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Mathematical Models: Unlocking Solution to Two Biological Frontiers 

 

Bevara Kondala Rao1,2   Biplab Kumar Rath1 and A. Manickam3* 
1Department of Mathematics, GIET University, Gunupur, Rayagada 765022, Odisha, India. 

2Department of Mathematics, SRI GCSR Degree College, Rajam 532127, Andhra Pradesh, India. 

3 School of Sciences, Division of Mathematics, SRM Institute of Science and Technology, 

Tiruchirappalli Campus, SRM Nagar, Trichy – Chennai Highway, Near Samayapuram, 

Tiruchirappalli –621105. Tamilnadu, India. 

Email: 1bevarakondala.rao@giet.edu, 2biplabkr@giet.edu, 3*manickammaths2011@gmail.com 

 

Article History: 

Received: 20-10-2024 

Revised: 04-12-2024 

Accepted: 11-12-2024 

Abstract: 

In this paper, we study two different biological problems by using mathematical 

formulations. In the first problem, our study is based on hyperprolactinemia of non-

cycling African elephant which is not associated with hyperestrogenism. There is no 

distinguishable pattern in the moderated and noticed hypoprolactiemic group, and 

moderate group is in base line. Time-based profile is absorbed in cyclic females, and 

elevations observed during follicular phase. In mathematical model, Weibull 

distribution is used. Medical results are analysed with corresponding mathematical 

model.In the second problem, a four-parameter generalized log-logistic distribution is 

introduced, using a quadrature rank transmutation map to create a transmuted four-

parameter log-logistic distribution. The reliability function for the four distributions is 

calculated. The standard model applies to the concentration of prolactin in dairy cows, 

with continuous probability and cumulative distribution functions for the four-

parameter log-logistic distributions. Finally, the implementation corresponds to the 

probability distribution, with results closely tied to medical reports. 

Keywords: Bromocriptine; Domperidone; Log-logistic distribution; Prolactin 

Hyperprolactinemia; non– Cycling elephant; Least square distribution; Simple 

percentile estimators 

Mathematical Subject Classification: 62𝐻𝑥𝑥 ; 62𝑁𝑂5 ; 90𝐵25. 

I) First problem: A mathematical model for hyperprolactinemia in noncycling African 

elephants 

 In applied mathematics and statistics, the Weibull distribution may be continued probability 

distribution. In 2020, Manickam [19] observed that the Weibull distribution is widely used in reliability 

and life data analysis due to its versatility. It was first identified by Frechet in 1927 [27]. In 1933, to 

describe a particle size distribution, researchers applied this distribution for the first time. It is named 

after Swedish mathematician Walodi Weibull, who described it in detail in 1951 [26]. It is a 

complementary cumulative distribution function which is also a stretched exponential function [18]. 

In 2021, Gayathri et al. in their study have said that depending on the values of the parameters, the 

distribution can be used to model a variety of life behaviour [12]. 

 



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1. Mathematical model 

 The probability density function of a two-parameter Weibull random variable is: 

𝑓(𝑥; 𝜌, 𝜏) =  
𝜏

𝜌
(

𝑥

𝜌
)

𝜏−1

𝑒𝑥𝑝 (− (
𝑥

𝜌
)

𝜏

), 

The Weibull distribution has two parameters: 𝜏 (shape) and 𝜌 (scale), 𝜏 > 0 and  𝜌 > 0. It is related to 

various other distributions, including the exponential and Rayleigh distributions. If 𝜏 < 1, the failure 

rate decreases over time; if 𝜏 = 1, the failure rate remains constant; and if 𝜏 > 1, the failure rate 

increases with time[6,11,20]. 

1.1 Estimation procedure: 

Estimation procedure is of 4 types: (a) method of moment estimators, (b) linear estimator like least – 

square type estimators, (c) estimators supported few order statistics and (d) maximum likelihood 

estimators [23]. 

1.1(a) Process of Moments 

Karl Pearson was the first person to introduce this method. Mathematical support for this procedure 

comes from the principal of moment [7]. This principle states that two distributions with a finite 

number of common lower moments will approximate each other. 

𝑘th moments is given by: 

𝑀𝑘 =  ∫ 𝑥𝑘𝑓(𝑥)𝑑𝑥

∞

−∞

 

The 𝑘th moment about the mean for the population is known to be 𝑘th central moment and is given 

by: 

𝐶𝑀𝑘 =  ∫(𝑥 − 𝑀1)𝑘𝑓(𝑥)𝑑𝑥

∞

−∞

 

A transformation of the extreme value distribution is commonly used to generate moment estimators 

as follows: If 𝑋 ~ Weibull (𝜌, 𝜏), then𝐹(𝑥) = 1 − exp [− (
𝑥

𝜌
)

𝜏

], then the variable 𝑇 = ln(𝑥) will 

have a extreme value scattering with distribution function. 

𝐹(𝑡) = 1 − 𝑒𝑥𝑝 [𝑒𝑥𝑝 {
𝑡 − 𝜌

 𝜏
}] 

The parameter 𝜌  and 𝜏 are parameters that be able to be estimated by the first two moments 𝑚1 and 

𝑐𝑚2 of the data on this scale [11,12,23]. 

1.1(b) Least Square Distribution 

To get standardized variable, the observed transformation is 𝑉𝑟 =  
(𝑇𝑟−𝜌 )

 𝜏
. 



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Let 𝑉𝑟 represent the rth order statistic from a sample of standardized variables. The expected mean, 

variance, and covariance are denoted by: 

𝐸(𝑉𝑟) =  𝐴𝑟 , 𝑉𝑎𝑟(𝑉𝑟) =  𝑊𝑟𝑟 and 𝐶𝑜𝑣(𝑉𝑖, 𝑉𝑗) = 𝑊𝑖𝑗, where 𝑖 is the row and 𝑗 is the column [8]. 

Then the least square estimate of the parameter 𝐵 are the ordinary weighted least square regression 

and is given by: 

𝐵𝑒𝑠𝑡 =  [𝑑′𝑊−1𝑑]−1𝑑′𝑊−1𝑇(𝑟) 

The resulting estimates will be best linear unbiased estimates and thus have very good properties [21]. 

The Weibull distributions were calculated for only limited sample size for an extreme value 

distribution [15,18,19,20]. 

1.1(c) Simple Percentile Estimators 

 Let 𝑡𝑖 , 𝑡𝑗 and 𝑡𝑘 be sample percentiles and 𝑝𝑖, 𝑝𝑗 and 𝑝𝑘 be approximate cumulative probability. 

Then 𝑡𝑠 =  𝛿 + 𝜌[−ln (1 − 𝑝𝑠)]
1

�̂�⁄ , where 𝑠 = 𝑖, 𝑗, 𝑘. 

After simplifying little, 
𝑡𝑘− 𝑡𝑗

𝑡𝑗− 𝑡𝑖
=

[−ln (1−𝑝𝑘)]
1

�̂�⁄ − [−ln (1−𝑝𝑗)]
1

�̂�⁄

[−ln (1−𝑝𝑗)]
1

�̂�⁄
−[−ln (1−𝑝𝑖)]

1
�̂�⁄
 

Further choose 𝑝𝑗 such that − ln(1 − 𝑝𝑗) =  {[−ln (1 − 𝑝𝑖)][−ln (1 − 𝑝𝑘)]}1 2⁄  

𝑝𝑗 = 1 − 𝑒𝑥𝑝{−[ln (1 − 𝑝𝑖) ln (1 − 𝑝𝑘)]1 2⁄ }  

Therefore the shape parameter �̂� =  
1

2
𝑙𝑛 (

[− ln(1−𝑝𝑘)]

[− ln(1−𝑝𝑘)]
) ln

𝑡𝑘− 𝑡𝑗

𝑡𝑗− 𝑡𝑖
⁄  

�̂� =  (𝑡𝑠 − 𝑡𝑟) {[−ln (1 − 𝑝𝑘)]
1

�̂�⁄ −  [−ln (1 − 𝑝𝑗)]
1

�̂�⁄
}⁄   

Where (𝑟 < 𝑠) ∈ (𝑖, 𝑗, 𝑘) 

1.1(d) Maximum Likelihood Estimation (MLE) 

The MLE is usually used in large sample efficiency. For two – parameter Weibull family, the density 

function is represented by:   

𝑓(𝑥; 𝜌, 𝜏) =  
𝜏

𝜌
(

𝑥

𝜌
)

𝜏−1

𝑒𝑥𝑝 (− (
𝑥

𝜌
)

𝜏

) 

Study a random trial of 𝑛  observations. The likelihood purpose of this trial is: 

𝐿(𝑥1, … … . , 𝑥𝑛;  𝜌, 𝜏) =  ∏ (
𝜏

𝜌
) (

𝑥

𝜌
)

𝜏−1
𝑛

𝑖=1

𝑒𝑥𝑝 (− (
𝑥𝑖

𝜌
)

𝜏

) 

By taking the logarithm of L and differentiating with respect to ρ and τ, then setting the derivatives to 

zero, we obtain two equations. 

𝜕 ln 𝐿

𝜕𝜏
=

𝑛

𝜏
+ ∑ ln 𝑥𝑖 −  

1

𝜌𝜏
∑ 𝑥𝑖

𝜏

𝑛

1

𝑛

1

ln 𝑥𝑖 



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𝜕 ln 𝐿

𝜕𝜌
=

𝑛

𝜌𝜏
+ ∑ ln 𝑥𝑖 −  

1

𝜌2𝜏
∑ 𝑥𝑖

𝜏

𝑛

1

𝑛

1

 

Eliminating ρ from the two equations and simplifying yields: 

[
∑ 𝑥𝑖

𝜏𝑛
1  𝑙𝑛𝑥𝑖

∑ 𝑥𝑖
𝜏𝑛

1

− 
1

𝜏
] =  

1

𝑛
∑ 𝑙𝑛𝑥𝑖

𝑛

1

 

This can then be solved iteratively to estimate the shape parameter, denoted by  �̂�. Then 

𝜌 ̂ =  ∑
𝑥𝑖

�̂�

𝑛

𝑛

1

 

2. Applications 

Prolactin is a protein and its main action is to convince caring behaviour and make the breast tissue for 

lactation [16], it is produced by lactotroph cells of the anterior pituitary [9].  PRL is concealed in a 

cyclic form in African elephants, with elevation in non – luteal phase [1] [3]. Hypothalamic dopamine 

controls PRL secretion by inhibitory mechanism via tuberoinfundibular dopaminergic [25]. Lyons and 

the team in 2012, Mohan Kumar and his team in 1997 and Simpkins and Gabriel in 1984 concluded 

that any intervention with dopamine’s mixture, release can control prolactin secretion [17]. 10 – 40% 

of the women’s foremost disorder of the hypothalamic – pituitary axis is hyperprolactinemia [22]. 

Same suspected problems are faced by zoo – managed elephants [1] [16] [28]. 

For long term survival, elephants are facing many threats. Zoo elephants are facing challenges 

associated with their population too [24]. Currently in US zoo, three calves are natural every year. To 

maintain this situation, six offspring are needed [2]. North American population may face a decline of 

2.3% in next 30 years and also female get aged and does not reproduce [24].  Thus, efforts are centred 

on increased reproductive output, decrease mortality and breeding al elephants which are 

reproductively viable may improve the populations long – term sustainability [2]. Breeding in zoo are 

logistical, a main problem because of inadequate housing to maintain large breeding group, expense 

of transporting female for breeding facilities [10].  

Most of the elephants face physiological problem associated with fertility [4] [13]. On observing the 

stable baseline concentrations of serum progestogens, most of the elephants are acyclic reveals 

Elephant Taxon Advisory Group/ Species Survival Plan [14]. In 2004, Brown and his team found that 

a third of elephants with abnormal cycles produced excessive prolactin, a condition known as 

hyperprolactinemia, which causes infertility in other species. A 2008 survey revealed that 11% of 

females had irregular ovarian cycles, while 31% did not cycle at all. Only 72% of the population were 

of reproductive age (11 to 35 years). In 2012, a study conducted by Dow and Brown [10] concluded 

that infertility problem has increased significantly. 71% of the American elephants are acyclic with 

hyperprolactinemia. After monitoring the African elephants for 20 years, the consequence of the study 

is only non – cycling elephants are identified through hyperprolactinemia [2] [10].  

  



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Figure 1 (a). PRL of three non-cycling 

African Elephant 

 
Figure 1 (b) Progestagen, PRL and 

Estrogen profile of three cycling African 

Elephant 

 
Figure 1 (c) Estrogen conjugate of three abnormal ovarian cycles 

In figure 1(a), PRL secretion remained varied with no distinct pattern in the reasonable and clear 

hyperprolactinemia group and concentration in low prolactin relatively stable baseline in non – cycling. 

In figure 1(b), there is elevation in the concentration were observed in follicular phase and also 

temporal profile observed in cycling female. In figure 1(c), secretion remained varied by no 

distinguished shape and infrequently got baseline. 

3. Mathematical result  

 
Figure 2(a) probability density function PRL of three non-cycling African Elephant 

The plot of probability density function of PRL of three non-cycling African Elephant in figure 2(a) 

indicates low prolactin shows its superiority than the functions and moderate hyperprolactinemia 



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control. The plot of initially low prolactin monotonically increasing up to t=7 hrs and low prolactin 

then decreasing monotonically. The rate of decreasing is comparatively good than and control 

functions.  

 
Figure 2(b) probability density function Progestagen, PRL and Estrogen profile of three 

cycling African Elephant 

The plot of probability density function of Progestagen, PRL and Estrogen profile of three cycling 

African Elephant in figure 2(b) indicates Progestagen, shows its superiority than the functions and 

prolactin control. The plot of initially Progestagen, monotonically increasing up to t=13 hrs and 

Progestagen, then decreasing monotonically. The rate of decreasing is comparatively good than and 

control functions.  

 
Figure 2(c) probability density function estrogen conjugate of three abnormal ovarian cycles 

The plot of probability density function of estrogen conjugate of three abnormal ovarian cycles in 

figure 2(c) indicates low prolactin shows its superiority than the functions and moderate 

hyperprolactinemia control. The plot of initially low prolactin, monotonically increasing up to t=9 

hrs and low prolactin, then decreasing monotonically. The rate of decreasing is comparatively good 

than and control functions.  

4. Conclusion 

 Individual PRL concentrations in hyperprolactinemic elephants were higher than those 

observed during gestation, although not as high. The examination of a putative temporal effect in non-

cycling, hyperprolactinemic elephants revealed no significant correlation between oestrogen conjugate 

and PRL secretion. Meanwhile, lag time effect models for cycling elephants discovered that oestrogen 

may influence subsequent PRL secretion, i.e., higher oestrogen conjugate concentration associated 

with increased PRL. The function's monotonicity has discovered using a mathematical model.  



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II) Second problem: Impact of mixing prolactin levels in dairy cows using a generalised Log-

Logistic distribution model 

 The logistics distribution is very useful in the field of analyzing survival. The 

probability density function (pdf) in its simplest form is given by  

∅(𝑧) =  
1

(1 + 𝑧)2
  𝑧 > 0                                                         (1) 

We studied [30,41,44,46], the logistical model with pdf given by the distribution  

∅(𝑧: 𝛾, 𝛿) =
𝛿𝛾𝛿𝑧𝛿−1

(𝛾𝛿 + 𝛾𝛿)2
𝑧 > 0 , 𝛾 > 0, 𝛿 ≥ 1                 (2) 

Where 𝛾 is the parameter of the scale and 𝛿is the shape parameter of the form. 

Four parameters widespread logistics distribution is given by probability density function 

𝜑(𝑧) =
𝛿𝜔𝛼𝜔𝑧𝛿−1

(𝛿 + 𝑧𝛿)𝜔+1
   𝑧 > 0 , 𝛾 > 0, 𝛿 ≥ 1 𝜔 > 0       (3) 

Where 𝜔 > 0 is the extra shape parameter. He established its properties and stated that some 

theorems relate it to some other distributions [30]. 

1. Log Logistic Model and descriptions 

The random variable Z implements four wide-ranging logistical classification parameters with the 

probability function noted in [42, 44] and the accumulated probability function F(z) in [44].   

 𝐹(𝑧) = 1 − (
𝛾

𝛾+ 𝛾𝛿)
𝜔

 𝛾 > 0                                                     (4) 

In the above equation, γ is the scale parameter whereas ω and δ are the shape parameters. Using the 

quadratic rand map, an appropriate transmuted four parameters widespread log-logistic distribution is 

obtained as 

𝑀(𝑧) = (1 + 𝜌) 𝑀(𝑧) − 𝜌𝑀2(𝑧), [𝜌] ≤ 1 − − − − − (5)is given by 

𝑀(𝑧) =
(𝛾 +  𝛾𝛿)

𝜔
𝑧𝛿−1{(1 − 𝜌)(𝛾 +  𝛾𝛿)

𝜔
+ 2𝜌𝛾𝜔

(𝛼 +  𝛼𝛽)2𝜔+1
, 𝑧 > 0 − − − −(6) 

and the corresponding pdf is given by 

𝑔(𝑧) =
𝛿𝜔𝛾𝜔 𝑧𝛿−1{(1 − 𝜌)(𝛾 +  𝛾𝛿)

𝜔
+ 2𝜌𝛾𝜔

(𝛼 + 𝛼𝛽)2𝜔+1
, 𝑧 > 0              − − − −(7) 

The parameter ρ > 0 is the transmuted exponential distribution [38,39 ]. 

The reliability function r(t) and cumulative distribution function F(t) represent the probability that a 

component will not fail before time t. 

    r (t) = 1- F (t). 

 



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The widespread log-logistic distribution’s reliability function (four parameters) is given by 

𝑟(𝑡) =
(1−𝜌)𝛾𝜔(𝛾+ 𝑡𝛿)

𝜔
+𝜆𝛼2𝜔

(𝛾+ 𝑡𝛿)2𝜔
[32,33] 

2. Applications of widespread log-logistic distribution 

In this study, the dataset for the numerical illustration is taken from the reference [47]. 

Prolactin (PRL) is required for the maintenance of lactation in most warm-blooded creatures, hence 

restricting PRL inhibits lactation [37]. In many well-developed species, including livestock, rodents, 

and primates, prolactin intake during lactation prevents or reduces production. But this isn't always the 

case with large lactating species; if the yield is minimal, it's as if it's a small sum. However, the 

importance of PRL in managing lactation in ruminants is uncertain, as short-term inhibition of PRL 

with bromocriptine had inconsistent effects on drain production [38]. 

In a subsequent stage, it appeared that daily infusions of dopamine antagonists domperidone for 5 

weeks improved PRL discharge. Drain generation increased continuously and was higher in 

domperidone-treated bovines. Both Quinagolide and cabergoline have been reduced drain generation 

on the first day of treatment in late-lactation bovines and generated more rapid alterations in a few 

udder indicators [40]. The role of prolactin in galactose production (PRL) in ruminants has been 

debated for decades. In dairy bovines, Knight (1993) found that restricting PRL with quinagolide (QN) 

reduced drain generation [36]. Meanwhile, Lacasse and Ollier discovered in 2015 that infusing the 

dopamine antagonist domperidone (DOMP) into dairy cows increased drain generation and baseline 

PRL concentration [38]. As a result, there is currently significant evidence that PRL is a film-forming 

specialist in dairy cows. 

Standard PRL concentrations are altered by the environment throughout the year, regardless of drain 

generation. In 1973, Koprowski and Trucker concluded in their research study that the affectability of 

the mammary organ to PRL is versatile [37]. Lacasse et al., 2014 discovered that less circulating PRL 

has increased the drain generation after the dry period [39,41]. In their inquiry ponder in 2013; Tao 

and Dahl came to a conflicting result. They discovered that a warm push increased PRL concentrations, 

and that cooling bovins during the drought period accelerated drain development [48]. McKinnon et 

al. in 1988 witnessed that the mammary gland’s capacity to bind PRL may be well expanded by 

expanding the recurrence of lactation. To extend the reactivity of the mammary organ to PRL, as it 

were an instrument to extend the number of PRL receptors [45]. Knight (1993) fed drain to goats and 

discovered that increasing one-way draining recurrence increased the goats' drain reaction to PRL 

organization [36].  



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Figure 3: CTL and QN cow milk production rates 

Before the study began, drain production levels were similar in QN and CTL cows. The figure over 

clearly appears that amid handling, drain abdicate diminished but was not influenced. Amid the DPKO 

period, bovines infused with QN created more drain than the control gather. Compared with control 

bovines, drain generation in dairy animals infused with QN was higher within the post-treatment 

period. The drugs had no effect on the drain fat substances during the investigation. During the 

treatment phase, QN diary animals had higher drain protein than CTL bovines, while the other periods 

were unaffected. There was no evident change in the lactose composition of the drain over the 

treatment period. Fat, protein, and lactose yields were compared during pre-treatment, treatment, and 

DPKO stages. Fat yield was higher in QN cows than in CTL cows, with protein and lactose also higher 

in QN cows post-treatment. Additionally, energy-corrected drain was higher in QN cows during the 

post-treatment period. 

 
Figure 4(A): The serum prolactin CTL and QN cows 

 In both bunches of bovines, pattern serum PRL concentrations were comparative earlier to the 

beginning of the dry. Figure 4(A) over clearly appears that PRL levels are moderately exceptionally 

more in QN bovines compared with CTL diary animals amid the treatment period. Serum PRL levels 

were higher during the DPKO period. During the post-treatment phase, No significant difference is 

observed between the CTL and QN groups. 



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Figure 4(B): The prolactin levels in the milk of CTL and QN cows 

Figure 4(B) clearly shows that the PRL concentration of milk was unaffected throughout the 

experiment. However, there was a shift in PRL levels in milk during DPKO and after treatment. 

 
Figure 4(C):  The Serum PRL levels of QN and CTL cow milk 

Figure 4(C) shows no difference in daily steady-state serum PRL proportions between QN and CTL 

cows during pre- and post-treatment. However, QN cows showed a significant increase during 

treatment, while CTL cows tended to be higher. During the DPKO period, PRL levels decreased in 

both groups. 

3. Mathematical Result  

The comparison results are shown in Figure 3(A) as a monotonic function to depict the ratio of milk 

production over time. The amount of milk production during the initial lactating period is same for 

both of the groups such as control group (CTL) and the quinagolide injected experimental group (QN). 

After time period 13, the rate of milk production for the QN group is higher than the CTL group. 

 
Figure 5(A): Probability density functions of production of milk: Control Group (water injected 

cow) Versus Experimental Group (quinagolide injected cow) 



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The comparison results are shown in Figure 5(B) as a monotonic function to depict the rate of serum 

prolactine production over time. The rate of serum prolactine production during the initial lactating 

period is the same for both of the groups such as the control group (CTL) and the quinagolide injected 

experimental group (QN). After time period 9, the rate of serum prolactine production for the CTL 

group is higher than the QN group. 

 
Figure 5(B): Probability density functions of Serum Prolactin Secretion:  Control Group (water 

injected cow) Versus Experimental Group (quinagolide injected cow) 

The comparison results are shown in Figure 5(C) as an exponential function to depict the rate of milk 

prolactine production over time. The rate of serum prolactine production during the initial lactating 

period in terms of function growth for both of the groups such as the control group (CTL) and the 

quinagolide injected experimental group (QN). After time period 13, the rate of milk prolactine 

production becomes a sudden drop in the CTL group than the QN group. 

 
Figure 5(C): Probability density functions of Milk Prolactin Secretion: Control Group (water 

injected cow) Versus Experimental Group (quinagolide injected cow) 

The comparison results are shown in Figure 5(D) as an exponential function to depict the steady-state 

serum PRL secreted milk proportions on a daily basis. No significant difference is observed between 

the CTL and QN groups. 

 
Figure 5(D): Probability density functions of daily steady-state serum PRL secretion in milk: 

Control (water-injected) vs. Experimental (quinagolide-injected) cows. 



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4. Discussion and Resolution 

The mammary gland is a complex tissue with roles in development, secretion, and growth. The organ 

is controlled by a complex system of hormones and endocrine components, as well as surrounding 

auto endocrine movement. Prolactin may be the major component of these complexes, in a few species 

the foremost critical single component. The study findings support the theory that PRL levels influence 

the mammary gland's response to the hormone. The numerical demonstration using a generalized log-

logistic distribution model, the mammary gland response for the quinagolide injected experimental 

group is gotten the PRL levels of theoretical study. Drain prolactin and the steady-state serum PRL 

proportion emitted day by day from cow's milk were also constant and mitotically reduced. This article 

will soon prove highly valuable in medicine and prolactin capacity development. 

Acknowledgement: NA 

Funding information: This research received no specific grant from any funding agency, commercial 

or non-profit sectors. 

Conflict of interest: The authors have no conflicts of interest to disclose.  

Ethical approval: This research did not require ethical approval. 

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