Impact of healthcare system collapse on hospital-acquired infection dynamics: a mathematical approach

The collapse of healthcare systems poses a serious threat to hospital environments, increasing the spread of nosocomial infections. Limited resources, overcrowding, and weakened infection control measures allow harmful pathogens to persist on surfaces, medical equipment, and in the air. Consequently, patients and healthcare workers face a higher risk of infection, leading to increased morbidity. In this study, we develop a mathematical model to examine how a failing healthcare system influences the spread of nosocomial infections in hospitals. The model properties, such as the positivity and boundedness of solutions, are established. The disease-free equilibrium is determined and its stability is analyzed using the basic reproductive number ${\mathcal{R}}_0$. Model simulations are performed to determine the influence of some important parameters on the reproduction number of the model. The findings show that a collapsing healthcare system significantly affects the control of nosocomial infections in hospitals, as inadequate infection control measures lead to a more contaminated environment. Therefore, strengthening healthcare systems is essential to reducing nosocomial infections and ensuring safer hospital environments. These results have huge implications for the management of nosocomial infections in hospitals.

2000 MSC: 34C11, 34C60, 92B05

1 Introduction

The history of Nosocomial infections (NIs) can be traced to the origins of hospitals themselves [1], and the World Health Organization (WHO) has defined it as infections that develop in a patient during their stay in a hospital or other clinical facilities, which were not present at the time of admission [2–4]. These infections usually become clinically apparent during hospitalization or after discharge, and the organisms that cause these infections are called nosocomial pathogens [5]. Nosocomial diseases can be transmitted from the environment, such as contaminated surfaces, medical equipment, or water systems, leading to infections such as Clostridioides difficile, Pseudomonas aeruginosa, and fungal infections. This infection can also be transmitted to patients by medical personnel through poor hygiene or improper practices, resulting in infections like Methicillin-Resistant Staphylococcus Aureus (MRSA), surgical site infections or hepatitis [6].

Most African countries [7, 8] struggle to meet the fundamental requirements for effective healthcare systems. Poor governance and human resource limitations are closely linked to the inefficient integration of healthcare services, particularly in resource-constrained nations. The stark environmental conditions described by Wendland [9] highlight systemic constraints that could be incorporated into stochastic models of healthcare efficiency or epidemic spread in resource-limited regions such as Malawi. A healthcare system is considered to be collapsing when it fails or is unable to meet the needs of its population [10]. This can result from a combination of factors, including overwhelmed facilities, lack of funding, workforce shortages, inefficiencies, corruption, political instability (such as wars or conflicts), and major crises like pandemics or natural disasters. Healthcare systems in Africa have long faced persistent challenges arising from man-made factors that span institutional, human resource, financial, technical, and political dimensions [10].

Research on Nosocomial Infections (NIs) through mathematical modeling has advanced significantly, with numerous studies examining various transmission dynamics in hospital settings. Wang et al. [11] developed a model that incorporates environmental contamination and volunteer involvement in the spread of NIs, building upon the work of Austin et al. [12]. Their model highlighted the significant role of volunteers in transmission, sometimes exceeding that of healthcare workers (HCWs), and underscored the importance of environmentalcleanliness and hand hygiene strategies. However, the model did not account for antibiotic use or variations in contamination levels within the hospital environment [11]. Sébille et al. [13] present a foundational framework for understanding the dynamics of nosocomial infections in hospital ICUs through mathematical modeling. Their study identifies healthcare workers as key vectors of transmission, emphasizing the crucial role of hygiene compliance in controlling pathogen spread. They also illustrate how patient turnover influences infection dynamics, with high turnover diluting colonization and low turnover prolonging it. Furthermore, the model highlights the importance of timely interventions, such as patient isolation, in preventing outbreaks. The work of [14] demonstrates how variations in system conditions can trigger qualitative changes in disease behavior, which is directly relevant for understanding how hospital-acquired infections may evolve when the healthcare system is under collapse.

Several studies have employed mathematical modeling to optimize control strategies for nosocomial infections in hospitals. Doan et al. [15] emphasize the importance of hand hygiene and its significant impact on colonization rates, though their model assumes stable hospital conditions, which may not apply in collapsing systems. Austin et al. [16] explore the dynamics of antibiotic misuse and resistance patterns, advocating for strong antibiotic stewardship programs. Related studies, by Darazirar and others [17], give useful ideas on how infections spread over space and time, which helps to understand how hospital-acquired infections might move through hospitals when the healthcare system is collapsing. Smith et al. introduce environmental contamination as a major reservoir for pathogen persistence, emphasizing the need for effective disinfection protocols. Beggs et al. [18] highlight the role of ventilation and airflow in airborne transmission, though their model assumes well-maintained systems. Bootsma et al. [19] focus on patient and healthcare workers' interactions, recommending targeted interventions, though reliance on detailed contact-tracing data limits its applicability in overburdened settings. Later models, such as those by [20, 21], refine the understanding of NIs by incorporating multiple compartments, varying healthcare settings, and stochastic elements, further advancing strategies for infection control. However, these models also face limitations in real-world applications, particularly in resource-limited and collapsing healthcare systems.

Despite these advancements, existing models have overlooked the collapse of healthcare systems, particularly how strained resources and increased environmental contamination, especially during crises such as pandemics or armed conflicts, affect NIs transmission. There is a notable gap in existing mathematical models of nosocomial infections, which typically assume stable healthcare infrastructure and constant infection control efficacy. Our study addresses this gap by incorporating the collapse of healthcare systems into a compartmental model framework. We focus on how overwhelmed healthcare systems increase environmental contamination and, consequently, the spread of NIs. Hence, the novelty of this work lies in integrating healthcare system failure in the modeling of nosocomial infections, which, to our knowledge, has not been explicitly done in the prior literature. And providing simulation-based information on how strengthening healthcare resources can mitigate hospital-acquired infection risks.

The paper is structured as follows: After the introductory section, the model formulation is presented in Section 2. We present and establish the basic properties of the proposed model in Section 3. The analysis of the model is presented in Section 4 with the uniform persistence, global stability and endemic equilibrium in Sections 4.1, 4.2, and 4.3 respectively. Numerical simulations, sensitivity analysis and some numerical results are presented in Section 5. The paper is then concluded by a discussion in Section 6.

2 Model formulation

Our model consists of four compartments representing patients and healthcare workers. Patients are either susceptible, denoted by P_s or infected, denoted by P_i. The total patient population is thus given by

$P=P_s+P_i.$

Healthcare workers are also either susceptible or infected, and the compartments are respectively denoted by H_s and H_i. The total number of healthcare workers is given by H = H_s + H_i.

In this model, we treat the total number of patients, P = P_s + P_i, and the total number of healthcare workers, H = H_s + H_i, as constants. This assumption is justified because we are studying the infection dynamics on a time scale where the overall hospital occupancy and staffing levels do not change significantly. Hospitals typically operate near fixed patient capacity, and staffing levels are maintained through routine replacement, so even if individuals move between susceptible and infected states, the total number of patients and the total number of healthcare workers remains approximately stable.

To capture the role of environmental contamination, we include a compartment E for the environment. The health care system, representing organizations, people other than healthcare workers, W, a hospital setting and resources that function collectively to provide healthcare services to a population, is represented by compartment W.

The susceptible healthcare workers, H_s become infected either through direct contact with infected patients at a rate $\frac{{\beta }_1{P}_i{H}_s}{a+W}$, or indirect transmission from the contaminated environment at the rate $\frac{{\beta }_1{\eta }_{1}E{H}_s}{a+W}$. However, infected healthcare workers can recover from infection at a rate of α₁, the recovery rate is influenced by the healthcare system that can provide good service to hospitals. Infected healthcare workers spread bacteria into the environment at a rate τ₁ or susceptible patients at a rate η₂. The susceptible patients, P_s become infected either through direct contact with infected healthcare workers at a rate $\frac{{\alpha }_2{P}_s{H}_i}{a+W}$, or indirect transmission from the contaminated environment at a rate $\frac{{\alpha }_2{\eta }_{2}E{P}_s}{a+W}$. Susceptible or infected patients, upon recovery, are discharged from the hospital at the rate ν₁ or ν₂, respectively. The admission of patients into the facility takes place at a rate Λ with a proportion ϕ of the patients admitted as susceptible, while the remainder will be infected. Infected patients can release bacteria either by transferring them to the surfaces they come into contact with, that is, to the environment at the rate τ₂ or to the susceptible healthcare workers at the rate α₁.

Infected healthcare workers (H_i) can release bacteria by transferring them to other surfaces with which they come in contact. Therefore, bacteria shed by infected patients and infected healthcare workers at a rate of τ₁ and τ₂, respectively, can be dispersed throughout the environment through H_i and P_i. For simplicity, we assume that the bacteria in the hospital are uniformly distributed within the hospital. Bacteria are cleared at a rate of μ₁ due to hospital sterilization procedures and the support of healthcare systems. Furthermore, we assume that once patients are infected, they remain infected for the duration of their stay in the hospital.

To model the healthcare system, we assume that W measures the impact on individuals by a system through resources per unit of time. The growth of W is a function of the number of hospital workers and patients. A system degrades depending on how it is managed. We therefore assume that the growth of the system is modeled by the function

${\theta }_{1}H+{\theta }_{2}P+\text{λ},$

where θ₁ and θ₂ are the rates at which the system grows through healthcare workers and patients, respectively. Note that the system is assumed to grow in response to the patient's or hospital workers' needs. The parameter λ measures the baseline growth from other factors such as donor funding. The degradation or decline of the healthcare system is modeled by μ₂W, with μ₂ being the rate of decline of the system. We assume that the rate of decay, μ₂, exceeds the growth rate, indicating a collapsing healthcare system. This assumption reflects real-world scenarios where resource depletion, increased patient burden, or inadequate infrastructure lead to system collapse.

The environment is assumed to be contaminated by infected patients at a rate τ₁ and infected healthcare workers at a rate τ₂. Unlike in the other models of nosocomial infections, the environment is decontaminated at a rate that is proportional to WE, i.e., the healthcare system impacts the decomposition through the provision of resources, at a rate μ₁.

Following the description of the model and the assumptions thereof, the model flow diagram is shown in Figure 1.

$\begin{array}{ll}\{\begin{array}{l}\frac{d{H}_s}{dt}={\alpha }_1{H}_{i}W-\frac{{\beta }_1(P_i+{\eta }_{1}E)H_s}{a+W},\hfill \\ \frac{d{H}_i}{dt}=\frac{{\beta }_1(P_i+{\eta }_{1}E)H_s}{a+W}-{\alpha }_1{H}_{i}W,\hfill \\ \frac{d{P}_s}{dt}=(1-\varphi )\text{Λ}-\frac{{\alpha }_2(H_i+{\eta }_{2}E)P_s}{a+W}-v_1{P}_s,\hfill \\ \frac{d{P}_i}{dt}=\varphi \text{Λ}+\frac{{\alpha }_2(H_i+{\eta }_{2}E)P_s}{a+W}-v_2{P}_i,\hfill \\ \frac{dE}{dt}={\tau }_1{P}_i+{\tau }_2{H}_i-{\mu }_{1}WE,\hfill \\ \frac{dW}{dt}={\theta }_{1}H+{\theta }_{2}P+\text{λ}-{\mu }_{2}W.\hfill \end{array}& (1)\end{array}$

Figure 1

Figure 1. The model diagram shows the flow of individuals between compartments and the interaction of the environment, the healthcare system, patients or healthcare workers.

The model consists of several components as described; therefore, we considered rescaling the model equations (Equations 1) TO normalize the model variables and parameters, which makes the relationships between the components easier to understand. This also allows us to work with dimensionless quantities, facilitating comparisons between different models or parameter sets. We now proceed to scale the equations. To do this, we let

$\begin{array}{l}{h}_s=\frac{H_s}{H};h_i=\frac{H_i}{H};p_s=\frac{P_s}{P};p_i=\frac{P_i}{P};e=\frac{E}{E_{max}};\\ w=\frac{W}{W_{max}},& (2)\end{array}$

where P = P_s+P_i and H = H_s+H_i.

By substituting equations given in Equation 2 in Equation 1 resulted in the following scaled model:

$\begin{array}{ll}\{\begin{array}{l}\frac{d{h}_s}{dt}={\beta }_2{h}_{i}w-\frac{{\sigma }_1(p_i+{\xi }_{1}e)h_s}{\psi +w},\\ \frac{d{h}_i}{dt}=\frac{{\sigma }_1(p_i+{\xi }_{1}e)h_s}{\psi +w}-{\beta }_2{h}_{i}w,\\ \frac{d{p}_s}{dt}=(1-\varphi )\gamma -\frac{{\sigma }_2(h_i+{\xi }_{2}e)p_s}{\psi +w}-{\nu }_1{p}_s,\\ \frac{d{p}_i}{dt}=\varphi \gamma +\frac{{\sigma }_2(h_i+{\xi }_{2}e)p_s}{\psi +w}-{\nu }_2{p}_i,\\ \frac{de}{dt}={\zeta }_1{p}_i+{\zeta }_2{h}_i-{\delta }_{1}we,\\ \frac{dw}{dt}=\theta -{\mu }_{2}w,\end{array}& (3)\end{array}$

where, β₂ = α₁W_max, ${\sigma }_1=\frac{{\beta }_{1}P}{W_{max}}$, ${\xi }_1=\frac{{\eta }_1{E}_{max}}{P}$, $\theta =\frac{{\theta }_{1}H+{\theta }_{2}P+\text{λ}}{W_{max}}$, ${\zeta }_1=\frac{{\tau }_{1}P}{E_{max}}$, ${\zeta }_2=\frac{{\tau }_{2}H}{E_{max}}$, δ₁ = μ₁W_max, ${\xi }_2=\frac{{\eta }_2{E}_{max}}{H}$, ${\sigma }_2=\frac{{\alpha }_{2}H}{W_{max}}$, $\psi =\frac{a}{W_{max}}$, $\gamma =\frac{\text{Λ}}{P}$,

with the initial conditions,

$\begin{array}{lll}{h}_s(0)& =h_{s_0}>0,h_i(0)=h_{i_0}\ge 0,p_s(0)=p_{s_0}>0,& (4)\end{array}$

$\begin{array}{ll}{p}_i(0)& =p_{i_0}\ge 0,w(0)=w_0>0,e(0)=e_0>0.\end{array}$

3 Model properties

3.1 Positivity of solutions

We now analyse the positivity of the model system (Equation 3), and show that all state variables remain non-negative and that every solution of the model system Equation 3 with positive initial conditions remains positive for all t > 0. We thus have the following theorem:

Theorem 3.1. Let X(t) = (h_s(t), h_i(t), p_s(t), p_i(t), e(t), w(t)) be the solution of the nosocomial infection model (Equation 3) with positive initial conditions (Equation 4). Then all components of X(t) remain positive for all time t > 0, that is, the region ${\mathcal{R}}_{+}^6$ is positively invariant.

Proof. Consider the first equation of model (Equation 3) given by

$\frac{d{h}_s}{dt}={\beta }_2{h}_{i}w-\frac{{\sigma }_1(p_i+{\xi }_{1}e)h_s}{\psi +w}.$

Since h_i ≥ 0 and w ≥ 0, we have

$\frac{d{h}_s}{dt}\ge -\frac{{\sigma }_1(p_i+{\xi }_{1}e)}{\psi +w}{h}_s.$

Integrating this first-order linear differential inequality, we obtain

$h_s(t)\ge h_{s_0}exp\left(-{\displaystyle {\int }_0^t}\frac{{\sigma }_1(p_i(t_1)+{\xi }_{1}e(t_1))}{\psi +w(t_1)}d{t}_1\right)>0.$

Therefore, h_s(t) > 0 for all t. From the second equation of the model system equations (Equation 3), we obtain

$\frac{d{h}_i}{dt}=\frac{{\sigma }_1(p_i+{\xi }_{1}e)h_s}{\psi +w}-{\beta }_2{h}_{i}w.$

Since $\frac{{\sigma }_1(p_i+{\xi }_{1}e)h_s}{\psi +w}\ge 0$, we have

$\frac{d{h}_i}{dt}>-{\beta }_{2}w{h}_i.$

Integrating yields

$h_i(t)\ge h_{i_0}exp(-{\displaystyle \underset{0}{\overset{t}{\int }}}{\beta }_{2}w(t_1)d{t}_1)>0.$

Similarly, using the same method for the other state variables, it can be shown that p_s(t), p_i(t), e(t), and w(t) remain positive for all t > 0, and this completes the proof.

3.2 Invariant region

To obtain the invariant region in which the model solution is bounded, we consider the total health worker population, h_s(t)+h_i(t), and the total patient population, p_s(t)+p_i(t). Recall that the healthcare worker variables h_s(t) and h_i(t) represent proportions of the total healthcare worker population H, hence h_s(t)+h_i(t) = 1 for all t ≥ 0. The patient variables p_s(t) and p_i(t) represent numbers (or densities) and their sum is not constant due to admissions and discharges.

Theorem 3.2. Let X(t) = (h_s(t), h_i(t), p_s(t), p_i(t), e(t), w(t)) be the solution of system (Equation 3) with the non-negative initial conditions (Equation 4). The compact set

$\begin{array}{r}\text{Ω}=\{(h_s,h_i,p_s,p_i,e,w)\in {ℛ}_{+}^6\mid h_s+h_i=1, p_s+p_i\le P^{*},\\ 0\le e\le E^{*}, 0\le w\le W^{*}\},\end{array}$

where $W^{*}=\frac{\theta }{{\mu }_2}$, $P^{*}=\frac{\gamma }{\nu }$, and $E^{*}=\frac{{\mu }_2({\zeta }_1{P}^{*}+{\zeta }_2)}{{\delta }_1\theta },$ is positively invariant and attracts all solutions in ${\mathcal{R}}_{+}^6.$

Proof. We add the differential equations of p_s(t) and p_i(t) to get the rate of change for the total scaled patient population, p(t) = p_s(t)+p_i(t):

$\frac{dp}{dt}=\gamma -{\nu }_1{p}_s-{\nu }_2{p}_i.$

Since p_s ≥ 0 and p_i ≥ 0, and p = p_s+p_i, let ν = min(ν₁, ν₂). Then:

$\frac{dp}{dt}=\gamma -({\nu }_1{p}_s+{\nu }_2{p}_i)\le \gamma -\nu p.$

Integrating the differential inequality yields:

$p(t)\le p(0)e^{-\nu t}+\frac{\gamma }{\nu }(1-e^{-\nu t}).$

Thus, $p_s(t)+p_i(t)\le max\left\{p(0),\frac{\gamma }{\nu }\right\}$. If $p(0)\le \frac{\gamma }{\nu }$, then $p_s(t)+p_i(t)\le \frac{\gamma }{\nu }$ for all t > 0. The long-term limit is $p\to P^{*}=\frac{\gamma }{\nu }$ as t → ∞.

Now, we consider the total scaled healthcare worker population, h = h_s(t)+h_i(t).

$\begin{array}{l}\frac{d{h}_s}{dt}+\frac{d{h}_i}{dt}=\left({\beta }_2{h}_{i}w-\frac{{\sigma }_1(p_i+{\xi }_{1}e)h_s}{\psi +w}\right)\\ +\left(\frac{{\sigma }_1(p_i+{\xi }_{1}e)h_s}{\psi +w}-{\beta }_2{h}_{i}w\right)=0.\end{array}$

Since $\frac{dh}{dt}=0,$ and given the scaling h_s(0)+h_i(0) = H_s(0)/H+H_i(0)/H = 1, we conclude that h_s(t)+h_i(t) = 1 for all t > 0.

Considering w(t), we have

$\frac{dw}{dt}=\theta -{\mu }_{2}w,$

which, upon integration, gives

$w(t)=\frac{\theta }{{\mu }_2}+\left(w(0)-\frac{\theta }{{\mu }_2}\right)e^{-{\mu }_{2}t}.$

Thus, w(t) is bounded, and $w(t)\le max\left\{w(0),\frac{\theta }{{\mu }_2}\right\}$. If $w(0)\le \frac{\theta }{{\mu }_2}$, then $w(t)\le W^{*}=\frac{\theta }{{\mu }_2}$ for all t > 0. The long-term limit is $\underset{t\to \infty }{lim}w(t)=W^{*}$.

Lastly, we consider the differential equation for e(t). We established that h(t) = 1 and $p(t)\to P^{*}=\frac{\gamma }{\nu }$. Since w(t) is bounded by W^*, for sufficiently large t, we have w(t)≈W^*, and p(t)≈P^*. The inequality for e(t) is:

$\frac{de}{dt}={\zeta }_1{p}_i+{\zeta }_2{h}_i-{\delta }_{1}we\le {\zeta }_{1}p(t)+{\zeta }_{2}h(t)-{\delta }_{1}we.$

Substituting the long-term limits p ≤ P^* and w ≥ W^*:

$\frac{de}{dt}\le {\zeta }_1{P}^{*}+{\zeta }_2(1)-{\delta }_1{W}^{*}e=q_1-q_{2}e,$

where $q_1={\zeta }_1{P}^{*}+{\zeta }_2$ and $q_2={\delta }_1{W}^{*}={\delta }_1\frac{\theta }{{\mu }_2}.$Integrating this yields

$e(t)\le \frac{q_1}{q_2}+\left(e(0)-\frac{q_1}{q_2}\right)e^{-q_{2}t}.$

The upper bound is $e(t)\le E^{*}=\frac{q_1}{q_2}=\frac{{\mu }_2({\zeta }_1{P}^{*}+{\zeta }_2)}{{\delta }_1\theta }.$

The analysis of the invariant region and the positivity of solutions provides the mathematical foundation for the model. We formally conclude that all state variables, X(t) = (h_s(t), h_i(t), p_s(t), p_i(t), e(t), w(t)), remain non-negative for all time t ≥ 0 (Theorem 3.2). Furthermore, the positive invariant set Ω is defined as:

$\begin{array}{l}\text{Ω}=\{(h_s,h_i,p_s,p_i,e,w)\in {ℛ}_{+}^6\mid h_s+h_i=1,p_s+p_i\le \frac{\gamma }{\nu },\\ 0\le e\le E^{*},0\le w\le W^{*}\}.\end{array}$

This establishes that the system is mathematically well-posed and that its solutions are biologically meaningful, as no population can become negative or grow infinitely large.

In summary, the model's solutions start in ${\mathcal{R}}_{+}^6$, are globally confined to the compact set Ω, and are thus ultimately bounded. This essential property ensures the stability and feasibility of the long-term dynamics to be analyzed in subsequent sections.

4 Analysis of the model

The disease-free equilibrium (DFE) is given by

$E_0=\left(h,0,\frac{(1-\varphi )\gamma }{{\nu }_1},0,0,\frac{\theta }{{\mu }_2}\right),$

obtained by setting p_i = h_i = e = 0.

The basic reproduction number, denoted as ${\mathcal{R}}_0$, is a fundamental concept in epidemiology that quantifies the average number of secondary infections generated by a single infected individual in a fully susceptible population. It serves as a critical threshold parameter for determining the potential for an infectious disease to spread within a population. To obtain the basic reproduction number, we used the next generation method, which is the spectral radius of the next generation matrix [22].

For this model, p_i, h_i and e are the disease- related compartments. The non-negative matrix $\mathcal{F}={\mathcal{F}}_1+{\mathcal{F}}_2$ represents the total contribution of new infections in the infected states p₁ and h_i. The spread of infection is influenced by the presence of infected patients (p_i), infected healthcare workers (h_i), and the contaminated environment (e). These factors collectively contribute to the transmission dynamics of the disease.

$\begin{array}{l}\mathcal{F}={\mathcal{F}}_1+{\mathcal{F}}_2=\left(\begin{array}{c}\frac{{\sigma }_1{p}_i{h}_s}{\psi +w}\\ \frac{{\sigma }_2{h}_i{p}_s}{\psi +w}\end{array}\right)+\left(\begin{array}{c}\frac{{\sigma }_1{\xi }_{1}e{h}_s}{\psi +w}\\ \frac{{\sigma }_2{\xi }_{2}e{p}_s}{\psi +w}\end{array}\right),\mathcal{V}=\left(\begin{array}{c}{\beta }_2{h}_{i}w\\ {\nu }_2{p}_i-\varphi \gamma \end{array}\right).\end{array}$

The Jacobian matrices of ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ evaluated at the disease-free equilibrium are given by F₁ and F₂:

$F_1=\left(\begin{array}{cc}0& \frac{{\mu }_2{\sigma }_{1}h}{\psi {\mu }_2+\theta }\\ \frac{{\sigma }_2{\mu }_2(1-\varphi )\gamma }{{\nu }_1(\psi {\mu }_2+\theta )}& 0\end{array}\right),F_2=\left(\begin{array}{c}\frac{{\mu }_2{\sigma }_1{\xi }_{1}h}{\psi {\mu }_2+\theta }\\ \frac{{\mu }_2{\sigma }_2{\xi }_2(1-\varphi )\gamma }{{\nu }_1(\psi {\mu }_2+\theta )}\end{array}\right).$

The Jacobian matrix of V evaluated at the disease-free equilibrium and the inverse for V are given by

$V=\left(\begin{array}{cc}\frac{{\beta }_2\theta }{{\mu }_2}& 0\\ 0& {\nu }_2\end{array}\right),V^{-1}=\left(\begin{array}{cc}\frac{{\mu }_2}{{\beta }_2\theta }& 0\\ 0& \frac{1}{{\nu }_2}\end{array}\right),$

thus, we have

$F_1{V}^{-1}=\left(\begin{array}{cc}0& \frac{{\kappa }_1\theta {\delta }_1}{{\kappa }_3{\xi }_1{\nu }_1{\mu }_2}\\ \frac{{\kappa }_2{\delta }_1}{{\kappa }_3{\xi }_2{\nu }_1{\beta }_2}& 0\end{array}\right),$

where,

$\begin{array}{ll}\begin{array}{c}{\kappa }_1={\mu }_2^2{\sigma }_1{\xi }_{1}h,{\kappa }_2={\mu }_2^2{\sigma }_2{\xi }_2(1-\varphi )\gamma ,{\kappa }_3=\theta {\delta }_1(\psi {\mu }_2+\theta ).\end{array}& (5)\end{array}$

The matrix FV⁻¹ gives the next generation matrix for the model with direct transmission from the infected health workers and the infected patients only.

Let é describe the rate of change in bacterial concentration in the environment.

$e^{\text{'}}=\left(\begin{array}{cc}{\zeta }_2& {\zeta }_1\end{array}\right)\left(\begin{array}{c}{h}_i\\ p_i\end{array}\right)-\left(\begin{array}{c}\frac{{\delta }_1\theta }{{\mu }_2}\end{array}\right)e.$

Then we have L, which determines how efficiently bacteria die off or are removed from the environment, potentially influenced by factors such as healthcare sanitation practices, and Q represents a linear combination of bacterial shedding from infected healthcare workers (h_i) and infected patients (p_i). The coefficients ζ₂ and ζ₁ determine how much bacteria are contributed by h₁ and p_i, respectively.

$L=\left(\begin{array}{c}\frac{{\delta }_1\theta }{{\mu }_2}\end{array}\right),L^{-1}=\left(\begin{array}{c}\frac{{\mu }_2}{{\delta }_1\theta }\end{array}\right),Q=\left(\begin{array}{cc}{\zeta }_2& {\zeta }_1\end{array}\right).$

The matrix $F_2{L}^{-1}Q{V}^{-1}$ gives the next generation matrix for the model with indirect transmission only, that is, infection from the environment. Thus, the next generation matrix of the model is:

$F_2{L}^{-1}Q{V}^{-1}=\left(\begin{array}{cc}\frac{{\zeta }_2{\mu }_2{\kappa }_1}{{\beta }_2\theta {\kappa }_3}& \frac{{\zeta }_1{\kappa }_1}{{\nu }_2{\kappa }_3}\\ \frac{{\mu }_2{\zeta }_2{\kappa }_2}{{\beta }_2\theta {\nu }_1{\kappa }_3}& \frac{{\zeta }_1{\kappa }_2}{{\nu }_2{\nu }_1{\kappa }_3}\end{array}\right).$

Thus, the basic reproduction number of the model is given by the highest eigenvalue of the following matrix

$\begin{array}{ll}\mathcal{M}=F_1{V}^{-1}+F_2{L}^{-1}Q{V}^{-1},& (6)\end{array}$

$\begin{array}{l}=\left(\begin{array}{cc}\frac{{\zeta }_2{\mu }_2{\kappa }_1}{{\beta }_2\theta {\kappa }_3}& \frac{{\zeta }_1{\kappa }_1}{{\nu }_2{\kappa }_3}+\frac{{\kappa }_1\theta {\delta }_1}{{\kappa }_3{\xi }_1{\nu }_1{\mu }_2}\\ \frac{{\mu }_2{\zeta }_2{\kappa }_2}{{\beta }_2\theta {\nu }_1{\kappa }_3}+\frac{{\kappa }_2{\delta }_1}{{\kappa }_3{\xi }_2{\nu }_1{\beta }_2}& \frac{{\zeta }_1{\kappa }_2}{{\nu }_2{\nu }_1{\kappa }_3}\\ \end{array}\right).\end{array}$

The basic reproduction number of model (Equation 3) is

$\begin{array}{ll}{\mathcal{R}}_0=\frac{{\mathcal{A}}_3{\mathcal{A}}_1+\sqrt{{\mathcal{A}}_3{\mathcal{A}}_2(\theta {\delta }_1{\nu }_2+{\zeta }_1{\mu }_2{\nu }_1{\xi }_1)+{\mu }_2{\xi }_2{\nu }_2{\mathcal{A}}_2{\mathcal{A}}_3+{\mathcal{A}}_1^2{\mathcal{A}}_3^2}}{2\theta {\beta }_2{\kappa }_3{\nu }_1{\nu }_2{\mathcal{A}}_3},& (7)\end{array}$

where,

$\begin{array}{l}{\mathcal{A}}_1=\theta {\beta }_2{\zeta }_1{\kappa }_2+{\zeta }_2{\kappa }_1{\mu }_2{\nu }_1{\nu }_2,{\mathcal{A}}_2=4{\theta }^2{\beta }_2{\delta }_1{\kappa }_1{\kappa }_2{\nu }_2,{\mathcal{A}}_3={\mu }_2{\xi }_1{\xi }_2.\end{array}$

Following [22], we have the following result on the local stability of the disease-free equilibrium.

Theorem 4.1. If ${\mathcal{R}}_0<1$, then the disease-free equilibrium of the system, that is,

$E_0=\left(h,0,\frac{(1-\varphi )\gamma }{{\nu }_1},0,0,\frac{\theta }{{\mu }_2}\right)$

is locally asymptotically stable. If ${\mathcal{R}}_0>1$, then the disease-free equilibrium is unstable.

Proof. The proof follows from the calculations of the next generation matrix. This approach was also demonstrated in [23]. We note that when ${\mathcal{R}}_0=1$, the system is at a critical point between infection elimination and infection persisting in the hospital. Linear analysis alone cannot predict the outcome, and small changes in conditions may cause the system to move toward either disease elimination or sustained outbreaks.

4.1 Uniform persistence

Theorem 4.2. Suppose that the invariant region Ω is positively invariant and attracts all solutions in ${ℝ}_{+}^6$. If the disease-free equilibrium (DFE) of system (Equation 3) is unstable, then the system exhibits uniform persistence; that is, there exists a positive constant ϵ > 0 such that every solution h_s(t), h_i(t), p_s(t), p_i(t), e(t), w(t) of Equation 3 with positive initial conditions satisfies:

$\begin{array}{l}{\displaystyle \underset{t\to \infty }{lim inf}}min\{h_i(t),p_i(t),e(t)\}\ge ϵ>0.\end{array}$

Proof. Let $\mathcal{X}$ be the state space given by the positively invariant region Ω. The boundary of $\mathcal{X}$, denoted $\partial \mathcal{X}$, consists of solutions where at least one of the disease-related variables (h_i, p_i, e) is zero. We thus define the persistence region as

$\begin{array}{l}{\mathcal{X}}_0=\mathcal{X}\backslash \partial \mathcal{X}=\{(h_s,h_i,p_s,p_i,e,w)\in {ℝ}_{+}^6:h_i>0,p_i>0,e>0\}.\end{array}$

The disease-free equilibrium (DFE) of Equation 3 is given by

$\begin{array}{l}{E}_0=(h_s^{*},0,p_s^{*},0,0,w^{*}),\end{array}$

where $h_s^{*}$, $p_s^{*}$, and w^* are the equilibrium values given in Theorem 4.3.

From the previous analysis, the basic reproduction number ${\mathcal{R}}_0$ determines the local stability of E₀. If ${\mathcal{R}}_0>1$, then E₀ is unstable, implying that small perturbations away from the DFE lead to growth in the infected populations (h_i, p_i, e). This instability ensures that solutions starting near E₀ eventually move into the interior of $\mathcal{X}$.

Define ${\mathcal{X}}_1=\overline{{\mathcal{X}}_0}$ as the closure of the persistence region. Since $\mathcal{X}$ is positively invariant and attracts all solutions, we conclude that there exists a compact, isolated invariant set $K\subset {\mathcal{X}}_1$ that excludes E₀.

By the standard uniform persistence theorem, [24], since E₀ is a repeller and there is no other equilibrium on the boundary $\partial \mathcal{X}$ attracting trajectories from ${\mathcal{X}}_0$, the system is uniformly persistent. Therefore, there exists a constant ϵ > 0 such that

$\begin{array}{l}{\displaystyle \underset{t\to \infty }{lim inf}}min\{h_i(t),p_i(t),e(t)\}\ge ϵ>0.\end{array}$

This completes the proof.

4.2 Global stability of the disease-free equilibria

A common approach in studying the global stability of the DFE is to construct a Lyapunov function. However, in this paper, we will apply the famous Lyapunov approach proposed by Castillo-Chavez et al. [25]. We list two conditions that, if met, guarantee the global asymptotic stability of the disease-free state.

Define the uninfected class by

$\mathcal{X}={\left(\begin{array}{c}{p}_s,h_s,w\end{array}\right)}^T\in {ℝ}^3,$

and the infected class by

$\mathcal{Z}={\left(\begin{array}{c}{p}_i,h_i,e\end{array}\right)}^T\in {ℝ}^3,$

and the disease-free equilibrium (DFE) by $E_0=({\mathcal{X}}^{*},0)$. Thus, Equation (3) become:

$\begin{array}{c}\frac{d\mathcal{X}}{dt}=F(\mathcal{X},\mathcal{Z}),\\ \frac{d\mathcal{Z}}{dt}=G(\mathcal{X},\mathcal{Z}),G(\mathcal{X},0)=0.\end{array}$

To guarantee global stability analysis of E₀, the following two conditions should be satisfied.

$\begin{array}{l}\left[\text{H}1:\right]\text{For}\frac{d\mathcal{X}}{dt}=F(\mathcal{X},0),{\mathcal{X}}^{*}\end{array}$

is globally asymptotically stable,

$\begin{array}{l}\left[\text{H}2:\right]G(\mathcal{X},\mathcal{Z})=D_{\mathcal{Z}}G({\mathcal{X}}^{*},0)\mathcal{Z}-\hat{G}(\mathcal{X},\mathcal{Z}),\end{array}$

such that $\hat{G}(\mathcal{X},\mathcal{Z})\ge 0for(\mathcal{X},\mathcal{Z})\in \text{Ω}$,

whereby the Jacobian of $G({\mathcal{X}}^{*},\mathcal{Z})$ forms a Metzler matrix M. Then the DFE is globally asymptotically stable provided that ${\mathcal{R}}_0<1$ and the conditions H1 and H2 are satisfied.

Theorem 4.3. The DFE, E₀, of the model system (Equation 3) is not globally asymptotically stable in the positive invariant region, Ω, when ${\mathcal{R}}_0<1$.

Proof. For the disease-free equilibrium, E₀ of system (Equation 3) to be globally asymptotically stable in the positively invariant region Ω, the system must satisfy the two conditions outlined by the Castillo-Chavez framework [25].

Thus, we show that condition H1 is satisfied, but condition H2 fails when ${\mathcal{R}}_0<1$.

From H1, we have the following system: $\frac{d\mathcal{X}}{dt}=F(\mathcal{X},0)=\left(\begin{array}{c}0\\ (1-\varphi )\gamma -{\nu }_1{p}_s\\ \theta -{\mu }_{2}w\end{array}\right).$ The result holds because $\underset{t\to \infty }{lim}||\mathcal{X}(t)||={\mathcal{X}}^{*}$.

For H2, we observe that the Metzler matrix M and the column vector $Ĝ(\mathcal{X},\mathcal{Z})$ are, respectively, defined as:

$\begin{array}{l}M=D_{\mathcal{Z}}G({\mathcal{X}}^{*},0)\\ =\left(\begin{array}{ccc}-\frac{{\beta }_2\theta }{{\mu }_2}& \frac{{\mu }_2{\sigma }_{1}h}{\psi {\mu }_2+\theta }& \frac{{\mu }_2{\sigma }_1{\xi }_{1}h}{\psi {\mu }_2+\theta }\\ \frac{{\sigma }_2{\mu }_2(1-\varphi )\gamma }{{\nu }_1(\psi {\mu }_2+\theta )}-{\nu }_2& \frac{{\sigma }_2{\mu }_2{\xi }_2(1-\varphi )\gamma }{{\nu }_1(\psi {\mu }_2+\theta )}\\ {\zeta }_2& {\zeta }_1-\frac{{\delta }_1\theta }{{\mu }_2}\\ \end{array}\right),\end{array}$

and

$\begin{array}{l}\widehat{G}(\mathcal{X},\mathcal{Z})\\ =\left(\begin{array}{c}{\hat{G}}_1\\ {\hat{G}}_2\\ {\hat{G}}_3\end{array}\right)\\ =\left(\begin{array}{c}{\beta }_2{h}_i\left(w-\frac{\theta }{{\mu }_2}\right)+{\sigma }_1(p_i+{\xi }_{1}e)\left(\frac{h{\mu }_2}{\psi {\mu }_2+\theta }-\frac{h_s}{\psi +w}\right)\\ {\sigma }_2(h_i+{\xi }_{2}e)\left(\frac{(1-\varphi )\gamma {\mu }_2}{{\nu }_1(\psi {\mu }_2+\theta )}-\frac{p_s}{\psi +w}\right)-\varphi \gamma \\ {\delta }_{1}e(w-\frac{\theta }{{\mu }_2})\end{array}\right).\end{array}$

In the invariant region Ω, we have that $w\le \frac{\theta }{{\mu }_2}$. Hence considering ${\hat{G}}_3$, we have

$w-\frac{\theta }{{\mu }_2}\le 0\Rightarrow {\hat{G}}_3(\mathcal{X},\mathcal{Z})={\delta }_{1}e(w-\frac{\theta }{{\mu }_2})\le 0.$

Moreover, the inequality is strict,

${\hat{G}}_3(\mathcal{X},\mathcal{Z})<0.$

Because H2 requires $\hat{G}(\mathcal{X},\mathcal{Z})\ge 0$ on the invariant region Ω, the negativity of ${\hat{G}}_3$ implies that the Castillo–Chavez decomposition does not hold on the given Ω. This implies that condition H2 of the Castillo-Chavez global stability framework is not satisfied. Consequently, the disease-free equilibrium is not globally asymptotically stable. This suggests that reducing ${\mathcal{R}}_0$ below unity may not be sufficient for disease elimination, thereby indicating that the condition ${\mathcal{R}}_0<1$ is necessary but not sufficient for disease eradication.

4.3 Endemic equilibrium

To determine the endemic (positive) equilibrium of the model using the classical Kermack–McKendrick approach [26], we set all time derivatives in system (Equation 3) to zero and solve for the steady-state values of the state variables

$E_1=(h_s^{*},h_i^{*},p_s^{*},p_i^{*},e^{*},w^{*}).$

That is, the system (Equation 3) reduces to the following algebraic equations:

$\begin{array}{ll}0={\beta }_2{h}_i^{*}{w}^{*}-\frac{{\sigma }_1(p_i^{*}+{\xi }_1{e}^{*})h_s^{*}}{\psi +w^{*}},& (8)\end{array}$

$\begin{array}{ll}0=\frac{{\sigma }_1(p_i^{*}+{\xi }_1{e}^{*})h_s^{*}}{\psi +w^{*}}-{\beta }_2{h}_i^{*}{w}^{*},& (9)\end{array}$

$\begin{array}{ll}0=(1-\phi )\gamma -\frac{{\sigma }_2(h_i^{*}+{\xi }_2{e}^{*})p_s^{*}}{\psi +w^{*}}-{\nu }_1{p}_s^{*},& (10)\end{array}$

$\begin{array}{ll}0=\phi \gamma +\frac{{\sigma }_2(h_i^{*}+{\xi }_2{e}^{*})p_s^{*}}{\psi +w^{*}}-{\nu }_2{p}_i^{*},& (11)\end{array}$

$\begin{array}{ll}0={\zeta }_1{p}_i^{*}+{\zeta }_2{h}_i^{*}-{\delta }_1{w}^{*}{e}^{*},& (12)\end{array}$

$\begin{array}{ll}0=\theta -{\mu }_2{w}^{*}.& (13)\end{array}$

Equations 8–13 form a system of algebraic constraints that describe the steady-state behavior of the hospital infection dynamics.

From Equation 13, we obtain the following result.

$\begin{array}{ll}{w}^{*}=\frac{\theta }{{\mu }_2}.& (14)\end{array}$

Substituting Equation 14 into Equation 12 gives the level of environmental contamination:

$\begin{array}{ll}{e}^{*}=\frac{{\zeta }_1{p}_i^{*}+{\zeta }_2{h}_i^{*}}{{\delta }_1{w}^{*}}=\frac{{\mu }_2({\zeta }_1{p}_i^{*}+{\zeta }_2{h}_i^{*})}{{\delta }_1\theta }.& (15)\end{array}$

From Equation 10, we have that:

$\begin{array}{ll}{p}_s^{*}=\frac{(1-\phi )\gamma (\psi +w^{*})}{{\sigma }_2(h_i^{*}+{\xi }_2{e}^{*})+{\nu }_1(\psi +w^{*})},& (16)\end{array}$

substituting Equation 14 and Equation 15, we have:

$\begin{array}{ll}{p}_s^{*}=\frac{(1-\psi )\gamma ({\mu }_2\psi +\theta ){\mu }_2{\delta }_1\theta }{{\mu }_2\left[{\sigma }_2{\mu }_2(h_i^{*}{\delta }_1\theta +{\xi }_2{\mu }_2({\zeta }_1{p}_i^{*}+{\zeta }_2{h}_i^{*}))+{\delta }_1\theta {\nu }_1({\mu }_2\psi +\theta )\right]}.& (17)\end{array}$

Substituting the expressions for w^* given in Equation 14 and e^* given in Equation 15, the systems for Equation 8 and Equation 11 reduce to the following equations in four variables, $p_i^{*},h_i^{*},h_s^{*}$ and $p_s^{*}$

$\begin{array}{l}{\sigma }_1{h}_s^{*}(p_i^{*}{\delta }_1\theta +{\xi }_1({\mu }_2{\zeta }_1{p}_i^{*}+{\zeta }_2{h}_i^{*}))-{\mu }_2{\beta }_2\theta ({\mu }_2\psi +\theta )h_i^{*}=0,\\ {\sigma }_2{p}_s^{*}(h_i^{*}{\delta }_1\theta +{\xi }_2({\mu }_2{\zeta }_1{p}_i^{*}+{\zeta }_2{h}_i^{*}))-{\nu }_2({\mu }_2\psi +\theta )p_i^{*}\\ -\phi \gamma ({\mu }_2\psi +\theta )=0.& (18)\end{array}$

Since $p_i^{*}+h_i^{*}+h_s^{*}+p_s^{*}+w^{*}+e^{*}=c^{**}$, then we can express $h_s^{*}$ in terms of $p_i^{*}$ and $h_i^{*}$:

$\begin{array}{ll}{h}_s^{*}=c^{**}-(h_i^{*}+p_i^{*}+p_s^{*}+w^{*}+e^{*}),& (19)\end{array}$

$\begin{array}{l}=c^{**}-(h_i^{*}+p_i^{*}+\frac{(1-\psi )\gamma ({\mu }_2\psi +\theta ){\mu }_2{\delta }_1\theta }{{\mu }_2[{\sigma }_2{\mu }_2(h_i^{*}{\delta }_1\theta +{\xi }_2{\mu }_2({\zeta }_1{p}_i^{*}+{\zeta }_2{h}_i^{*}))+{\delta }_1\theta {\nu }_1({\mu }_2\psi +\theta )]}\\ +\frac{\theta }{{\mu }_2}+),\end{array}$

where

$\begin{array}{l}\mathcal{P}=\frac{{\mu }_2({\zeta }_1{p}_i^{*}+{\zeta }_2{h}_i^{*})}{{\delta }_1\theta }.\end{array}$

Substituting the results for $h_s^{*}$ as given in Equation 19, and $p_s^{*}$ as given in Equation 17 in the Equation 18, leads to two equations with two unkowns ($h_i^{*}$ and $p_i^{*}$). However, to the best of our efforts, we are not able to analytically obtain the endemic equilibrium point(s). Our numerical analysis, in the next section (Equation 5) of our study, confirms its existence and stability for ${\mathcal{R}}_0>1$. All subsequent results regarding the endemic state are based on values computed through numerical simulation of the model.

5 Simulations

In this section, numerical simulations of the model system (Equation 3) are performed using Matlab and a set of parameter values given in Table 1, to support our theoretical findings. The following table presents the parameter values used in the simulation. Most of these values were derived from the literature. As the model is not fitted to empirical data, parameter values that were not available in the literature were assumed. However, the goal was to have the model trajectories exhibit behavior that is consistent the analytical results.

Table 1

Table 1. Table of parameter values.

5.1 Sensitivity analysis

Sensitivity analysis is a crucial method for assessing the impact of parameter variations on the model output, providing insights into which parameters most significantly influence the basic reproduction number. In this paper, Partial Rank Correlation Coefficient (PRCC) analysis was employed to quantify the strength and direction of the relationship between input parameters and ${\mathcal{R}}_0.$

The sensitivity analysis results, see Figure 2, indicates that the parameters ζ₁, ζ₂, μ₂, ϕ, ξ₁, ξ₂, σ₁ and σ₂ have a positive correlation with the basic reproduction number, ${\mathcal{R}}_0$. This suggests that increasing these parameters would enhance disease transmission. Specifically, ζ₁ represents the shedding of bacteria from infected patients, while ζ₂ accounts for bacterial shedding from infected healthcare workers. Higher values of these parameters indicate an increase in environmental contamination, leading to a greater risk of infection. The parameter μ₂ reflects the impact of a collapsed healthcare system, which can exacerbate disease spread by reducing effective treatment and containment measures. The recruitment rate, ϕ, influences the influx of susceptible individuals into the population, contributing to potential outbreaks. Additionally, σ₁ and σ₂, both representing transmission rates, determine how efficiently the disease spreads from infected individuals to susceptible ones. These findings highlight the critical role of infection control measures, healthcare system stability, and environmental decontamination in mitigating disease transmission and reducing ${\mathcal{R}}_0.$

Figure 2

Figure 2. Partial Rank Correlation Coefficient (PRCCs) showing the effects of parameters variation on ${\mathcal{R}}_0$ using ranges in Table 1.

In addition, parameter such as θ has a negative PRCC value, which suggests that an increase in healthcare system support reduces ${\mathcal{R}}_0$, emphasizing the importance of adequate medical resources and infrastructure. The negative PRCC value associated with β₂ implies that a higher recovery rate leads to a decline in ${\mathcal{R}}_0$. This is because infected individuals spend less time transmitting the disease. Therefore, these findings highlight the critical role of environmental sanitation, healthcare capacity, and effective treatment strategies in mitigating disease spread and reducing its impact on public health facilities.

5.2 Numerical results

We shall present in this subsection the numerical results of the compartments of the contaminated environment, the healthcare system, the infected patients and the infected health workers. We shall also present contour plots showing how some parameters affect the basic reproduction number ${\mathcal{R}}_0$.

Figure 3b shows a collapsing healthcare system, which resulted in worsening environmental contamination in hospitals, increasing the risk of nosocomial infections. When resources are limited, essential infection control measures, such as proper cleaning, waste disposal, and equipment sterilization, become ineffective. As a result, failure to maintain a clean hospital environment leads to more infections (see Figure 3a), higher patient mortality, and additional strain on already overwhelmed healthcare facilities. Furthermore, as a result of the collapse of the healthcare system, the number of infected patients (see Figure 4a) as well as the number of infected health workers (see Figure 4b) increase. Expectedly, it can be seen that the impact is more on the patients.

Figure 3

Figure 3. (a) An increase in pathogens in the environment, and (b) the decay of the healthcare system.

Figure 4

Figure 4. Graphs of (a) infected patients, and (b) the infected health workers.

In Figure 5, we show the result when the healthcare system is collapsing, with the impact on the infected healthcare workers. As the healthcare system declines from a good state of 0.8, the number of health workers who contract the infection increases from a state of almost no infection to more infections. As expected, this number will increase if the health system deteriorates further, until almost all health workers are infected. This is not a good situation for any healthcare facility.

Figure 5

Figure 5. The impact of a collapsed healthcare system on the infected health workers.

Figure 6, shows the result of the impact of a collapsing healthcare system on the infected patients. As the healthcare system declines, the number of infected patients increases linearly due to growth in nosocomial infection in the hospital. As observed, this number increases until it reaches a point of intersection. Around day 68, the two curves intersect, meaning the healthcare system's capacity and the infected patients population become approximately equal, that is, a critical point that might represent total collapse of the healthcare system. Beyond this point, infections (as well as infected patients) have now outpace the healthcare capacity, leading to further deterioration of control efforts.

Figure 6

Figure 6. The impact of a collapsed healthcare system on the infected health workers.

Figures 7a, b show the effect of contamination from patients and health workers respectively. As expected, the contribution to the contamination of the environment is more from the patients than from the health workers. This is because, the patients are more in population than the health workers. Also, the patients are not trained in hygienic practices as the health workers. In Figure 8, we show the effect of cleaning efficiency, δ₁, on the contaminated environment. The plot demonstrates an inverse relationship between δ₁ and e(t), that is, as the cleaning efficiency increases, the environmental contamination decreases. This means that implementing stronger or more frequent cleaning interventions will significantly reduces the environmental contamination and, consequently, may help control the spread of NIs.

Figure 7

Figure 7. Graphs of (a) effect of contamination from patients, and (b) the effect of contamination from the health workers.

Figure 8

Figure 8. The effect of a cleaning efficiency on the environment.

Figure 9 examines how δ₁ (the rate at which pathogens decay) and μ₂ (the collapse of the healthcare system) affect the basic reproduction number, ${\mathcal{R}}_0$. The results show that when δ₁ increases, ${\mathcal{R}}_0$ decreases, meaning that faster pathogen decay helps reduce the spread of infections. On the other hand, a higher μ₂ leads to an increase in ${\mathcal{R}}_0,$ indicating that as the healthcare system weakens, infections spread more easily. This highlights the importance of both effective infection control, such as proper sterilization, and a stable healthcare system in preventing nosocomial infections. These findings reinforce the main focus of this study by showing how both pathogen persistence and healthcare system collapse can contribute to hospital infection outbreaks.

Figure 9

Figure 9. Contour plot of ${\mathcal{R}}_0$ vs. δ₁ and μ₂. All other parameters are as shown in Table 1.

Figure 10 illustrates the effects of θ (the supply to the healthcare system) and δ₁ (the decay rate of pathogens) on the basic reproduction number, ${\mathcal{R}}_0$. The results indicate that as θ increases, ${\mathcal{R}}_0$ decreases, suggesting that a well-supplied healthcare system helps control nosocomial infections. Conversely, a lower θ corresponds to a rise in ${\mathcal{R}}_0$, highlighting the potential risk of infection spread when healthcare resources are insufficient. Similarly, an increase in δ₁, which represents the rate at which pathogens decay, leads to a reduction in ${\mathcal{R}}_0,$ emphasizing the importance of effective sterilization and infection control measures. These findings align with the central theme of this study, which examines how the collapse of healthcare systems can worsen nosocomial infections in hospital settings.

Figure 10

Figure 10. Contour plot of ${\mathcal{R}}_0$ versus δ₁ and θ. All other parameters are as shown in Table 1.

Figure 11 shows how θ (the supply to the healthcare system) and μ₂ (the collapse of the healthcare system) affect the basic reproduction number, ${\mathcal{R}}_0.$ The results indicate that when θ increases, ${\mathcal{R}}_0$ decreases, meaning that a well-supported healthcare system helps reduce the spread of infections. On the other hand, a higher μ₂ leads to an increase in ${\mathcal{R}}_0$, showing that as the healthcare system collapses, infections spread more easily. This highlights the importance of maintaining adequate healthcare resources to control nosocomial infections. The findings support the main focus of this study by showing how a failing healthcare system can worsen infection outbreaks in hospitals.

Figure 11

Figure 11. Contour plot of ${\mathcal{R}}_0$ versus θ and μ₂. All other parameters are as shown in Table 1.

In Figure 12, we show how β₂ (the recovery rate of infected health workers) and σ₁ (the transmission rate of infections to health workers) affect the basic number of reproduction, ${\mathcal{R}}_0$. The figure indicates that when β₂ increases, ${\mathcal{R}}_0$ decreases. That is, a higher recovery rate helps reduce the spread of infections. However, a higher σ₁ with a lower β₂ produces an increase in ${\mathcal{R}}_0$. That is, a higher infection transmission rate with a lower recovery rate leads to an increase in the spread of infections.

Figure 12

Figure 12. Contour plot of ${\mathcal{R}}_0$ versus β₂ and σ₁. All other parameters are as shown in Table 1.

Figure 13 shows that when the transmission rate (σ₂) increases, the reproduction number, ${\mathcal{R}}_0$ also increases. This means that higher transmission rates make infections more likely to spread and persist in hospitals. On the other hand, when the exit rate of infected patients (ν₂) increases, meaning patients recover or are removed from the hospital more quickly, ${\mathcal{R}}_0$ decreases. This suggests that improving patient discharge or isolation can help reduce hospital infections.

Figure 13

Figure 13. Contour plot of ${\mathcal{R}}_0$ versus ν₂ and σ₂. All other parameters are as shown in Table 1.

The figure also highlights the balance between these two factors. If the transmission rate is high and the exit rate is low, Figure 13 shows that the infections spread more easily, especially in overcrowded hospitals. However, if the exit rate is high, even with some transmission, the infection can be controlled. This finding shows why hospitals need strong infection control measures, especially in places with limited resources. Quick patient recovery, proper hygiene, and strict infection prevention rules are all important to stop the spread of infections when healthcare systems are struggling.

6 Discussion and conclusion

In this paper, we developed a mathematical model to examine the impact of collapsed healthcare systems on the spread of nosocomial infections in hospitals. Our analysis focuses on key parameters, including healthcare system capacity, pathogen decay rate, and system collapse, to assess their influence on the basic reproduction number ${\mathcal{R}}_0$. The results indicate that a well-supported healthcare system with adequate resources is crucial in controlling infection transmission. A higher level of healthcare resources effectively reduces ${\mathcal{R}}_0$, suggesting that sufficient medical supplies, staffing, and appropriate healthcare infrastructure contribute to minimizing nosocomial infections. In contrast, as the healthcare system collapses, ${\mathcal{R}}_0$ increases, highlighting the elevated risk of infection spread in hospitals facing resource shortages, overwhelmed staff, and inadequate infection control measures. This finding aligns with real-world observations, where healthcare system failures, particularly during crises such as pandemics, lead to a surge in hospital-acquired infections.

Furthermore, our study demonstrates that an increased pathogen decay rate significantly reduces ${\mathcal{R}}_0,$ emphasizing the importance of effective sterilization and sanitation in hospitals. When pathogens persist in the environment for extended periods, they contribute to sustained infection transmission. This underscores the necessity of routine disinfection and infection control strategies to limit nosocomial outbreaks. The combined effects of healthcare system collapse and pathogen persistence reinforce the urgent need for strong healthcare policies that ensure hospitals remain well-staffed and capable of managing infection risks. Governments and healthcare organizations must prioritize investments in hospital infrastructure and infection prevention programs to mitigate the impact of nosocomial infections, particularly during healthcare crises.

A collapsing healthcare system greatly impairs infection control in hospitals. Resource-strained hospitals struggle to contain infections, leading to a more contaminated environment. Poor cleaning, inadequate sterilization, and overcrowding facilitate the spread of harmful pathogens, putting both patients and healthcare workers at increased risk. To reduce nosocomial infections, hospitals require better resources, proper training, and stronger infection control measures. Strengthening healthcare systems is essential for maintaining safe hospital environments and protecting public health.

Overall, this paper underscores the critical relationship between healthcare system stability and infection dynamics in hospital settings. Without adequate healthcare resources and infection control measures, nosocomial infections spread more rapidly, endangering both patients and medical staff. Future research could expand on this work by incorporating real-world data, considering additional factors such as patient movement and antimicrobial resistance, and exploring targeted intervention strategies. By reinforcing healthcare infrastructure and improving hospital infection control, the risks associated with healthcare system collapse can be minimized, ultimately leading to better patient outcomes and reduced disease transmission in healthcare facilities.

The following are the limitations of the study. The mathematical model relies on some simplified assumptions about hospital environments, pathogen transmission, and healthcare system dynamics, which may not fully capture the complexity of real-world hospital settings. Factors such as patient demographics, comorbidities, and staff compliance with infection control protocols were not explicitly modeled. The study assumes that the collapse of the healthcare system occurs at a certain rate, but in reality, hospital resources fluctuate due to policy interventions, emergency aid, or staff mobilization. A more dynamic approach could better reflect real-world conditions.

Data availability statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author/s.

Author contributions

FN: Conceptualization, Supervision, Writing – review & editing. MS: Methodology, Writing – original draft. LJ: Formal analysis, Methodology, Writing – original draft.

Funding

The author(s) declared that financial support was received for this work and/or its publication. The authors acknowledge the support of their respective institutions in the production of the manuscript. LJ acknowledges support from the National Research Foundation (NRF) of South Africa under the postdoctoral research fellowship number PSTD23040489299.

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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References

1. Liu JY, Dickter JK. Nosocomial infections: a history of hospital-acquired infections. Gastrointest Endosc Clin. (2020) 30:637–52. doi: 10.1016/j.giec.2020.06.001

PubMed Abstract | Crossref Full Text | Google Scholar

2. Lemiech-Mirowska E, Kiersnowska ZM, Michałkiewicz M, Depta A, Marczak M. Nosocomial infections as one of the most important problems of healthcare system. Annals of Agricultural and Environmental medicine. (2021) 28:122629. doi: 10.26444/aaem/122629

PubMed Abstract | Crossref Full Text | Google Scholar

3. Kollef MH, Torres A, Shorr AF, Martin-Loeches I, Micek ST. Nosocomial infection. Crit Care Med. (2021) 49:169–87. doi: 10.1097/CCM.0000000000004783

PubMed Abstract | Crossref Full Text | Google Scholar

4. Raoofi S, Pashazadeh Kan F, Rafiei S, Hosseinipalangi Z, Noorani Mejareh Z, Khani S, et al. Global prevalence of nosocomial infection: A systematic review and meta-analysis. PLoS ONE. (2023) 18:e0274248. doi: 10.1371/journal.pone.0274248

PubMed Abstract | Crossref Full Text | Google Scholar

5. Konde-Lule J, Gitta SN, Lindfors A, Okuonzi S, Onama VO, Forsberg BC. Private and public health care in rural areas of Uganda. BMC Int Health Hum Rights. (2010) 10:1–8. doi: 10.1186/1472-698X-10-29

PubMed Abstract | Crossref Full Text | Google Scholar

6. Caini S, Hajdu A, Kurcz A, Böröcz K. Hospital-acquired infections due to multidrug-resistant organisms in Hungary, 2005-2010. Eurosurveillance. (2013) 18:10. doi: 10.2807/ese.18.02.20352-en

PubMed Abstract | Crossref Full Text | Google Scholar

7. Oleribe OO, Momoh J, Uzochukwu BS, Mbofana F, Adebiyi A, Barbera T, et al. Identifying key challenges facing healthcare systems in Africa and potential solutions. Int J General Med. (2019) 12:395–403. doi: 10.2147/IJGM.S223882

PubMed Abstract | Crossref Full Text | Google Scholar

8. Marais DL, Petersen I. Health system governance to support integrated mental health care in South Africa: challenges and opportunities. Int J Ment Health Syst. (2015) 9:1–21. doi: 10.1186/s13033-015-0004-z

PubMed Abstract | Crossref Full Text | Google Scholar

9. Wendland CL. A Heart for the Work: Journeys Through an African Medical School. Chicago, IL: University of Chicago Press. (2010).

Google Scholar

10. World Health Organization (WHO). Everybody's Business: Strengthening Health Systems to Improve Health Outcomes: WHO's Framework for Action (2007). Available online at: https://www.who.int/publications/i/item/everybody-s-business–strengthening-health-systems-to-improve-health-outcomes (Accessed January 23, 2025).

Google Scholar

11. Wang X, Xiao Y, Wang J, Lu X. A mathematical model of effects of environmental contamination and presence of volunteers on hospital infections in China. J Theor Biol. (2012) 293:161–73. doi: 10.1016/j.jtbi.2011.10.009

PubMed Abstract | Crossref Full Text | Google Scholar

12. Austin DJ, Bonten MJ, Weinstein RA, Slaughter S, Anderson RM. Vancomycin-resistant enterococci in intensive-care hospital settings: transmission dynamics, persistence, and the impact of infection control programs. Proc Nat Acad Sci. (1999) 96:6908–13. doi: 10.1073/pnas.96.12.6908

PubMed Abstract | Crossref Full Text | Google Scholar

13. Sébille V, Chevret S, Valleron AJ. Modeling the spread of resistant nosocomial pathogens in an intensive-care unit. Infect Cont Hosp Epidemiol. (1997) 18:84–92. doi: 10.1086/647560

PubMed Abstract | Crossref Full Text | Google Scholar

14. Abdulkadhim MM, Mohsen AA, Al Husseiny HF, Hattaf K, Zeb A. Stability analysis and bifurcation for an bacterial meningitis spreading with stage structure: mathematical modeling. Iraqi J Sci. (2024) 2630–2648. doi: 10.24996/ijs.2024.65.5.23

Crossref Full Text | Google Scholar

15. Doan TN, Kong DC, Kirkpatrick CM, McBryde ES. Optimizing hospital infection control: the role of mathematical modeling. Infect Cont Hosp Epidemiol. (2014) 35:1521–30. doi: 10.1086/678596

PubMed Abstract | Crossref Full Text | Google Scholar

16. Austin DJ, Kakehashi M, Anderson RM. The transmission dynamics of antibiotic-resistant bacteria. PNAS. (1999) 96:1152–6.

Google Scholar

17. Darazirar R, Yaseen RM, Mohsen AA, Khan A, Abdeljawad T. Minimal wave speed and traveling wave in nonlocal dispersion SIS epidemic model with delay. Bound Value Prob. (2025) 2025:67. doi: 10.1186/s13661-025-02055-1

Crossref Full Text | Google Scholar

18. Beggs CB, Shepherd SJ, Kerr KG. Potential for airborne transmission of infection in hospitals. Emerg Infect Dis. (2006) 12:1003–10.

Google Scholar

19. Bootsma MCJ, Diekmann O, Bonten MJM. Controlling methicillin-resistant Staphylococcus aureus: Quantifying the effects of interventions and rapid diagnostic testing. Proc. Natl. Acad. Sci. U.S.A. (2006) 103:5620–5. doi: 10.1073/pnas.0510077103

PubMed Abstract | Crossref Full Text | Google Scholar

20. Cheng Z, Jia H, Sun J, Wang Y, Zhou S, Jin K, et al. Multiple transmission routes in nosocomial bacterial infections—a modeling study. Commun. Nonlinear Sci. Numer. Simul. (2024) 139:108265. doi: 10.1016/j.cnsns.2024.108265

Crossref Full Text | Google Scholar

21. Wang L, Teng Z, Huo X, Wang K, Feng X. A stochastic dynamical model for nosocomial infections with co-circulation of sensitive and resistant bacterial strains. J. Math. Biol. (2023) 87:41. doi: 10.1007/s00285-023-01968-8

PubMed Abstract | Crossref Full Text | Google Scholar

22. Van den Driessche P, Watmough J. Further notes on the basic reproduction number. In: Mathematical Epidemiology. Cham: Springer (2008). p. 159–178.

Google Scholar

23. Mohsen AA, Naji RK. Dynamical analysis within-host and between-host for HIV\AIDS with the application of optimal control strategy: dynamical analysis within-host and between-host for an HIV\AIDS. Iraqi J Sci. (2020) 2020:1173–89. doi: 10.24996/ijs.2020.61.5.25

Crossref Full Text | Google Scholar

24. Butler Geoffrey FHI, Paul W. Uniformly persistent systems. Proc Am Mathem Soc. (1986) 96:425–30. doi: 10.1090/S0002-9939-1986-0822433-4

Crossref Full Text | Google Scholar

25. Castillo-Chavez C, Feng Z, Huang W. On the computation of R. On the computation of R(o) and its role on global stability. Mathematic. (2002) 1:229. doi: 10.1007/978-1-4757-3667-0_13

Crossref Full Text | Google Scholar

26. Anderson RM. Discussion: the Kermack-McKendrick epidemic threshold theorem. Bull Math Biol. (1991) 53:1–32. doi: 10.1007/BF02464422

PubMed Abstract | Crossref Full Text | Google Scholar

27. Wang L, Ruan S. Modeling nosocomial infections of methicillin-resistant Staphylococcus aureus with environment contamination. Sci Rep. (2017) 7:580. doi: 10.1038/s41598-017-00261-1

Crossref Full Text | Google Scholar