , I , B and f i ( x , t ) = 1 are bounded on [ 0 , T ] so F i ( • , x u ) ∈ ℒ 1 .

Now, we have to show that the running cost function

C ( t , x , D ) = I ( t ) + B ( t ) + P [ D 11 2 ( t ) + D 12 2 ( t ) + D 13 3 ( t ) ] + Q [ D 21 2 ( t ) + D 22 2 ( t ) + D 23 3 ( t ) ] + R [ ∣ D 3 ( t ) ∣ 2 ]

satisfies conditions ( C 1 )–( C 5 ) of Theorem 2.2 of [5].

Here, C : [ 0 , T ] × X × U → R .

C1: We see that C ( t , • , • ) is a continuous function as it is a sum of continuous functions, which are the functions of t ∈ [ 0 , T ] .

C2: S , I , and B and all D i j ’s are bounded, implying that C ( • , x , D ) is bounded and hence measurable for each x ∈ X and D i j ∈ U .

C3: Consider Ψ ( t ) = κ such that κ = min { I ( 0 ) , B ( 0 ) } , then, Ψ will bounded such that for all t ∈ [ 0 , T ] , x ∈ X and D i j ∈ U , we have

C ( t , x , D ) ≥ Ψ ( t )

C4: Since C ( t , x , D ) is the sum of the functions that are convex in U for each fixed ( t , x ) ∈ [ 0 , T ] × X ; therefore, C ( t , x , D ) follows the same.

C5: Using the similar type of argument, we can easily show that for each fixed ( t , x ) ∈ [ 0 , T ] × X , C ( t , x , D ) is a monotonically increasing function.

Hence, we have shown that the optimal control problem satisfies all the hypothesis of Theorem 2.2 of [5]. Therefore, there exists a 8-tuple of optimal controls ( D 1 * ( t ) , D 2 * ( t ) , D 3 * ( t ) ) in the set of admissible controls U such that the cost function is minimized.□

5 Numerical studies for the optimal control problem

5.1 Theory

Here, we discuss the technique to evaluate the aforementioned optimal control problem. To evaluate the optimal control variables and the optimal state variables, we use the forward backward sweep method [26] and the Pontryagin maximum principle [22].

The Hamiltonian of the control Systems (1)–(10) is given by:

(12) ℋ ( I , V , D 1 , D 2 , D 3 , λ ) = I ( t ) + B ( t ) + P [ D 11 2 ( t ) + D 12 2 ( t ) + D 13 3 ( t ) ] + Q [ D 21 2 ( t ) + D 22 2 ( t ) + D 23 3 ( t ) ] + R [ D 31 2 ( t ) + D 33 2 ( t ) ] + λ 1 d S d t + λ 2 d I d t + λ 3 d B d t + λ 4 d I γ d t + λ 5 d T α d t + λ 6 d I 10 d t + λ 7 d I 12 d t + λ 8 d I 15 d t + λ 9 d I 17 d t ,

where λ = ( λ 1 , λ 2 , λ 3 , λ 4 , λ 5 , λ 6 , λ 7 , λ 8 , λ 9 ) is the as co-state variable/adjoint vector.

Now, using the Pontryagin maximum principle with D * = ( D 1 * , D 2 * , D 3 * ) and X * = ( x 1 , x 2 , … , x 9 ) being the optimal control and state variable, respectively, there exits an optimal co-state variable for which

(13) d x i d t = ∂ ℋ ( X * , D * , λ * ) ∂ λ i ,

(14) d λ i d t = − ∂ ℋ ( X * , D * , λ * ) ∂ x i .

Clearly, (13) will be equivalent to the control Systems (1)–(10) and system of ordinary differential equations for co-state variables should satisfy the following:

(15) d λ 1 d t = ( β B + μ 1 + γ + D 11 + D 21 − D 31 ) λ 1 − ( β B ) λ 2 , d λ 2 d t = ( μ 1 + δ + D 12 + D 22 ) λ 2 − ( α − D 23 2 − D 33 ) λ 3 − α I γ λ 4 − β T α I γ λ 5 − α I 10 λ 6 − β I 12 I γ λ 7 − β I 15 I γ λ 8 − β I 17 I γ λ 9 − 1 , d λ 3 d t = ( β S ) λ 1 − ( β S ) λ 2 + ( y + μ 2 + D 13 2 ) λ 3 − 1 , d λ 4 d t = μ I γ I λ 4 − β T α I λ 5 + δ I γ I 10 λ 6 − β I 12 I λ 7 − β I 15 I λ 8 − β I 17 I λ 9 , d λ 5 d t = δ T α I γ I λ 4 + μ T α λ 5 , d λ 6 d t = μ I 10 λ 6 , d λ 7 d t = δ I 12 I γ I λ 4 + μ I 12 λ 7 , d λ 8 d t = δ I 15 I γ I λ 4 + μ I 15 λ 8 , d λ 9 d t = δ I 17 I γ I λ 4 + μ I 17 λ 9 ,

and the transversality condition λ i ( T ) = ∂ ϕ ∂ t t = T = 0 for all i = 1 , 2 , … , 9 (where ϕ is the final cost function and here ϕ ≡ 0 ).

Now, to obtain the optimal value of the controls, we use the Newton’s gradient method for optimal control problem [9]. In the following, we discuss briefly on the go about of this method dealing with the iterative update of the control inputs for either minimizing or maximizing the objective function.

This process typically involves the following steps:

Calculate the gradient of the objective function with respect to the control inputs ( ∇ J ) , which provides information about how the objective function changes as the control inputs are adjusted. This calculation is made in (16).
Update the control inputs by moving in a direction that reduces the objective function value. This is performed through the line search optimization technique in this work. The equation is given in (17).
Solve the state equations for the new control inputs to obtain the corresponding state trajectories. We solved these equations using the Runge-Kutta fourth-order method.
Evaluate the objective function for the updated control inputs and state trajectories.
Check for convergence by comparing the change in the objective function value with a predefined tolerance. If the change is small, the optimization is considered complete.

As mentioned earlier for our case, we use the following recursive formula for updating the control in each step of numerical simulation, i.e.,

(16) D i j k + 1 ( t ) = D i j k ( t ) + θ k d k ,

where D i j k ( t ) is the value of the control at k th iteration at time instance t , d k is the direction, and θ is the step size. Usually, direction in Newton’s gradient method is evaluated by the negative of the gradient of the objective function, i.e., d k = − g i j ( D i j k ) , and here, we take g i j ( D i j k ) = ∂ ℋ ∂ D i j D i j k ( t ) as described in [9]. The step size θ is evaluated at each iteration by linear search technique that minimizes the Hamiltonian, ℋ . Therefore, the previous formula (16) will be as:

(17) D i j k + 1 ( t ) = D i j k ( t ) − θ k ∂ ℋ ∂ D i j D i j k ( t )

Now, to execute the aforementioned idea, we have to calculate the gradient for each control, i.e., g i j ( D i j ) , which are as follows:

g 11 ( D 11 ) = 2 P D 11 ( t ) − λ 1 S ( t ) , g 12 ( D 12 ) = 2 P D 12 ( t ) − λ 2 I ( t ) , g 13 ( D 13 ) = 3 P D 13 2 ( t ) − 2 λ 3 D 13 ( t ) B ( t ) , g 21 ( D 21 ) = 2 Q D 21 ( t ) − λ 1 S ( t ) , g 22 ( D 22 ) = 2 Q D 22 ( t ) − λ 2 I ( t ) , g 23 ( D 23 ) = 3 Q D 23 2 ( t ) − 2 λ 3 D 23 ( t ) I ( t ) , g 31 ( D 31 ) = 2 R D 31 ( t ) + λ 1 S ( t ) , g 33 ( D 33 ) = 2 R D 33 ( t ) − λ 3 I ( t ) .

5.2 Numerical simulations

In this section, we perform the numerical simulations to study the correlation of cytokine levels in Type 1 lepra reaction and the drugs involved in MDT in a qualitative manner.

The values of the parameters used are collected from various clinical articles and appropriate references are cited in Table 1. For some of the parameters such as μ , γ , and δ , the doubling time was available; hence, they were estimated using the following formula:

rate % = log ( 2 ) doubling time × 100 ≈ 70 doubling time .

We then divide these rate percentages by 100 to obtain the values of these parameters. Some of the cases we have taken average of the result yield from different medium such as 7 − A A D and T U N N E L as described in [30]. Some parameters are finely tuned in order to satisfy certain hypothesis/assumptions for convenience of numerical simulation.

For these simulations, we consider the time duration of 100 days, i.e., ( T = 100 ), and the parameter values are chosen as ω = 20.9 , β = 0.3 , μ 1 = 0.00018 , γ = 0.01795 , δ = 0.2681 , α = 0.2 , y = 0.3 , μ 2 = 0.57 , α I γ = 0.0003 , δ T α I γ = 0.00554 , δ I 12 I γ = 0.00903 , δ I 15 I γ = 0.00625 , δ I 17 I γ = 0.00499 , μ I γ = 2.16 , β T α = 0.004 , μ T α = 1.112 , α I 10 = 0.044 , δ I γ I 10 = 0.00146 , μ I 10 = 16 , β I 12 = 0.011 , μ I 12 = 1.88 , β I 15 = 0.025 , μ I 15 = 2.16 , β I 17 = 0.029 , μ I 17 = 2.34 , Q I γ = 0.1 , Q T α = 0.14 , Q I 10 = 0.15 , Q I 12 = 1.11 , Q I 15 = 0.2 , and Q I 10 = 0.317 .

First, we have solved the system numerically without any drug intervention. All the numerical calculations were made in MATLAB, and we used the fourth-order Runge-Kutta method to solve the system of ODEs and to find the value of the θ in each iteration, we used the fminsearch() function of MATLAB. Here, we consider the initial value of the state variables as S ( 0 ) = 520 , I ( 0 ) = 275 , B ( 0 ) = 250 , I γ ( 0 ) = 50 , T α ( 0 ) = 50 , I 10 ( 0 ) = 75 , I 12 ( 0 ) = 125 , I 15 ( 0 ) = 125 , and I 17 ( 0 ) = 100 as in [12,21].

Furthermore, to simulate the system with controls, we use the forward-backward sweep method starting with the initial value of the controls as zero and estimate the state variables forward in time. Since the transversality conditions have the value of adjoint vector at end time T , so the adjoint vector was calculated backward in time.

Using the value of state variables and adjoint vector, we calculate the control variables at each time instance that get updated in each iteration. The strategy to update controls is followed by implementing the Newton’s gradient method as expressed in equation (17). We continue this till the convergence criterion is met as in [9].

The weights P , Q , and R in the cost function J min are chosen based on their hazard ratio of the corresponding drugs. We chose the weights directly proportional to the hazard ratios. In Table 2, the hazard ratios of the different drugs are enlisted. We have chosen the weights ( P , Q , and R ) proportional to the hazard ratios, i.e., P = 1 , Q = 1.99 , and R = 7.1 .

Table 2

Hazard ratio of the drugs

Drugs	Hazard ratio	Source
Rifampin	0.26	[2]
Dapsone	0.99	[8]
Clofazimine	1.85	[8]

We now numerically simulate the S , I , and B populations and the cytokine levels with single control intervention, with two control interventions, and finally with three control interventions of MDT. In each of these plots, we also depict the no control intervention case for comparison purpose. In the next part, we will discuss the findings of these simulations.

5.3 Findings

In each of the figures discussed in this section, the first three frames (1–3) depict the dynamics of the respective compartments in Models (1)–(9) with individual drug administration, a combination of administration of two drugs, and all the three drugs in MDT administration, respectively, as control variables/interventions. The further frames in the following are the magnified versions of either Frame-1 and Frame-2 or Frame-1, Frame-2 and Frame-3 and are depicted for better clarity purpose to the reader.

Figure 2 depicts the dynamics of the susceptible cells S ( t ) with individual drug administration, a combination of two drugs in MDT administration, and all the three drugs in MDT administration, respectively. From Frame-1 and its magnification, it can be seen that the drug combination of clofazimine and dapsone has a positive impact on susceptible cell count, i.e., these drugs increase the count of susceptible cell, and among them, clofazimine acts most effectively. On the other hand, rifampin decreases the count of the susceptible cells. In case of combination of two drugs, a combination of rifampin and dapsone decreases the number of susceptible cells. But other two combinations increase the count, and among them, a combination of clofazimine and dapsone has more impact in increment of the cell count. Finally, Frame-3 depicts that the combination of three drugs increases the susceptible cell count.

$Figure 2 Plot depicting the dynamics of the susceptible cells S ( t ) S\left(t) on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1 and Frame-2 are also depicted for better clarity purpose to the reader.$

Figure 2

Plot depicting the dynamics of the susceptible cells S ( t ) on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1 and Frame-2 are also depicted for better clarity purpose to the reader.

Figure 3 depicts the dynamics of the infected cells I ( t ) on the administration of MDT drugs. Analyzing the magnified Frame-1, we see that each of the MDT drugs is helpful in reducing the infected cells. In the context of effectiveness, we see that the clofazimine takes the top position followed by dapsone and then by rifampin. From the magnified Frame-2, we see that among the combination of two drugs, the combination consisting of clofazimine and dapsone acts most effectively, whereas the combination of rifampin and dapsone has the least impact. All the three drugs of MDT when administered in combination reduce the infected cell count the best.

$Figure 3 Plot depicting the dynamics of the infected cells I ( t ) I\left(t) on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1 and Frame-2 are also depicted for better clarity purpose to the reader.$

Figure 3

Plot depicting the dynamics of the infected cells I ( t ) on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1 and Frame-2 are also depicted for better clarity purpose to the reader.

From the magnifications in Figure 4, we see that clofazimine is the most effective and rifampin is the least effective drug in reducing the bacterial load when drugs are administered individually. In case administration of combination of two drugs, dapsone and clofazimine combination, has the most impact and rifampin and dapsone has the least impact, all the three drugs of MDT when administered in combination reduce the bacterial load the best.

$Figure 4 Plot depicting the dynamics of the bacterial load B ( t ) B\left(t) on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1 and Frame-2 are also depicted for better clarity purpose to the reader.$

Figure 4

Plot depicting the dynamics of the bacterial load B ( t ) on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1 and Frame-2 are also depicted for better clarity purpose to the reader.

Figure 5 provides us the most important information that without any intervention of drugs, the level of IFN- γ goes on decreasing during lepra reaction, and upon the administration of drugs in MDT, the levels of the IFN- γ get enhanced. In the case of administration of the drugs individually, we see that clofazimine enhances the levels of IFN- γ the highest followed by dapsone and rifampin. In case of combination of administration of two drugs, we see that dapsone and clofazimine enhance the levels of IFN- γ the highest followed by rifampin and clofazimine and then by rifampin and dapsone.

$Figure 5 Plot depicting the dynamics of the IFN- γ \hspace{0.1em}\text{IFN-}\hspace{0.1em}\gamma on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1, Frame-2, and Frame-3 are also depicted for better clarity purpose to the reader.$

Figure 5

Plot depicting the dynamics of the IFN- γ on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1, Frame-2, and Frame-3 are also depicted for better clarity purpose to the reader.

Figure 6 clearly depicts that the level of TNF- α increases during lepra reaction. But the different combinations of drugs in MDT slow down the rate of increment of the level of TNF- α . In case of individual administration of drugs, clofazimine is the best for suppressing the increment of the levels of TNF- α followed by dapsone and then by rifampin. In case of combination of administration of two drugs, dapsone and clofazimine combination works the best followed by rifampin and clofazimine combination and then by rifampin and dapsone combination. Similar behaviours can be observed for the cytokines IL-15, IL-17 which are depicted in Figures 7 and 8, respectively.

$Figure 6 Plot depicting the dynamics of the TNF- α \alpha on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1 and Frame-3 are also depicted for better clarity purpose to the reader.$

Figure 6

Plot depicting the dynamics of the TNF- α on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1 and Frame-3 are also depicted for better clarity purpose to the reader.

$Figure 7 Plot depicting the dynamics of the IL- 15 \hspace{0.1em}\text{IL-}\hspace{0.1em}15 on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1 and Frame-2 are also depicted for better clarity purpose to the reader.$

Figure 7

Plot depicting the dynamics of the IL- 15 on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1 and Frame-2 are also depicted for better clarity purpose to the reader.

$Figure 8 Plot depicting the dynamics of the IL- 17 \hspace{0.1em}\text{IL-}\hspace{0.1em}17 on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1 and Frame-2 are also depicted for better clarity purpose to the reader.$

Figure 8

Plot depicting the dynamics of the IL- 17 on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1 and Frame-2 are also depicted for better clarity purpose to the reader.

From Figures 9 and 10, it can be seen that during lepra reaction, the levels of both IL- 10 and IL- 12 cytokines decrease. But the different combinations of drug interventions of MDT can further enhance the rate of decrement. When drugs are applied individually, rifampin has the less impact in enhancing the decrement as compared to other drugs. Clofazimine has the most impact on enhancing the decrement. The degree of enhancement of the rate of decrement of the cytokine levels in case of combination of two drugs follows the order, rifampin and dapsone < rifampin and clofazimine < dapsone and clofazimine. Finally, the combination of three drugs does the same impact on cytokine level, which can be seen in Frame-3 of Figures 9 and 10, respectively.

$Figure 9 Plot depicting the dynamics of the IL- 10 \hspace{0.1em}\text{IL-}\hspace{0.1em}10 on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1, Frame-2, and Frame-3 are also depicted for better clarity purpose to the reader.$

Figure 9

Plot depicting the dynamics of the IL- 10 on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1, Frame-2, and Frame-3 are also depicted for better clarity purpose to the reader.

$Figure 10 Plot depicting the dynamics of the IL- 12 \hspace{0.1em}\text{IL-}\hspace{0.1em}12 on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1 and Frame-2 are also depicted for better clarity purpose to the reader.$

Figure 10

Plot depicting the dynamics of the IL- 12 on the administration of MDT drugs individually (Frame-1), in combination of two (Frame-2) and all the three MDT drugs together (Frame-3). Magnified versions of Frame-1 and Frame-2 are also depicted for better clarity purpose to the reader.

6 Discussion and conclusions

The novel thing about this article is that it deals with a model that includes the dynamics of the levels of crucial cytokines that are involved in Type 1 lepra reaction, and also, this work studies the impact of different drugs in MDT on the levels of these cytokines. The findings of the studies includes the following:

Among the drugs used in MDT for treating leprosy, clofazimine and dapsone increase the susceptible cell count, whereas rifampin has a negative impact on it.
The two drug combination of rifampin and dapsone has the negative impact on susceptible cell count.
The MDT drug combinations decrease both the infected cell count and the bacterial load. Clofazimine works the best in reduction when each of the drugs is administered individually, and in combination of two drug administration, a combination of clofazimine and dapsone reduces the best.
During the Type I lepra reaction, the levels of IFN- γ , IL- 10 , and IL- 12 decrease, whereas the levels of TNF- α , IL- 15 , and IL- 17 increase.
Each of the drugs used in MDT enhances the IFN- γ levels in host body. Clofazimine enhances the best when each of the drugs is administered individually, and in combination of two drug administration, clofazimine and dapsone enhances the best.
The levels of both the cytokines IL- 10 and IL- 12 decrease on the administration of the drugs in MDT. Rifampin works the least in reduction when each of the drugs is administered individually, and in combination of two drug administration, rifampin-dapsone combination impacts the least.
In the case of TFN- α , IL- 15 , and IL- 17 , the drugs in MDT reduce the rate of increment of these bio-markers. Clofazimine is the best for suppressing the increment of these bio-markers when each of the drugs is administered individually, and in combination of two drug administration, clofazimine and dapsone turns out to be the best.
In summary, this is a novel and first of its kind work wherein we have discussed the natural history and dynamics of crucial bio-markers in a Type I lepra reaction and also studied in detail the influence of different combinations of drugs in MDT used for treating leprosy on the levels of these bio-makers. This study can be of important help to the clinician in early detection of the leprosy and avoid and control the disease from going to lepra reactions and help in averting major damages.

Acknowledgements

The authors dedicate this article to the Founder Chancellor of SSSIHL, Bhagawan Sri Sathya Sai Baba. The corresponding author also dedicates this article to his loving elder brother D.A.C. Prakash who still lives in his heart.

Funding information: This research was supported by Council of Scientific and Industrial Research (CSIR) under project grant – Role and Interactions of Biological Markers in Causation of Type1/Type 2 Lepra Reactions: A In Vivo Mathematical Modelling with Clinical Validation (Sanction Letter No. 25(0317)/20/EMR-II).
Conflict of interest: The authors declare no conflict of interest for this research work.
Ethical approval: This research did not require any ethical approval.
Data availability statement: We do not analyze or generate any datasets, because our work proceeds within a theoretical and mathematical approach.

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Received: 2023-07-17

Revised: 2023-11-13

Accepted: 2023-12-13

Published Online: 2023-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/cmb-2023-0117

Keywords for this article

Hansen’s disease; Type I lepra reaction; bio-markers; multidrug therapy; Newton’s gradient method; optimal drug regimen

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