Nosocomial infections, such as those caused by carbapenem-resistant Klebsiella pneumoniae (CRKP) transmitted into hospitals, worsen patient outcomes, strain healthcare infrastructure, increase treatment costs, prolong hospital stays, and complicate medical treatment due to their high level of antibiotic resistance. Taking into account these detrimental aspects, this study deals with a dynamic system of nonlinear differential equations with maintenance preventive strategies, and treatment applying combination therapy has been considered to prevent CRKP transmission and reduce antibiotic resistance while accounting for two hospital populations. The analytical analysis includes positivity, boundedness, existence of a unique solution, stability at equilibrium points, reproduction number (R0), sensitivity, and convergence of state variables. And, the numerical simulations were performed using the fourth-order Runge–Kutta method. We find that the proper implementation of hygiene practices among hospitalized patients and the consistent use of personal protective equipment (PPE) by healthcare workers play an important role in preventing CRKP transmission in hospitals. Additionally, applying combination therapy as a treatment approach for CRKP reduces antibiotic resistance. Consequently, this improves hospital safety and ensures a healthier environment by minimizing the spread of infectious agents and reducing the risks of nosocomial infections.
1. Introduction
The infections that are acquired while receiving medical care in healthcare settings are defined as nosocomial infections, also known as healthcare-associated infections. These infections can occur in various environments, such as hospitals, ambulatory care centers, and long-term care facilities. Additionally, nosocomial infections can manifest even after a patient has been discharged from a hospital [1]. These infections usually occur within 48 h of being admitted to the hospital, 3 days after discharge, or 30 days after surgery and afflict one in every 10 patients hospitalized in the hospital [2]. Common Gram-negative organisms associated with nosocomial infections include Klebsiella pneumoniae (KP), Klebsiella oxytoca, Escherichia coli, Proteus mirabilis, and Enterobacter species. These infections developed rapidly within a short time period but became a serious health problem during the antibiotic period, contributing to greater antibiotic consumption, longer hospital stays, and rising healthcare costs [3].
Resistance to antibiotics occurs when bacteria, viruses, fungi, and parasites develop to resist medicines, providing ineffective treatments, making infections more difficult to control, and increasing the risk of disease transmission, severe illness, and mortality [4]. The World Health Organization [5] reported that KP has shown resistance to common antibiotics, such as carbapenems. This is a Gram-negative bacterium that belongs to the family of Enterobacteriaceae with the Klebsiella genus [6]. In recent years, Klebsiella species have emerged as critical pathogens responsible for nosocomial infections. A significant number of antibiotic resistance genes are carried by KP. These genes are exchanged with other human pathogens via plasmids. One common recipient of AMR genes from KP is Salmonella [7]. Nowadays, the KP bacterium has developed resistance to antibiotics, and the overuse or misuse of antibiotics has contributed to the emergence of carbapenem-resistant Klebsiella pneumoniae (CRKP) [8]. CRKP has emerged as a significant public health concern due to its high resistance to carbapenem antibiotics, which are commonly used to treat bacterial infections. The emergence of CRKP is attributed to a variety of causes, including antibiotic abuse and misuse, horizontal transfer of resistance genes via plasmids, and worldwide travel, which assists in the spread of resistant bacteria. The majority of infections affect hospitalized patients, and there have been multiple outbreaks noted in hospitals [9]. In hospitals, CRKP transmission can occur through [10] direct person-to-person contact, contact with contaminated water and soil, contact with contaminated equipment, or a wound caused by injury or surgery. Over the last decade, the global outbreak of CRKP has emerged as a major public health concern, owing to its fast spread in healthcare settings and the rising prevalence of multidrug-resistant infections. The discovery of this infection in the United States was in 1996 [11]. From 2008 to 2013, CRKP infection in Italy increased to 60%. In 2014, this plasmid-mediated carbapenemase gene was identified in China after being horizontally spread around the world [12]. In 2016, the nationwide transmission rate of CRKP was 8.7%, with a prevalence ranging from 0.9% to 23.6% in various regions of China [13]. According to statistics from the Centers for Disease Control and Prevention, 13,100 hospitalized patients experienced CRKP infections in 2017 [14]. Carbapenem resistance rates are troubling, as the incidence of CRKP in China rose from 6.4% to 11.3% between 2014 and 2021, highlighting that CRKP continues to be a major multidrug resistance issue in the country [15]. As mentioned in some studies [16–19], some major infections that are associated with CRKP are bloodstream infections, pneumonia, UTIs, wound infections, meningitis, osteomyelitis, endocarditis, liver abscess, and colonization of the gastrointestinal tract.
Sypsa et al. [20] explored the transmissibility of CRKP in a tertiary care hospital and examined the effects of different infection control interventions. A hypothetical transmission map was carefully constructed to illustrate potential pathways of spread from epidemiological data, and a range of preventive and control strategies were thoughtfully proposed to reduce the transmission risks [21]. Another study [22] examined the prevalence and evolutionary traits of ST11-KL64 hv-CRKP isolates, stressing the immediate necessity for actions to monitor and control the spread of these highly resistant and virulent superbugs, which presented a significant danger to healthcare. Combination therapy plays a significant role in managing infections and combating antibiotic resistance [23]. According to them, the combination of CAZ/AVI and tigecycline increased the likelihood of effectively eliminating pathogens, as it targets bacteria from multiple angles, reducing their ability to survive or develop resistance. Numerous studies incorporate mathematical models to analyze and predict the epidemiological dynamics of infectious and noninfectious diseases. In a study [24], a mathematical model has been presented to evaluate the combined effects of chemotherapy and surgery, which have helped determine the optimal treatment strategy for lung cancer. Also, another study [25] developed a fractional model that incorporates the effects of lockdown measures to evaluate their role in controlling the spread of COVID-19. El-Mesady and Ali [26] proposed an optimal control approach to manage and reduce the spread of chickenpox outbreaks. Other related disease models include HIV [27], Nipah virus [28], lumpy skin disease [29, 30], hepatitis B [31], Zika virus [32], and many more.
In a study [6], they formulated a mathematical model for CRKP infection and presented the effects of various control measures and applying an antibiotic treatment systematically. However, they did not account for pathogen transmission between patients and hospital workers, and the treatment strategies took more time. Another study [33] demonstrated a transmission model of CRKP based on the Ross–Macdonald model. They introduced AMS and IPC interventions using clinical data but did not address the treatment of the disease.
Motivated by the growing threat of CRKP infections, this study proposes a mathematical model to describe the epidemiology of CRKP infection between two major populations in a hospital and shows how abuse of antibiotics makes resistance to antibiotics of the disease. Our main aim is to present a prevention strategy to minimize the spread of CRKP in hospitals and to consider combination therapy as an effective treatment to reduce antibiotic resistance.
The organization of the paper is as follows: Section 2 presents a mathematical model on the transmission dynamics of CRKP. In Section 3, an analytical analysis of this model is conducted. The positivity, boundedness, equilibrium points, reproduction number (R0), stability, sensitivity, and convergence of state variables have been included in this section. Also, the sensitivity analysis is presented in Section 4. Section 5 contains the numerical simulations, and the result has been illustrated with graphical representations. Further, Section 6 presents the findings and outcomes of our work. Finally, concluding remarks are provided in Section 7.
2. Model Formulation
The transmission dynamics of CRKP occur in a hospital between two different human populations, such as patients and healthcare workers. In this newly formed disease model, we have considered the whole human population into six major compartments: susceptible (SP), healthcare workers (SH), exposed (E), infected (I), resistant (IR), and recovered (R). The model flow diagram is illustrated in Figure 1. The total population can be defined as follows:
where N(t) represents the whole population of this compartmental model. The susceptible patients SP(t), the people who come to a hospital for treatment, are highly susceptible to infections caused by CRKP. The pathogen is transmitted to susceptible patients through direct physical contact or by contaminated equipment that has been in contact with infected individuals or resistant populations. This is defined by the following ordinary differential equation:
()
where the recruitment rate of patients is denoted by Λ, while the transmission rates of the disease from infected and resistant populations are represented by β1 and β2, respectively. And the fractional rate of santitation is ε, 0 ≤ ε ≤ 1. Patients who have recovered from their illness are discharged from the hospital at a rate of ξ. The parameter τ1 indicates the rate at which the recovered patients become susceptible again.
The diagram illustrates the transmission dynamics of CRKP within a hospital.
Healthcare workers SH(t), including doctors, nurses, surgeons, and hospital staff, are at risk of being infected by CRKP when treating patients. They can be affected by the pathogen through physical contact with infected patients or contaminated surfaces and equipment. Susceptible healthcare workers can be shown by the following equation:
()
where α represents the newly appointed healthcare workers. In this equation, γ1 and γ2 represent the rates of transmission of the disease from infected and resistant populations, respectively. The parameter ϕ denotes the fractional rate of personal protective equipment (PPE) in preventing transmission, 0 ≤ ϕ ≤ 1. The parameter τ2 refers to the rate at which recovered healthcare workers become susceptible again.
The pathogen has already been transmitted to certain susceptible patients and healthcare workers, but they have not yet shown any symptoms or developed the disease, exposed population E(t). The following ordinary differential equation represents this population:
()
where the rate at which individuals in the exposed population develop symptoms of the disease is represented by the parameter ρ.
The infected population I(t) in CRKP disease models includes individuals who have contracted the infection and can spread the pathogen to susceptibles. The following equation denotes this:
()
where η reflects the rate of how quickly patients receive antibiotic treatment and c indicates the rate at which antibiotics are consumed in the population, which can impact the development of antibiotic resistance. A portion of the population can recover from the disease due to antibiotic treatment and strong immunity at a rate represented by σ. The death due to disease is denoted by ω1.
CRKP disease is highly resistant to multiple drugs, making it challenging to treat effectively with standard antibiotics. The failure of antibiotic treatment contributes to the persistence of resistant populations IR(t) in hospital environments, increasing the risk of transmission and complications. The following ordinary differential equation represents this population:
()
where the parameter δ represents the rate of combination therapy, an effective treatment strategy against resistant infections like CRKP. The death caused by the disease due to antibiotic resistance is denoted by ω2.
The recovered population R(t) in CRKP disease models represents individuals who have successfully overcome the infection through effective treatment, such as combination therapy, natural immune response, and antibiotic treatment. Then, our last equation will be,
()
The parameter μ represents the rate of natural death of all populations.
We have considered several assumptions to formulate a dynamical mathematical model for the transmission of CRKP between two populations and its treatment using combination therapy.
•
Patients who appear at the hospital are considered susceptible to CRKP.
•
Every healthcare worker is also considered susceptible to CRKP.
•
We consider only horizontal transmission.
•
Natural death rate is the same.
Therefore, we get the following system of equations:
()
The system of Equation (7) is subjected to the initial conditions: SP(0) ≥ 0, SH(0) ≥ 0, E(0) ≥ 0, I(0) ≥ 0, IR(0) ≥ 0, and R(0) ≥ 0.
The flow diagram of a newly formulated model of CRKP disease is illustrated in Figure 1.
3. Analytical Analysis
3.1. Positivity Analysis and Boundedness of the Model
Theorem 1 (positivity). All solutions SP(t), SH(t), E(t), I(t), IR(t), and R(t) are non-negative satisfying SP(0) ≥ 0, SH(0) ≥ 0, E(0) ≥ 0, I(0) ≥ 0, IR(0) ≥ 0, and R(0) ≥ 0 at any instant t ≥ 0.
Proof 1. At first, we consider the first equation of Model (7).
Therefore, SP(t), SH(t), E(t), I(t), IR(t), and R(t) of the proposed Model (7) is non-negative ∀t ≥ 0.
Theorem 2 (boundedness). The solution trajectories (SP(t), SH(t), E(t), I(t), IR(t), and R(t)) of the system are bounded.
Proof 2. We have the total population of this epidemiological model.
()
Now, differentiating Equation (14) with respect to t,
()
Adding all the equations of System (7) and then simplifying, we get
When the death rate due to disease of infected and resistant populations is absent (ω1 and ω2 are zero), neglecting the term ξSP.
Using initial conditions and solving the above equation,
When t⟶∞, then
Therefore, the model is bounded, and also the model of System (7) is well-posed.
3.2. Existence and Uniqueness of the Model
Theorem 3. Let D be the domain defined in such a way that Lipschitz conditions are satisfied. Then, for all non-negative initial conditions, the solution of the system exists, and they are also unique at the same time for all time T ≥ 0 in the domain D.
Proof 3. According to the study [34], we have to show that
()
is continuous and bounded in the domain D. From the first equation of System (7), we can write,
()
Differentiating with respect to SP, we get
()
Again, differentiating with respect to SH, E, I, IR, and R, respectively, we get
In this same manner, we can easily show that,
Thus, all the partial derivatives are bounded and continuous, satisfying the Lipschitz condition.
Therefore, a unique solution to System (7) exists in the region D.
3.3. Equilibrium Points of the System
The equilibrium point of the system of differential equations can be defined by taking the rate of change of the population is equal to zero. Hence, we can write Equation (7) of the model as dSP/dt = 0, dSH/dt = 0, dE/dt = 0, dI/dt = 0, dIR/dt = 0, and dR/dt = 0. Therefore,
()
()
()
()
()
()
In the disease model, the point where there are no diseases is defined as a disease-free equilibrium point. Let E0 the disease-free equilibrium point and the point is .
The endemic equilibrium point is defined as a steady-state solution where the disease exists in the population.
The endemic equilibrium point is denoted as . By solving equations from (19) to (24), then we get the endemic equilibrium point.
The existence of an endemic equilibrium point of this model arises.
3.4. Reproduction Number R0
Reproduction number R0 serves as a threshold indicator for determining the stability of a disease-free state [35]. We have used the next-generation matrix approach to determine the basic reproduction number [36]. The new infection matrix and its transfer matrix can be denoted by F and V, respectively.
The basic reproduction number R0 is the spectral radius of the next-generation matrix FV−1. Therefore, the reproduction number in our model is
()
3.5. Stability at Disease-Free and Endemic Equilibrium Point
The stability of a mathematical model is its ability to maintain consistent, bounded behavior despite small changes. In this study, the Routh–Hurwitz criterion was applied to demonstrate the local stability at the disease-free equilibrium point [37]. The Lyapunov function [38] has been considered for global stability at endemic equilibrium.
Theorem 4. The model is asymptotically stable at the disease-free equilibrium point if R0 < 1, otherwise unstable.
Proof 4. Let us consider a Jacobian matrix J. Then, this matrix at the disease-free equilibrium point can be written as,
()
where
Then, the characteristic equation of the Jacobian matrix is |J0 − λI| = 0.
Hence,
()
From equation characteristics equation, it can easily be seen that the first, second, and third eigenvalues λ1, λ2, and λ3 are λ1 = −ξ − μ, λ2 = −μ, and λ3 = −τ1 − τ2 − μ. Then, the determinant can be written as:
()
After solving Equation (28), we get a cubic equation of λ.
()
where
and
()
According to the Routh–Hurwitz criterion, since a1 > 0 and a2 > 0, a3 will possitive when R0 < 1. As a result, the model is asymptotically stable at the disease-free equilibrium point.
Theorem 5. If R0 > 1, the endemic equilibrium point is asymptotically stable in ℝ6.
Proof 5. Consider a logarithmic function will be in the form of , where Ci represents the positive coefficient and Xi and define the state variable of the ith compartment and the equilibrium point, respectively. Then,
()
The constants C1, C2, C3, C4, C5, and C6 are non-negative. The Lyapunov function F, along with the selected constant, is designed to guarantee that it F remains continuous and differentiable within the state space, which aids in the assessment of System (7) stability.
Now, differentiating with respect to t and using (7), we get
()
From endemic equilibrium point,
Using this in Equation (32) and after simplification, it becomes
()
The function G(SP, SH, E, I, IR, R) is negative. As a result, dF/dt ≥ 0 for all SP, SH, E, I, IR, and R, and when , , E = E∗, I = I∗, , and R = R∗, it becomes zero. As after the study [39], the endemic equilibrium point is globally stable when R0 > 1.
3.6. Convergence of Infected Population When Exposed Population Are Constant
In this scenario, the term αEc represents the fourth equation of System (7), the constant influx of the exposed population into the infected group. As the exposed population remains constant, the rate of new infections becomes directly proportional to this constant Ec.
According to the study [40], the fourth equation of System (7) is as follows:
As a result, the sequence of infected population I(t) converses to
3.7. Convergence of Resistant Population When Infected Population Are Constant
We also examine the model while keeping the infected population as constant in System (7). The term ηcIc denotes the constant appearance of infected population through consumption of antibiotics.
According to the study [40], we also take the fifth equation from System (7).
Therefore, it is clear that the sequence of resistant population IR(t) converses to
4. Sensitivity Analysis
The sensitivity index quantifies the extent to which a specific parameter influences the behavior of the model in reducing the prevalence of disease within the CRKP framework.
According to [41, 42], the sensitivity index of reproduction number R0 with respect to a parameter p is defined as
For our model,
The sensitivity index is shown graphically in Figure 2 from Table 1. In epidemiological modeling, if R0 > 1, the disease is expected to spread within the population. And if R0 < 1, the disease is unlikely to sustain transmission and will eventually disappear. The sensitivity indices of the parameters β1, β2, γ1, γ2, ρ, η, and c are positive, indicating that reducing these parameters can lead to disease extinction. Conversely, the sensitivity indices of the parameters ε, ξ, ϕ, σ, δ, ω1, and ω2 are negative, suggesting that increasing these parameters can also result in disease extinction.
Table 1.
Sensitivity index of R0 concerning model parameters.
Parameter
Sensitivity index
Λ
0.94258
β1
0.55473
ε
−1.75978
γ1
0.02676
ϕ
−0.15728
η
0.14770
σ
−0.57386
ω1
−0.13670
μ
−0.90280
α
0.05243
β2
0.39284
ξ
−0.09374
γ2
0.02567
ρ
0.01904
c
0.14770
δ
−0.37514
ω2
−0.0206
5. Numerical Simulation and Visualization
The model was numerically simulated using MATLAB R2018a, with the fourth order Runge–Kutta technique used to investigate the dynamics of the state variables. For this conduct, Table 2 shows the values of key parameters. Some parameter settings were derived from previous research, while others were calculated to match the simulation’s environment.
The initial values for the state variables are considered as follows: SP(0) = 250, SH(0) = 130, E(0) = 200, I(0) = 90, IR(0) = 20, and R(0) = 100, which have been analyzed based on several literature sources [6, 33, 44]. These initial values are the starting point for the model’s simulation, providing information about the progression of each variable over time. The simulation is executed for 20 days, allowing us to observe the model’s behavior during that time period.
5.1. Dynamic Changes in State Variables Influenced by Variation in Parameters
Figure 3 shows the population dynamics within six compartments of the model to illustrate the consequence of variation in the hygiene rate ε. In the case of the susceptible patient (Figure 3a), as ε increases, the population gradually grows from its initial values, with the susceptible population comparatively increasing due to the limited spread of the diseases. Though ε does not directly affect susceptible healthcare workers, the increase in their population is due to the reduced disease spread from infected patients as ε increases, which is demonstrated in Figure 3b. Furthermore, Figure 3c displays a general downward trend, indicating a decrease in the number of exposed individuals over time for diminishing infection. After day nine, the population slightly tends to increase. Likewise, Figure 3d illustrates a declining trend in the more rate ε over time, with the infected population peaking on Day 1, declining by Day 9, and then slightly increasing, reaching values of 80, 73, and 63 for ε = 0.60, ε = 0.65, and ε = 0.70, respectively. Reducing ε leads to a larger resistant population, initially peaking at Day 4, declining, and then rising again after Day 12 as infections increase, from Figure 3e. Therefore, the recovered population increases over time due to more infected and resistant patients for ε = 0.60, as shown in Figure 3f.
Promoting patient awareness of hygiene practices reduces infection risk in hospitals and helps prevent the spread of pathogens. (a) The number of susceptible patients increases with improved maintenance of hygiene rates ε. (b) Enhanced hygiene contributes to an increase in susceptible healthcare workers for less transmission of disease from patients. (c) By raising hygiene rates ε, the population becomes exposed. (d) The number of infected populations is decreasing due to the increase in hygiene rate ε. (e) Due to fewer infections, there is a declining tendency of resistant population. (f) Recovery is increasing as fewer people are getting infected, leading to a decline in disease spread.
Promoting patient awareness of hygiene practices reduces infection risk in hospitals and helps prevent the spread of pathogens. (a) The number of susceptible patients increases with improved maintenance of hygiene rates ε. (b) Enhanced hygiene contributes to an increase in susceptible healthcare workers for less transmission of disease from patients. (c) By raising hygiene rates ε, the population becomes exposed. (d) The number of infected populations is decreasing due to the increase in hygiene rate ε. (e) Due to fewer infections, there is a declining tendency of resistant population. (f) Recovery is increasing as fewer people are getting infected, leading to a decline in disease spread.
Promoting patient awareness of hygiene practices reduces infection risk in hospitals and helps prevent the spread of pathogens. (a) The number of susceptible patients increases with improved maintenance of hygiene rates ε. (b) Enhanced hygiene contributes to an increase in susceptible healthcare workers for less transmission of disease from patients. (c) By raising hygiene rates ε, the population becomes exposed. (d) The number of infected populations is decreasing due to the increase in hygiene rate ε. (e) Due to fewer infections, there is a declining tendency of resistant population. (f) Recovery is increasing as fewer people are getting infected, leading to a decline in disease spread.
Promoting patient awareness of hygiene practices reduces infection risk in hospitals and helps prevent the spread of pathogens. (a) The number of susceptible patients increases with improved maintenance of hygiene rates ε. (b) Enhanced hygiene contributes to an increase in susceptible healthcare workers for less transmission of disease from patients. (c) By raising hygiene rates ε, the population becomes exposed. (d) The number of infected populations is decreasing due to the increase in hygiene rate ε. (e) Due to fewer infections, there is a declining tendency of resistant population. (f) Recovery is increasing as fewer people are getting infected, leading to a decline in disease spread.
Promoting patient awareness of hygiene practices reduces infection risk in hospitals and helps prevent the spread of pathogens. (a) The number of susceptible patients increases with improved maintenance of hygiene rates ε. (b) Enhanced hygiene contributes to an increase in susceptible healthcare workers for less transmission of disease from patients. (c) By raising hygiene rates ε, the population becomes exposed. (d) The number of infected populations is decreasing due to the increase in hygiene rate ε. (e) Due to fewer infections, there is a declining tendency of resistant population. (f) Recovery is increasing as fewer people are getting infected, leading to a decline in disease spread.
Promoting patient awareness of hygiene practices reduces infection risk in hospitals and helps prevent the spread of pathogens. (a) The number of susceptible patients increases with improved maintenance of hygiene rates ε. (b) Enhanced hygiene contributes to an increase in susceptible healthcare workers for less transmission of disease from patients. (c) By raising hygiene rates ε, the population becomes exposed. (d) The number of infected populations is decreasing due to the increase in hygiene rate ε. (e) Due to fewer infections, there is a declining tendency of resistant population. (f) Recovery is increasing as fewer people are getting infected, leading to a decline in disease spread.
Figure 4 illustrates the dynamics of population changes within each compartment as a result of slight differences in healthcare workers’ PPE maintenance procedures. According to Figure 4a, the number of susceptible patients increases as disease transmission from healthcare workers to patients is reduced. Upon observing Figure 4b, the increasing trend of PPE use among healthcare workers reduces disease transmission, leading to a rise in susceptible patients. However, for the exposed population in Figure 4c, less population becomes exposed for increasing ϕ. Figure 4d shows that infections peak after Day 1, decline to a minimum on Day 9, and then rise again, with lower ϕ leading to higher infections. Subsequently, Figure 4e demonstrates that as ϕ decreases, the resistant population rises, peaking above 60 on Day 4, decreasing, and then showing an increasing nature. At smaller ϕ values, the disease spreads to a larger population, as shown in Figure 4f, resulting in a relatively higher number of recovered individuals.
PPE plays a crucial role in preventing infections, protecting health, ensuring safety, and upholding ethical care standards among healthcare workers. (a) High PPE compliance among healthcare workers contributes less transmission of disease to susceptible patients. (b) Improved PPE use among healthcare workers increases the susceptible HCW population due to diminished transmission. (c) The exposed population is falling down quickly as PPE usage rates ϕ increase. (d) Amplify PPE usage among healthcare workers, which results in a decrease in the infected population. (e) Impact of PPE usage rates ϕ trends among healthcare workers reduces the resistant population. (f) A reduction in PPE usage correlates with an increase in the recovered population, as both infected and resistant populations rise comparatively.
PPE plays a crucial role in preventing infections, protecting health, ensuring safety, and upholding ethical care standards among healthcare workers. (a) High PPE compliance among healthcare workers contributes less transmission of disease to susceptible patients. (b) Improved PPE use among healthcare workers increases the susceptible HCW population due to diminished transmission. (c) The exposed population is falling down quickly as PPE usage rates ϕ increase. (d) Amplify PPE usage among healthcare workers, which results in a decrease in the infected population. (e) Impact of PPE usage rates ϕ trends among healthcare workers reduces the resistant population. (f) A reduction in PPE usage correlates with an increase in the recovered population, as both infected and resistant populations rise comparatively.
PPE plays a crucial role in preventing infections, protecting health, ensuring safety, and upholding ethical care standards among healthcare workers. (a) High PPE compliance among healthcare workers contributes less transmission of disease to susceptible patients. (b) Improved PPE use among healthcare workers increases the susceptible HCW population due to diminished transmission. (c) The exposed population is falling down quickly as PPE usage rates ϕ increase. (d) Amplify PPE usage among healthcare workers, which results in a decrease in the infected population. (e) Impact of PPE usage rates ϕ trends among healthcare workers reduces the resistant population. (f) A reduction in PPE usage correlates with an increase in the recovered population, as both infected and resistant populations rise comparatively.
PPE plays a crucial role in preventing infections, protecting health, ensuring safety, and upholding ethical care standards among healthcare workers. (a) High PPE compliance among healthcare workers contributes less transmission of disease to susceptible patients. (b) Improved PPE use among healthcare workers increases the susceptible HCW population due to diminished transmission. (c) The exposed population is falling down quickly as PPE usage rates ϕ increase. (d) Amplify PPE usage among healthcare workers, which results in a decrease in the infected population. (e) Impact of PPE usage rates ϕ trends among healthcare workers reduces the resistant population. (f) A reduction in PPE usage correlates with an increase in the recovered population, as both infected and resistant populations rise comparatively.
PPE plays a crucial role in preventing infections, protecting health, ensuring safety, and upholding ethical care standards among healthcare workers. (a) High PPE compliance among healthcare workers contributes less transmission of disease to susceptible patients. (b) Improved PPE use among healthcare workers increases the susceptible HCW population due to diminished transmission. (c) The exposed population is falling down quickly as PPE usage rates ϕ increase. (d) Amplify PPE usage among healthcare workers, which results in a decrease in the infected population. (e) Impact of PPE usage rates ϕ trends among healthcare workers reduces the resistant population. (f) A reduction in PPE usage correlates with an increase in the recovered population, as both infected and resistant populations rise comparatively.
PPE plays a crucial role in preventing infections, protecting health, ensuring safety, and upholding ethical care standards among healthcare workers. (a) High PPE compliance among healthcare workers contributes less transmission of disease to susceptible patients. (b) Improved PPE use among healthcare workers increases the susceptible HCW population due to diminished transmission. (c) The exposed population is falling down quickly as PPE usage rates ϕ increase. (d) Amplify PPE usage among healthcare workers, which results in a decrease in the infected population. (e) Impact of PPE usage rates ϕ trends among healthcare workers reduces the resistant population. (f) A reduction in PPE usage correlates with an increase in the recovered population, as both infected and resistant populations rise comparatively.
Over antibiotic use has a bad effect, as the CRKP disease has demonstrated in Figure 5. It increases disease transmission to patients and healthcare workers, as illustrated from Figure 5a,b, respectively. As a result, from Figure 5c, there arises a more exposed population to CRKP disease. Also, Figure 5d shows that increased antibiotic misuse, with higher η and c, leads to a higher tendency for the population to develop resistance. So, the overuse of antibiotic treatment increases the resistant population, Figure 5e. On the other hand, less use of antibiotics for CRKP disease decreases the resistance population. The initially recovered population gradually increases until Day 6, then remains constant as reduced antibiotic use decreases the percentage of resistance and infected population Figure 5f.
Reducing antibiotic misuse and overuse in CRKP treatment helps prevent resistance, preserve antibiotic effectiveness, and improve patient outcomes. (a) Increasing antibiotic treatment rate η and consumption rate c tends to decrease the number of susceptible patients. (b) Susceptible healthcare workers also decline as antibiotic treatment rate η and consumption rate c rise. (c) The exposed population decreases as the rate of antibiotic use decreases. (d) Initially, a large number of the population has experienced infection, but this is declining as time varies. (e) Proper maintenance of antibiotic use reduces the likelihood of resistance. (f) Due to the proper use of antibiotics, the recovery becomes comparatively slower for reducing disease.
Reducing antibiotic misuse and overuse in CRKP treatment helps prevent resistance, preserve antibiotic effectiveness, and improve patient outcomes. (a) Increasing antibiotic treatment rate η and consumption rate c tends to decrease the number of susceptible patients. (b) Susceptible healthcare workers also decline as antibiotic treatment rate η and consumption rate c rise. (c) The exposed population decreases as the rate of antibiotic use decreases. (d) Initially, a large number of the population has experienced infection, but this is declining as time varies. (e) Proper maintenance of antibiotic use reduces the likelihood of resistance. (f) Due to the proper use of antibiotics, the recovery becomes comparatively slower for reducing disease.
Reducing antibiotic misuse and overuse in CRKP treatment helps prevent resistance, preserve antibiotic effectiveness, and improve patient outcomes. (a) Increasing antibiotic treatment rate η and consumption rate c tends to decrease the number of susceptible patients. (b) Susceptible healthcare workers also decline as antibiotic treatment rate η and consumption rate c rise. (c) The exposed population decreases as the rate of antibiotic use decreases. (d) Initially, a large number of the population has experienced infection, but this is declining as time varies. (e) Proper maintenance of antibiotic use reduces the likelihood of resistance. (f) Due to the proper use of antibiotics, the recovery becomes comparatively slower for reducing disease.
Reducing antibiotic misuse and overuse in CRKP treatment helps prevent resistance, preserve antibiotic effectiveness, and improve patient outcomes. (a) Increasing antibiotic treatment rate η and consumption rate c tends to decrease the number of susceptible patients. (b) Susceptible healthcare workers also decline as antibiotic treatment rate η and consumption rate c rise. (c) The exposed population decreases as the rate of antibiotic use decreases. (d) Initially, a large number of the population has experienced infection, but this is declining as time varies. (e) Proper maintenance of antibiotic use reduces the likelihood of resistance. (f) Due to the proper use of antibiotics, the recovery becomes comparatively slower for reducing disease.
Reducing antibiotic misuse and overuse in CRKP treatment helps prevent resistance, preserve antibiotic effectiveness, and improve patient outcomes. (a) Increasing antibiotic treatment rate η and consumption rate c tends to decrease the number of susceptible patients. (b) Susceptible healthcare workers also decline as antibiotic treatment rate η and consumption rate c rise. (c) The exposed population decreases as the rate of antibiotic use decreases. (d) Initially, a large number of the population has experienced infection, but this is declining as time varies. (e) Proper maintenance of antibiotic use reduces the likelihood of resistance. (f) Due to the proper use of antibiotics, the recovery becomes comparatively slower for reducing disease.
Reducing antibiotic misuse and overuse in CRKP treatment helps prevent resistance, preserve antibiotic effectiveness, and improve patient outcomes. (a) Increasing antibiotic treatment rate η and consumption rate c tends to decrease the number of susceptible patients. (b) Susceptible healthcare workers also decline as antibiotic treatment rate η and consumption rate c rise. (c) The exposed population decreases as the rate of antibiotic use decreases. (d) Initially, a large number of the population has experienced infection, but this is declining as time varies. (e) Proper maintenance of antibiotic use reduces the likelihood of resistance. (f) Due to the proper use of antibiotics, the recovery becomes comparatively slower for reducing disease.
The dynamical changes of the state variables with the rate of combination therapy are shown in Figure 6 and have shown positive results in the treatment of resistant patients with CRKP disease. As a result, both susceptible patients and susceptible healthcare workers increase with increasing rate δ, as highlighted in Figure 6a,b. Since the exposed population generally exhibits a decrease at δ = 0.9826, this decline becomes significantly more obvious, demonstrating a marked reduction in the general tendency for pathogen transmission in Figure 6c. Figure 6d illustrates the dynamics of the infected population, which peaks at 120 individuals on Day 1, then comparatively declines for more use of combination therapy δ. In Figure 6e, we observe that an increase in δ significantly reduces the resistant population compared to other approaches. Specifically, at rates of δ = 0.1986, δ = 0.2536, and δ = 0.9826, the resistant population decreases to 73, 55, and 8, respectively. From this phenomenon, it is clear that combination therapy is a key strategy for managing antibiotic resistant infections, enhancing effectiveness, reducing resistance, and improving patient outcomes. Consequently, as the rate δ increases, firstly, the recovery increases due to good outcomes, then the appearance of a smaller population in both the infected and resistant compartments demonstrates a declining trend shown in Figure 6f.
Combination therapy is an effective treatment strategy that not only prevents CRKP infections but also reduces antibiotic resistance. (a) The number of susceptible patients increases as the rate of combination therapy δ rises. (b) The outcomes of combination therapy rate δ also positively affect susceptible healthcare workers. (c) This therapy has a beneficial impact on the disease, resulting in fewer individuals becoming exposed. (d) The infected population decreases as combination therapy rate δ increases, comparatively. (e) Combination therapy rate δ shows the best outcome in the presence of a treatment procedure for antibiotic resistance in CRKP infections. (f) Though initially the recovered population increases, it is reducing for less population infected and resistance to antibiotics as time passing.
Combination therapy is an effective treatment strategy that not only prevents CRKP infections but also reduces antibiotic resistance. (a) The number of susceptible patients increases as the rate of combination therapy δ rises. (b) The outcomes of combination therapy rate δ also positively affect susceptible healthcare workers. (c) This therapy has a beneficial impact on the disease, resulting in fewer individuals becoming exposed. (d) The infected population decreases as combination therapy rate δ increases, comparatively. (e) Combination therapy rate δ shows the best outcome in the presence of a treatment procedure for antibiotic resistance in CRKP infections. (f) Though initially the recovered population increases, it is reducing for less population infected and resistance to antibiotics as time passing.
Combination therapy is an effective treatment strategy that not only prevents CRKP infections but also reduces antibiotic resistance. (a) The number of susceptible patients increases as the rate of combination therapy δ rises. (b) The outcomes of combination therapy rate δ also positively affect susceptible healthcare workers. (c) This therapy has a beneficial impact on the disease, resulting in fewer individuals becoming exposed. (d) The infected population decreases as combination therapy rate δ increases, comparatively. (e) Combination therapy rate δ shows the best outcome in the presence of a treatment procedure for antibiotic resistance in CRKP infections. (f) Though initially the recovered population increases, it is reducing for less population infected and resistance to antibiotics as time passing.
Combination therapy is an effective treatment strategy that not only prevents CRKP infections but also reduces antibiotic resistance. (a) The number of susceptible patients increases as the rate of combination therapy δ rises. (b) The outcomes of combination therapy rate δ also positively affect susceptible healthcare workers. (c) This therapy has a beneficial impact on the disease, resulting in fewer individuals becoming exposed. (d) The infected population decreases as combination therapy rate δ increases, comparatively. (e) Combination therapy rate δ shows the best outcome in the presence of a treatment procedure for antibiotic resistance in CRKP infections. (f) Though initially the recovered population increases, it is reducing for less population infected and resistance to antibiotics as time passing.
Combination therapy is an effective treatment strategy that not only prevents CRKP infections but also reduces antibiotic resistance. (a) The number of susceptible patients increases as the rate of combination therapy δ rises. (b) The outcomes of combination therapy rate δ also positively affect susceptible healthcare workers. (c) This therapy has a beneficial impact on the disease, resulting in fewer individuals becoming exposed. (d) The infected population decreases as combination therapy rate δ increases, comparatively. (e) Combination therapy rate δ shows the best outcome in the presence of a treatment procedure for antibiotic resistance in CRKP infections. (f) Though initially the recovered population increases, it is reducing for less population infected and resistance to antibiotics as time passing.
Combination therapy is an effective treatment strategy that not only prevents CRKP infections but also reduces antibiotic resistance. (a) The number of susceptible patients increases as the rate of combination therapy δ rises. (b) The outcomes of combination therapy rate δ also positively affect susceptible healthcare workers. (c) This therapy has a beneficial impact on the disease, resulting in fewer individuals becoming exposed. (d) The infected population decreases as combination therapy rate δ increases, comparatively. (e) Combination therapy rate δ shows the best outcome in the presence of a treatment procedure for antibiotic resistance in CRKP infections. (f) Though initially the recovered population increases, it is reducing for less population infected and resistance to antibiotics as time passing.
5.2. Visualizations of Relationship Among State Variables
In this part, the phase portrait is derived from simulations of our model to analyze and describe its qualitative dynamics. The phase portrait visually represents the interactions and relationships between the state variables, which are illustrated in Figure 7.
(a) The exposed population decreases as the number of susceptible populations increases due to the infectious agent being less. (b) The pathogen is being transmitted to susceptible patients, causing new infections and reducing the number of susceptible patients. (c) While the resistant population initially grows, there is a slight increase in the susceptible population due to the reduction in resistance. (d) A larger proportion of the population recovers as effective treatment is administered, increasing the number of susceptible patients.
(a) The exposed population decreases as the number of susceptible populations increases due to the infectious agent being less. (b) The pathogen is being transmitted to susceptible patients, causing new infections and reducing the number of susceptible patients. (c) While the resistant population initially grows, there is a slight increase in the susceptible population due to the reduction in resistance. (d) A larger proportion of the population recovers as effective treatment is administered, increasing the number of susceptible patients.
(a) The exposed population decreases as the number of susceptible populations increases due to the infectious agent being less. (b) The pathogen is being transmitted to susceptible patients, causing new infections and reducing the number of susceptible patients. (c) While the resistant population initially grows, there is a slight increase in the susceptible population due to the reduction in resistance. (d) A larger proportion of the population recovers as effective treatment is administered, increasing the number of susceptible patients.
(a) The exposed population decreases as the number of susceptible populations increases due to the infectious agent being less. (b) The pathogen is being transmitted to susceptible patients, causing new infections and reducing the number of susceptible patients. (c) While the resistant population initially grows, there is a slight increase in the susceptible population due to the reduction in resistance. (d) A larger proportion of the population recovers as effective treatment is administered, increasing the number of susceptible patients.
The susceptible population increases, and the exposed population decreases, indicating the spread of the CRKP infection is reducing in the hospital over time. The susceptible population rises from 250 to 493, while the exposed population decreases from 200 to 80. The susceptible and infected populations are inversely related; as the number of susceptible individuals decreases due to infection, the infected population increases. This occurs because susceptible individuals who encounter the pathogen are at risk of becoming infected in Figure 7b. On Day 1, the infected population reaches its peak as the susceptible population increases because a larger portion of the population becomes infected on this day. After that, the susceptibility rises to reduce the infection over time. Finally, the susceptible population reaches 493, while the infected population stands at 73. Figure 7c illustrates that as the number of susceptible patients increases, the resistance initially rises but gradually decreases as the susceptible population continues to grow. This trend reflects the impact of antibiotic abuse and misuse, where a large portion of the population becomes resistant to treatment first. However, with the implementation of combination therapy, this upward trend in resistance begins to decline over time, demonstrating that a strategic approach to treatment can help mitigate the development of resistance. Since population in the recovered population from CRKP infection and resistance do not develop permanent immunity, they become susceptible to reinfection over time. As a result, an increase in the recovered population leads to a rise in the susceptible population, as seen in Figure 7d.
6. Results and Discussion
Hospitals are intended to be places of recovery from any illness; however, they also pose a significant threat as environments where CRKP can spread, making it one of the most concerning hospital-acquired infections.
This study develops a mathematical model to understand CRKP transmission dynamics between patients and healthcare workers, focusing on risk factors and potential interventions to control disease spread and antibiotic resistance. The model was analyzed both analytically and numerically to validate and assess its effectiveness. For qualitative analysis, positivity, boundedness, existence of a unique solution, and steady state solution have been performed. The reproduction number R0 is crucial in determining disease transmission when R0 > 1 an outbreak is likely, as each infected population spreads the disease to more than one person, while R0 < 1 suggests the disease will eventually die out. The stability at disease-free and endemic equilibrium points has also been analyzed. The model is stable at disease-free equilibrium point when R0 < 1 and stable at the endemic equilibrium point when dF/dt ≥ 0.
The dynamic behavior of the state variables in the model can be analyzed through numerical simulations by exploring variations in regulation parameters and treatment strategies. It can be observed that maintaining hygiene among patients and the use of PPE among healthcare workers has been shown to significantly contribute to preventing disease transmission. When the hygiene rate among patients is below ε < 0.65, disease transmission increases, while a reduction is seen when ε > 0.65, highlighting the importance of hygiene in preventing bacteria spread. Similarly, lower PPE usage ϕ < 0.75 among healthcare workers leads to higher transmission, while higher usage reduces transmission. Antibiotic treatment for CRKP is challenging due to its high levels of drug resistance, which identify with increased antibiotic use. As resistance exceeds 45% with a 31% treatment rate and 17% consumption rate, the risk and difficulty in treating infections grow, raising concerns about treatment effectiveness. Combination therapy is a ray of hope for treatment in the case of resistance populations. With the rate of combination therapy δ > 0.2526, it significantly reduces the resistance population, offering a potential solution to combat antibiotic resistance.
To manage CRKP risk in hospitals, disease transmission prevention protocols, including hygiene practices among patients and proper use of PPE among healthcare workers, are essential. Combination therapy is effective for resistant patients, while proper antibiotic use and timing are key to reducing resistance. Antimicrobial stewardship, monitoring, and education ensure antibiotics are used appropriately to preserve their efficacy and reduce infection and resistance.
7. Conclusion
Nowadays, hospital management systems are facing significant challenges. Limited resources and overcrowded healthcare facilities have significantly increased the risk of nosocomial infections. Additionally, dense populations and frequent patient turnover further exacerbate the likelihood of disease transmission within hospitals.
This work highlights the growing concern about nosocomial infections and antibiotic resistance by unraveling the dynamics of resistant pathogens such as CRKP and identifying key factors driving disease transmission and resistance within hospitals. Hence, a mathematical model was proposed to study the transmission dynamics of CRKP, incorporating preventive strategies and combination therapy as potential treatment approaches. We have performed both analytical and numerical analyses of the proposed model. The validity of the model has been confirmed through positivity and boundedness, and the existence of a unique solution has been established. Furthermore, we have determined the basic reproduction number R0, which represents the transmissibility of the disease to the population. At the disease-free equilibrium point E0, the system is locally stable when R0 < 1, while at the endemic equilibrium point E1, the system is globally stable. Next, a sensitivity analysis was performed, showing the sensitivity indices of the parameters in the model. After that, the model solution is also represented through numerical simulation using parameter values. Improved hygiene practices among patients (represented by an increase in ε) and increased use of PPE by healthcare workers (represented by an increase in ϕ) are seen to play a critical role in reducing the prevalence of disease within the hospital setting. These measures collectively contribute to breaking the transmission chain, thus improving overall infection control. An important observation is that antibiotic treatment not only led to worse outcomes for CRKP disease but also accelerated the emergence of a highly resistant population. Due to the presence of resistance genes, CRKP rapidly developed antibiotic resistance, making disease management and treatment increasingly difficult. To address this problem, combination therapy has been identified as an effective treatment approach. By increasing the rate of this therapy (denoted by δ), the antibiotic resistance of the disease is significantly reduced, leading to a decrease in CRKP transmission. This approach highlights the potential of combination therapy to control both the spread of infection and the challenge of antibiotic resistance. In this study, we focus mainly on exploring various preventive strategies for infectious diseases, such as CRKP. In addition, we emphasize the implementation of effective treatment measures specifically adapted to the nature of the disease.
Based on observations, it is essential for both government and private hospitals to raise awareness about nosocomial infections, such as CRKP, and to improve infection prevention measures. In addition, antimicrobial management is essential in mitigating the risk of antibiotic resistance and in ensuring that effective treatment options remain available.
Key components of antimicrobial stewardship include:
•
Preserve antibiotic effectiveness: use antibiotics responsibly, prevent misuse, and implement infection control measures.
•
Extend antibiotic lifespan: combine antibiotics, invest in research, and explore alternative treatments to reduce reliance on antibiotics.
•
Prevent resistant infections: promote vaccination, rapid diagnostics, and public awareness to reduce unnecessary use.
•
Avoid side effects: educate patients about proper use, personalize treatments, and monitor for adverse reactions.
By following these practices, the hospital environment can be protected from the severe consequences of infections such as CRKP while also implementing the proper use of antibiotics to reduce the tendency to resistance. In the future, we will focus on developing a delay differential mathematical model to analyze the transmission dynamics of different CRKP variants.
Ethics Statement
The authors have nothing to report.
Conflicts of Interest
The authors declare no conflicts of interest.
Author Contributions
Saiful Islam Rokib: conceptualization, formal analysis, methodology, data collection, software, writing–original draft. Shahadat Hossain: conceptualization, formal analysis, methodology, data collection, software, writing–original draft. Maria Binte Malek: conceptualization, formal analysis, methodology, data collection, software, writing–original draft. Fariha Akter Ruhi: conceptualization, formal analysis, methodology, data collection, software, writing–original draft. Uzzwal Kumar Mallick: conceptualization, methodology, software, writing–review and editing, supervision, validation.
Funding
We declare that this research did not receive any specific funding from public, private, or not-for-profit sectors. However, the corresponding author, motivated by personal interest, collected relevant information from various websites.
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