Marburg disease is a rare but severe hemorrhagic fever caused by the Marburg virus, a member of the Filoviridae family. This disease can lead to serious illness and has a high fatality rate, with case fatality rates varying from 24% to 88% depending on the virus strain and case management [1, 2]. The virus was first identified in 1967 in West Germany and is transmitted through direct contact with infected individuals or animals, not through the air. Symptoms include sustained high fevers, confusion, irritability, and, in advanced stages, orchitis [3]. Since its emergence, Marburg virus outbreaks have been reported in several countries, including Angola, Kenya, and Uganda, with a recent outbreak in Ghana in June 2022 [2]. These incidents highlight the ongoing risk of transmission, particularly through unprotected contact in healthcare settings and communities. The disease’s impact varies significantly across regions. For instance, Uganda reported a total of 19 cases from 2012 to 2017, resulting in a mortality rate of 42%. Angola recorded the highest number of cases in 2005, with 374 cases and a mortality rate of 88%. The Democratic Republic of the Congo documented 154 cases between 1998 and 2000, exhibiting an 83% mortality rate. In contrast, both the United States and Yugoslavia showed significantly lower case counts and fatalities, with mortality rates of 0% [2]. This variability underscores the importance of context in understanding the disease’s epidemiology. This study employs a mathematical model to investigate the transmission dynamics of Marburg virus disease (MVD), focusing on two time-dependent control strategies that prioritize the isolation of infected individuals. Utilizing optimal control strategies based on Pontryagin’s Maximum Principle [4, 5], we calculate the basic reproduction number to identify disease-free and endemic equilibrium points. Our findings indicate that efective control measures, including isolation of infected individuals and personal protective practices, can significantly reduce disease transmission. Additionally, we explore the role of robust immune systems in influencing infection dynamics, providing valuable insights for efective MVD management and highlighting the importance of both individual behaviors and broader healthcare strategies in combating infectious diseases.

In this study, we constructed a deterministic mathematical model to analyze the transmission dynamics of Marburg Virus Disease (MVD). The model incorporates two time-dependent control strategies, primarily focusing on the isolation of infected individuals. By employing Pontryagin’s Maximum Principle [6, 7, 8], we determined optimal control strategies that minimize infection levels [9]. The main goal of our research is to explore how isolating infected individuals afects the spread of MVD [10, 11]. We assessed the influence of individuals with natural immunity and the importance of supportive hospital care in controlling the overall infection rate. To achieve this, we calculated the basic reproduction number a crucial epidemiological metric representing the average number of secondary infections generated by a single infected individual in a fully susceptible population. This calculation helps identify both disease-free and endemic equilibrium points [12, 13], providing insights into the conditions required for disease eradication or persistence. Our model also accounts for personal protective measures, such as wearing masks and practicing good hygiene, which are vital for managing infection spread. Furthermore, we explored complementary and alternative medicine (CAM) practices, including herbal remedies and nutritional support, as adjuncts to conventional care [14, 15]. While these treatments can improve health and well-being, they should be used under the guidance of qualified practitioners to ensure safety and efectiveness. Simulations of the optimal control model consistently demonstrate a significant reduction in infection levels when these strategies are applied [16, 17]. This highlights the importance of a multifaceted approach to combating

MVD, where isolation, natural immunity [18], personal protective measures, and supportive medical care work together to mitigate disease impact. Maintaining a healthy diet is crucial for a strong immune system, which is essential for eradicating viral infections [19, 20]. Key recommendations include:

Additionally, adequate sleep, stress management techniques (such as meditation and yoga), and good hygiene practices are vital for overall health. Avoiding smoking and limiting alcohol intake further enhance immune function. Spending time outdoors for sunlight exposure helps produce vitamin D, crucial for immune health. Numerous herbal formulations have demonstrated efectiveness in treating viral infections [21, 22]. Herbal supplements like echinacea, elderberry, and garlic are believed to enhance immunity. Maintaining a healthy weight, fostering social connections, and engaging in regular exercise contribute to overall health and improved immune responses. This study ofers valuable insights into the efective management of Marburg Virus Disease, emphasizing the significance of individual actions in strengthening the immune system and the role of systemic healthcare strategies in controlling infectious diseases.

Mathematical modeling provides representations of real-world situations, enabling predictions and insights into viral diseases [23, 24, 25]. In this study, we develop a deterministic SEIR (Susceptible, Exposed, Infected, Recovered) model that incorporates control strategies [26]. The total human population at any time t is divided into five groups: for susceptible individuals, for exposed individuals, for isolated individuals (either asymptomatic or symptomatic), for currently infected individuals, and for recovered individuals. The total population is expressed as:

The model can be explained as follows,

All parameters in the model are positive. The parameter α represents the source rate of the human population, while denotes the force of infection afecting exposed individuals. The number of exposed individuals can become infected through asymptomatic individuals in isolation and symptomatic individuals at rates and , respectively. The recovery rate for those infected with Marburg Virus Disease (MVD) is denoted by r. To enhance the realism of our model, we incorporate optimal control interventions that respond to infection levels and disease management. Thus, we introduce time-variant control functions aimed at minimizing infection rates, which are crucial for addressing infectious disease challenges [9, 12]. The control functions are defined as follows:

The assumptions regarding the diferential equations of our model can be outlined as follows:

with the initial conditions given by The following Table 1 shows the values of the parameters.

The basic reproduction number of (2.2) is defined as follows (refer to Appendix A for detailed derivation):

In the model system described by equation (2.2), we define the control set as:

The objective is to minimize the function:

where are positive constants. The terms and represent the costs associated with controls and . To minimize infections, we seek optimal controls such that:

The Hamiltonian is defined as:

The specific form of the Hamiltonian is given by:

Here, the objective Function represents the total cost associated with infection levels and control measures. The Hamiltonian combines the costs of infections and control eforts with the dynamics of the population states, and costate variables reflect the marginal value of each state variable

Lemma 1.Assuming the set minimizes we then have adjoint variables that satisfy the adjoint equations

Therefore,

The characterized control sets are:

where,

Proof. Consider and being the associated solutions. Pontryagin’s maximum principle [4, 5, 12] is applied such that there exist adjoint all variables (2.6) satisfying

with transversality conditions Within the interior of the set, where . We determine the time variant control functions when . Applying these using (2.5) we get optimal contro

Hence,

Hence completes the proof.

Remark 1.The qualitative analysis of the model (2.2) is presented in Appendix A, which includes a detailed examination of the disease-free equilibrium, endemic equilibrium, and basic reproduction number. Additionally, the results of the parameter estimation and the sensitivity analysis are provided in Appendix A.1 and A.2, respectively.

In this section, we investigate the impacts of isolation rate , infection rate natural death rate and control variable on basic reproduction number which is calculated in (A.4), moreover these parameters are directly related in The values for all parameters are obtained from Table 1. Note that this section is similar to [4]. Here, we consider birth rate and natural death rate .

The following figures illustrate the model forecasting of (2.2) using birth rate 0.25, and natural death rate and all the values from Table 1. Here, four cases may arise, namely-

To identify the optimal approach to putting the best control in place to lower MVD in the community, we conducted a cost-efectiveness analysis in this section. We use the incremental cost-efectiveness ratio (ICER) to measure the variations in the expenses and health results of these three strategies. ICER is utilized to evaluate two intervention options, and for mitigating disease transmission, helping to prevent the ineficient use of limited resources. The ICER formula is given below [4, 27].

. The total number of averted infections is calculated as the diference between the overall number of infected individuals without any controls and those with controls. Meanwhile, the objective function (2.4) in Appendix is utilized to evaluate the overall cost associated with each method. To compute the total cost and the number of infections prevented, we refer to the parameter values provided in Table 1, excluding α and We consider and . Consequently, Table 2 is organized with the total infection averted ratio (IAR) listed in ascending order. To identify the most cost-efective saving strategy, we calculate the ICER based on the total cost and IAR values.

In Table 2, to calculate the total averted infections (TA), we subtract the final values of the state variables from their initial values. This gives us the total number of infections that were averted over the simulation time. The similar study was found on Section 4.4 in [27]. To calculate the total cost (TC), we assume a cost of 0.5 per unit of control for the isolation strategy and a cost of 0.7 per unit of control for the medical treatment strategy . We then sum up the total cost over the simulation time. The time frame analyzed extends up to 100 units (years). This study is comparable to the findings in [4].

To calculate the total cost (TC) associated with the control strategies for MVD, we assume a cost of 0.5 per unit of control for the isolation strategy and a cost of 0.7 per unit of control for the medical treatment strategy (u ). The total cost can be expressed mathematically as:

where and are the control eforts, and is the total simulation time. Summing these costs over the simulation period allows for an assessment of the financial implications of the implemented strategies. These cost assumptions are grounded in empirical research that highlights the economic aspects of public health interventions [28]. The study by [28] discusses the costefectiveness of isolation and treatment strategies in controlling infectious diseases, emphasizing that investments in these areas can lead to significant reductions in disease transmission and overall healthcare costs [28].

Note that the strategy that has the highest ICER values is excluded from the ICER computations at each stage.

The results obtained from the ICER(2) and ICER(1) values show that strategy (1) is cheaper than strategy (2). Hence, strategy (2) is eliminated from the set of alternatives, and then strategies (1) and (3) are compared.

Ultimately, the values of ICER(1) and ICER(3) indicate that strategy (3) is more cost-efective than strategy (1). Therefore, strategy (1) is removed from the list of options.

Thus, we conclude that strategy (3) is the most cost-efective option among all the strategies for controlling MVD interventions.

The ICER values provide crucial insights into the cost-efectiveness of different health interventions. A lower ICER indicates that a strategy ofers more health benefits per unit of cost, making it a more favorable option for public health decision-making. In the context of the strategies analyzed, the negative ICER values suggest that certain interventions not only save costs but also provide significant health benefits, which is vital for resource allocation in public health. These results can guide policymakers in selecting the most cost-efective strategy by highlighting options that maximize health outcomes while minimizing expenditures. By comparing the ICER values, decision-makers can prioritize interventions that provide the greatest return on investment, ensuring eficient use of limited healthcare resources. The elimination of less efective strategies ensures that funding is directed toward interventions that yield the best health outcomes. In real-world settings, thresholds or benchmarks for determining whether a strategy is considered ”costefective” often vary by country and health system. Commonly cited thresholds range from $50,000 to $100,000 per Quality-Adjusted Life Year (QALY) gained in the United States, while other countries may use lower thresholds based on economic conditions. These benchmarks help establish a standard for evaluating the relative value of healthcare interventions and facilitate informed decision-making in public health policy [29, 30, 31].

The proposed model demonstrates that efective control of Marburg Virus Disease (MVD) necessitates the isolation of infected individuals, which is critical for preventing transmission. Our findings indicate that achieving a 99% eficacy rate in prevention eforts can significantly mitigate the spread of the virus. The most efective strategy identified combines isolation, personal protective measures, and the enhancement of natural immunity. Simulations underscore the significance of improving the natural immunity of isolated individuals through dietary modifications, hygiene practices, and supplementary medicine. This multifaceted approach is essential for the potential eradication of MVD. Our findings are consistent with existing literature on Ebola and other hemorrhagic fevers, reinforcing the eficacy of isolation as a primary control measure. Previous studies have similarly emphasized the role of isolation in reducing transmission rates. The integration of natural immunity dynamics within our model further enriches the discourse on control strategies, suggesting that individual health practices can efectively complement public health interventions. Furthermore, real-world interventions during past outbreaks have demonstrated that rigorous isolation protocols, combined with community engagement and health education, have successfully contained similar viral infections. By comparing our model with these interventions, we highlight the necessity of a holistic approach that incorporates both individual behavioral changes and systemic healthcare strategies. In conclusion, the prevention of MVD relies on the dual strategies of isolating infected individuals and harnessing natural immunity. Our deterministic mathematical model underscores that a combination of isolation, supportive healthcare, and healthy lifestyle promotion is critical in combating MVD. Prioritizing these strategies is essential for efectively managing and ultimately eliminating this deadly disease.

First here we explain the boundedness and positivity of the model (2.2). The region is positive invariant set. From the total population size over a time then

Since is considered to remain constant over time we have . If the disease-induced death rate is zero, then equation (A.1) simplifies to for . Consequently, Ω is a positively invariant set, and within this set, the model (2.2) is both epidemiologically and mathematically well-defined.

Let represent the disease-free equilibrium point (DFE) of the model (2.2). To find DEF we have,

Since, infection is not found at DFE then i.e. and . Therefore, the disease-free equilibrium point is given by Let denote the endemic equilibrium point (EEP) of the model (2.2). To find EEP we have,

Solving the model (2.2) using the above equation (A.3) we get

where,

Also the model (2.2) has a basic reproduction number given by

Using the parameter values from Table 1, we calculate .1694 [11]), indicating that the disease can be efectively controlled through the optimal control measures outlined in the model (2.2).

In this section, we conducted numerical simulations of the system (2.2) using the ODE45 solver in MATLAB to validate our analytical results. The parameter values utilized in the simulation are presented in Table 1, which includes some estimated, assumed, and best-fit values for the model. The initial conditions are as follows: , and . All populations are estimated in millions.

In the figure above, we notice a rapid rise in the number of isolated individuals, which coincides with a sharp decrease in the number of infected individuals. This suggests that prompt isolation of individuals has a substantial impact on reducing the infection rate.

By comparing the two (Figure 15 and Figure 16) above, we observe that increasing the number of isolated individuals while implementing control measures in our model leads to a significant decrease in infections within the overall population. Even as the number of isolated individuals declines over time, the sustained application of control measures continues to result in a steady reduction in infections. This illustrates the efectiveness of these control measures in lowering the infection rate.

In Figure 17, when we stopped using the control in our model, the infection continued to spread in the population without diminishing. The infection rate never decreased. Meanwhile, even though the isolation rate remained steady, the recovery rate started to decline. This suggests that without ongoing control measures, the situation worsens over time.

From Figure 18, we noted that at first, when full controls and isolation were implemented, the infection rate exhibited continuous fluctuations. After a period, the recovery rate increased significantly. However, upon reducing the isolation rate, the infection rate also declined and was nearly eradicated over time with the application of two steady control measures. In contrast, when we lessened the use of controls in our model, the infection rate surged rapidly. This highlights the critical need to maintain control measures for efectively managing the infection rate. The initial fluctuations in the infection rate indicate that control measures require time to stabilize the situation. Over time, consistent application of these controls results in a marked increase in the recovery rate and a decrease in the infection rate.