Mathematical modeling and analysis of COVID-19 and typhoid fever co-dynamics with treatment

Abstract

A deterministic mathematical model for COVID-19 and typhoid fever co-infection was developed and analyzed to assess the effects of treatment on the transmission and control of COVID-19 and typhoid fever in the community. The model incorporates both human and pathogen populations and is subjected to analytical scrutiny, including investigations into the existence, boundedness, and positivity of the solution. The study employs the Next-Generation Matrix Method to calculate the effective reproduction number , and the global stability of the disease-free equilibrium point is examined using the Lyapunov function method. Sensitivity analyses were conducted by computing Partial Rank Correlation Coefficients (PRCC) to determine the impact of each parameter on the spread or control of the diseases. The necessary conditions for the existence of optimal control and the optimality system for the model are established using Pontryagin’s maximum principle. These include three control measures which are: improved COVID-19 treatment, improved typhoid fever treatment, and improved co-infection treatment. Numerical simulations of the model show that, treatment control can effectively reduce the spread of the diseases. Furthermore, increasing the treatment and screening rates can minimize the transmission of COVID-19 and typhoid fever in a population.

Introduction

Global pandemics such as COVID-19 and typhoid fever significantly influence health and the economy worldwide^1,2. COVID-19, caused by the highly contagious SARS-CoV-2 virus³, induces respiratory illness and has become a pervasive global threat since its identification in Wuhan, China, in 2019^2,4,5. SARS-CoV-2 is a respiratory virus that primarily transmits through droplets of saliva generated by an infected person when coughing or sneezing, leading to symptoms ranging from mild to severe, including fever, cough, fatigue, and respiratory distress^3,6. COVID-19 involves both symptomatic individuals and asymptomatic carriers. Individuals who develop typical symptoms of COVID-19, such as fever, cough, shortness of breath, and fatigue, are considered infectious and can transmit the SARS-CoV-2 virus to others^3,7. On the other hand, individuals infected with the SARS-CoV-2 virus who do not develop any symptoms can still transmit the virus to others, even though they are not showing any signs of illness. As of February 22, 2023, the WHO reported over 670 million confirmed COVID-19 cases globally since the start of the pandemic. The total number of reported COVID-19 deaths worldwide has reached over 6.8 million as of February 22, 2023, and 13,375,580,553 COVID-19 vaccine doses have been administered²⁹.

Before initiating treatment for COVID-19, it is essential to assess the patient’s risk factors for severe disease. These risk factors include older adults, especially those aged 65 and older, unvaccinated individuals, and patients with certain medical conditions such as chronic lung diseases, heart diseases, diabetes, or a weakened immune system¹². Several treatments for COVID-19 symptoms include hospitalization, oxygen therapy, antiviral medications, and monitoring for complications.

Typhoid fever, caused by Salmonella Typhi, presents a distinct biological challenge⁸. However, the one transmitted through contaminated food and water, it exclusively affects humans. This manifests itself through symptoms like prolonged fever, headache, vomiting, and diarrhea. It mainly affects organs such as the liver, spleen, gut, kidneys, and gallbladder, and can cause joint pain⁶. The global burden of typhoid fever remains significant, especially in low and middle income countries. According to the World Health Organization (WHO), there are an estimated 11 to 20 million cases of typhoid fever annually, leading to approximately 128,000 to 161,000 deaths worldwide¹⁰. The incidence is highest in South Asia, South east Asia, and sub-Saharan Africa. The primary treatment for typhoid fever is antibiotic therapy. The choice of antibiotics may depend on the region where the infection was acquired due to the varying resistance patterns of Salmonella Typhi strains²⁶. In addition to antibiotics, supportive care is essential for recovery, including hydration, nutritional support, and monitoring for complications. While treatment is critical for those infected, vaccination remains a key preventive measure for individuals traveling to areas where typhoid fever is endemic¹⁶.

Co-infection with COVID-19 and typhoid fever adds complexity, potentially leading to severe symptoms and complicated treatment^10–12. Individuals with typhoid fever face an elevated risk of severe illness and death from COVID-19^8,12. The co-occurrence of symptoms makes accurate diagnosis challenging, emphasizing the importance of prompt and appropriate treatment^10,11. Both COVID-19 and typhoid fever are infectious diseases. Therefore, measures to control their spread include the isolation of infected individuals, contact tracing, and the implementation of public health protocols such as hand hygiene, social distancing, and the use of personal protective equipment (PPE)¹⁵. Vaccines are available for both diseases; however, COVID-19 vaccines are more widely available and distributed globally compared to typhoid fever vaccines¹⁴. Furthermore, COVID-19 and typhoid fever share common symptoms, including fever, cough, fatigue, headache, and muscle aches¹⁰.

Mathematical modeling is essential for monitoring, controlling, and predicting infectious disease outbreaks. It also aids policymakers and decision-makers in public health in creating and implementing control strategies¹⁰. Mathematical models have been developed to understand the control strategies for COVID-19, typhoid fever, and co-infections⁹. For instance,²⁹ developed a mathematical model for the transmission dynamics of COVID-19 and dengue fever, incorporating optimal controls and cost-effectiveness analysis. The results revealed that either a dengue-only or a COVID-19-only control strategy significantly reduces new cases of co-infection.³⁰ developed a mathematical model for the co-infection of COVID-19 and cholera, which included control measures. The results showed that social distancing and water purification can be effective in preventing the spread of these diseases.²⁵ established a nonlinear mathematical model for the co-infection of typhoid fever and cholera, revealing that controlling both diseases in tandem is necessary for the greatest impact on disease control.³¹ developed a mathematical model for the co-infection of COVID-19, HIV, and dengue, which included an optimal control strategy. The results showed that the combined strategy for COVID-19 and HIV would be the most effective and optimal approach to controlling the spread of these diseases.³² presents a nonlinear deterministic mathematical model for the co-infection of pneumonia and COVID-19 transmission dynamics with optimal control. The results revealed that a combination of treatment and prevention is the most effective way to minimize these diseases.

However, in all these studies, no work has been done to investigate the co-infection of COVID-19 and typhoid fever, along with their treatments. Therefore, the goal of this study is to ascertain how treatment, used as a control measure, affects the co-dynamics of the transmission of COVID-19 and typhoid fever.

Model formulation

In this section, the mathematical model for Typhoid fever and COVID-19 co-infection is formulated. The model depicts two main populations: the human population and the pathogen population. The human population, denoted as , is further categorized into seven distinct subclasses. Susceptible humans are those at risk of developing Typhoid fever, COVID-19, or both . Exposed humans are those infected but incapable of transmitting the infection. Infected humans, denoted as , , and , are those showing symptoms of the diseases and can transmit them to others, Treatment class denoted by , are those who have been treated for the diseases. Recovered humans are those who have temporary immunity and have recovered from the diseases. The pathogen population is classified into two subclasses: the concentration of Salmonella Typhi in the environment, which is found in contaminated food and water, including raw fruits and vegetables, raw meat and seafood, sewage, and rivers, denoted by . The concentration of SARS-CoV-2 viruses in the environment, which is found in enclosed indoor spaces and on high-touch surfaces like doorknobs, handrails, and elevator buttons, is denoted by . Hence, the total human and pathogen population is given by:

1

The model assumes that the susceptible class increases due to birth or immigration at a constant rate, denoted as , and also from the recovered class by losing temporary immunity at a rate of . The entire human population in each compartment experiences a natural death rate of . Susceptible individuals in class acquire Typhoid fever and COVID-19 and move to the exposed class through contact with infected individuals and a contaminated environment, influenced by the force of infection denoted by , where

2

The susceptible individuals acquire Typhoid fever through contact with active Typhoid fever patients, co-infected patients, and a contaminated environment containing Salmonella Typhi, through the forces of infection and , given by:

3

In this expression, denotes the effective contact rate for person-to-person transmission of Typhoid fever, while denotes the effective contact rate due to transmission from a contaminated environment to a person. Additionally, represents the carrying capacity for Salmonella typhi. Susceptible individuals acquire COVID-19 through contact with individuals infected with COVID-19, co-infected patients, and through contaminated environments containing SARS-CoV-2, via the forces of infection and , as given by

4

In this expression, denotes the effective contact rate for person-to-person transmission of COVID-19, while represents the effective contact rate due to environmental contamination (environment-to-person transmission). Additionally, is the carrying capacity for the SARS-CoV-2 virus. Moreover, it is assumed that all individuals in the exposed class undergo screening and double screening to identify those who acquire Typhoid fever at a rate of , COVID-19 at a rate of, and co-infected individuals at a rate of , with a transition rate of per day. They then move to classes , , and , respectively.

A co-infected population increases when individuals infected with COVID-19, denoted as , come into contact with individuals infected with typhoid fever, denoted as , or with an environment contaminated with Salmonella typhi, denoted as .

Additionally, the co-infected population increases when individuals infected with Typhoid fever come into contact with individuals infected with COVID-19 or an environment contaminated with SARS-CoV-2 .

The individuals infected by Typhoid fever or COVID-19 recover naturally at the rates of and , respectively, or through treatment . It is assumed that the number of infected individuals in the classes and reduces through treatment at the transition rates of , , and , with induced deaths at rates , , and , respectively. Furthermore, we assume that individuals in compartment leave for compartments and at rates of and , or at a rate of . Finally, individuals in the compartment decrease when they return to the susceptible compartment due to loss of immunity at a rate of .

Additionally, in the pathogen environment, the population decreases due to decay rates of for Salmonella Typhi and for the SARS-CoV-2 virus, respectively. Furthermore, Salmonella Typhi and SARS-CoV-2 in the environment grow logistically at rates of and , respectively, but also through shedding rates of , ,, and.

The mathematical model describing the transmission dynamics of Typhoid fever and COVID-19 co-infection under the specified assumptions is represented by the following set of non-linear differential equations:

The schematic diagram in Fig. 1 represents the model. The dashed arrow indicates contamination of the environment by infected humans.

Figure 1.

The transmission dynamics of Typhoid fever and COVID-19 co-infection.

The dynamics of the disease can be described by the following system of non-linear differential Eq. (5):

5

with initial conditions

,   ,  ,  ,   ,  ,  ,

  and   .

Table 1 provides the description of variables, and Table 2 are the parameters used in the proposed model.

Table 1.

Variables of the co-infection model of typhoid fever and COVID-19.

Symbol	Description
	The total human population at time
	The susceptible human population for both COVID-19 and Typhoid Fever and co-infection
	The number of individuals exposed to COVID-19, Typhoid Fever, and co-infections
	Individuals infected only with Typhoid Fever
	Individuals infected only with COVID-19
	Individuals infected with both Typhoid Fever and COVID-19
	Treated individuals infected with Typhoid Fever, COVID-19, and co-infections
	Individuals who have recovered from Typhoid Fever, COVID-19, and co-infections
	The concentration of Salmonella Typhi bacteria in the environment
	The concentration of SARS-CoV-2 viruses in the environment

Table 2.

Description of model parameters with units for the proposed model.

Par.	Description	Units
	Constant human recruitment rate	–
	Human-to-human Typhoid fever transmission rate	–
	Transmission rate from the environment to humans for Salmonella typhi
	Human-to-human COVID-19 transmission rate
	Transmission rate from the environment to humans for SARS-CoV-2
	in the contaminated environment
	Rate of losing immunity for recovered humans
	Natural death rate of humans
	Decay rate of the SARS-CoV-2 virus
	Decay rate of Salmonella typhi
	Treatment rate for Typhoid fever patients
	Treatment rate for co-infected patients
	Treatment rate for COVID-19 patients
	Shading rate of Salmonella typhi by Typhoid fever-infected humans
	Shading rate of Salmonella typhi by co-infected humans
	Shading rate of SARS-CoV-2 by co-infected humans
	Shading rate of SARS-CoV-2 by COVID-19-infected humans
	Typhoid fever-induced death rate
	Co-infection-induced death rate
	COVID-19-induced death rate
	Natural recovery rate from one disease among co-infections
	Fraction of co-infected individuals who naturally recovered from COVID-19
	Fraction of co-infected individuals who naturally recovered from Typhoid fever
	Fraction of co-infected individuals who received treatment
	Transition rate per day to infected classes
	Rate of screening for exposed individuals with COVID-19
	Rate of screening for exposed individuals with Typhoid fever
	Rate of screening for exposed individuals with co-infection diseases
	Growth rate of Salmonella typhi bacteria
	Growth rate of the SARS-CoV-2 virus
	Natural recovery rates for Typhoid fever and COVID-19, respectively
	Recovery rate for co-infection after treatment
	Carrying capacity for Salmonella typhi and SARS-CoV-2, respectively

Mathematical analysis of the proposed model

Positivity and boundedness

Since the system (5) represents the population in each class and all model parameters are positive, it lies in a region defined by

For the model (5) to be mathematically and biologically meaningful, it is necessary to prove that its solutions are all positive if their initial values are non-negative. In this subsection, we investigate the positivity of the solutions of the model and establish their bounds.

Positivity of the solution of the model

Theorem 1

The solutions of the model (5) are positive in the region if its initial values are non-negative for all .

Consider the ordinary differential equation for susceptible human of the model (5),

and observe that

6

Upon separating variables and integrating (6), we obtain

By the same process, we obtain

This implies that all solutions of the model (5) are positive for all . This ends the proof of Theorem 1.

Boundedness of the model’s solutions

The boundedness of the solutions of the model (5) in the region is demonstrated by considering the total rate of change of the human population, , governed by

and observe that

7

Using Birkhoff and Rota’s theorem¹⁷ on differential inequalities, along with the method of separation of variables applied to the inequality (7), and after substituting the initial conditions, we obtain

8

As , inequality (8) reduces to

Since human population is non-negative, implies that

Similarly, it follows that the upper bound for the bacterial concentration of salmonella Typhi is bounded above by

9

From inequality (Eq. 7),

If , then

10

After differentiate (Eq. 10)

11

The constant

12

is the upper bound for the differential inequality (11) since (11) is the logistic growth model with carrying capacity (12). For some , is an upper bound for where (10) is false, while is bounded above by (12) for the rest of the time points in the domain if ( 10) is true. The constant is the maximum shedding rate from the Typhoid fever infected individuals and dually infected individuals. In both cases

13

Within the feasible region

14

Hence, we can conclude that the solutions of the developed model are bounded in the positive region . In this case, the developed model is mathematically and biologically meaningful, and it is considered for further analysis.

Equilibrium points and the effective reproduction number

Disease free equilibrium point (DFE)

A disease-free equilibrium point refers to a state in which a community, population, or system is entirely free from COVID-19 and typhoid fever. To analyze the disease-free equilibrium point, it is necessary to examine both the conditions required for its formation and the factors that contribute to the absence of disease transmission.

The DFE is established by setting the right-hand side of the model system (5) to zero and substituting zero values for the infectious states and the force of infection (i.e., ).

Upon solving and simplifying the resulting system, we get

The formulated DFE is used to compute a state where there are no causative agents of COVID-19, Typhoid fever, or both. Furthermore, the DFE is used to compute the effective reproduction number of the co-infection of COVID-19 and Typhoid fever, which describes the persistence or extinction of the disease(s) within the population.

The effective reproductive number

The effective reproductive number, denoted as , is a crucial metric in the study of epidemiological models for disease management. By estimating , epidemiologists can assess the potential effects of an outbreak and predict its progression. Specifically, if , it indicates that each infected individual, on average, causes fewer than one secondary infection, leading to the eventual extinction of the disease within the population. When , the average number of secondary infections remains stable, it means that the disease will neither spread nor die out. Conversely, if , it suggests that a single infected individual generates more than one secondary infection resulting in the disease potentially spreading throughout the entire population.

Adopting the next-generation matrix approach by^18,19 to formulate the formula for of the model (5) is given by

where

By inspection, it is observed that depends on sub-effective reproductive numbers: and These respectively define the expected number of newly infected individuals produced by one infected person with Typhoid fever, COVID-19, and co-infection; the secondary number of infected Typhoid fever patients generated by ingesting Salmonella Typhi shed by one infected Typhoid fever individual; and the secondary number of infected COVID-19 patients generated by ingesting SARS-CoV-2 shed by one infected COVID-19 individual.

Global stability of disease free equilibrium (DFE)

The disease-free equilibrium is considered stable if it can maintain its state despite perturbations or initial conditions. This concept, known as global stability, ensures that a system remains at the disease-free equilibrium regardless of initial circumstances. Global stability analysis aims to assess whether a population can stay disease-free without external influences, such as infections or changes in dynamics. Additionally, it offers insights into the effectiveness of control measures in preventing disease recurrence and replication.

Theorem 2

DFE is globally asymptotically stable whenever as and unstable otherwise.

Let the infectious state of the system (5) be denoted by

Adopting the comparison principle by¹⁸ we get:

15

where matrices and are evaluated at disease-free equilibrium in is a vector containing the rate of change of infected states, where

16

Applying the Perron-Frobenius theorem¹⁹ for Metzler matrices and , we observe that has a dominant eigenvalue that corresponds to a non-negative vector . It follows that

Motivated by²⁰, we use the Lyapunov function:

17

Differentiating the function (17) with respect to along with the infected states of the system (5), and using Eq. (15), we get:

The stability criteria is attained only for .

If for , then . Because the Perron-Frobenius eigenvector has non-negative elements, then

18

Substituting the coordinates of (18) into the model (5), we get the DFE .

When , it implies that the infected variables , and lose their energy to transmit the disease to others, and therefore the system turns to DFE as . The two scenarios suggest that every solution of the system (5) converges to DFE as for . By LaSalle’s invariant principle²¹, DFE is globally asymptotically stable whenever .

Global stability of endemic equilibrium

The endemic equilibrium is a state where Salmonella typhi and SARS-Cov-2 infections persist among individuals. During the endemic state, the infected individuals and pathogens transmit the diseases to susceptible individuals, and therefore, the number of infected individuals remains relatively constant over time. That is, the forces of infections , , , and . The endemic equilibrium is said to be globally stable if, in the short or long run, the disease becomes endemic (does not die out). During this state, any invariant set in close to remains close to it and eventually converges to over an indefinite time.

Definition 1

A function is called a Lyapunov function if it is continuous and differentiable for all values of satisfying the conditions:

• i)
• ii)
• iii)

Theorem 3

The endemic equilibrium of the model system (5) is globally asymptotically stable if the effective reproduction number . To prove theorem (3.3), we adopt the approach in Osman et al. (2020) by considering the non-linear function.

19

It can be observed that the function in (i) in the definition above is valid for all where and is the endemic equilibrium. Hence, is a Lyapunov function.

To prove if condition (iii) holds, we differentiate along the solution of the model (5) to obtain:

20

Substituting system (5) into Eq. (21) yields

21

Now substitute

we get

22

Now, collecting positive and negative terms together in the system (22):

23

Let,

24

and

25

Then,

26

and

Therefore, the maximum compact invariant set in is a singleton , which is the endemic equilibrium of the model system (5). Then, by LaSalle’s invariant principle, it implies that is globally asymptotically stable in the interior of if .

Global sensitivity of model parameters on the state variables using LHS–PRCC

In this study, we utilize the Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficient (PRCC) methods to conduct uncertainty and global sensitivity analyses²². The LHS method samples model parameters while accounting for uncertainty by treating them as probabilistic variables that are uniformly distributed across specified ranges. The PRCC method calculates the correlation coefficients between model outputs (state variables) and model inputs (parameters), assessing the strength of their relationship. A positive PRCC value indicates that as the input parameter increases, the output state variable also tends to increase. The PRCC of the parameters with respect to the state variables and is computed using the basic model (5), LHS parameters, a large sample size of , initial conditions for simulating the basic model, and an algorithm from²³. This relationship is illustrated in the following figures.

The results in Fig. 2 indicate that , , , , and are positively correlated to the infected Typhoid fever humans , meaning that they are responsible for increasing the number of infected humans . All these parameters should be reduced throughout the epidemic of Typhoid fever, with the exception of shedding behavior and loss of immunity, which should be reduced during the first twenty (20) days. Through various control measures such as sanitation practices, boosting individuals’ immune systems, educational campaigns, and safe environmental and food management systems, efforts should be made to reduce the number of humans infected by Typhoid fever. On the other hand, , , and are inversely proportional to , meaning that they lower the number of infected humans whenever they increase.

Figure 2.

PRCC values for LHS parameters of model (5) in the typhoid infected human population.

Figure 3 indicates that , , , , , and are significantly positively correlated to . These parameters are responsible for the increase in the number of SARS-CoV-2 infections in the human population when they increase. The influence of parameters , , , , , and is high at the beginning of the epidemic and decreases slightly, stabilizing after 20 days. The influence of increases with time because the more individuals are recruited, the more interactions occur, resulting in more physical transmission among humans. On the other hand, parameters such as , , , and are significantly negatively correlated with , implying that they are responsible for lowering the infections among the human population when they increase. Control measures that speed up recovery rates among humans are ideal for reducing human infections. The parameters such as , , and are observed to have little impact on the dynamics of infected humans.

Figure 3.

PRCC values for LHS parameters of model (5) in the COVID-19 infected human population.

Figure 4 shows that , , , , and are directly proportional to the number of . The increase or decrease of these parameters at specific times influences the increase or decrease in the number of co-infection cases. All these parameters should be reduced throughout the co-infection epidemics, except for the contact rate of COVID-19 through human-to-human transmission and the transition rate , which should be reduced during the first 18 days and 30 days, respectively. To achieve this, control measures such as vaccinations, water sanitation, and treatment should be introduced to reduce the spread of co-infection diseases. On the other hand, , , , , and are inversely proportional to . This implies that an increase or decrease in these parameters results in a decrease or increase in co-infection cases.

Figure 4.

PRCC values for LHS parameters of model (5) in the typhoid, and COVID-19 co-infected human population.

Analysis of optimal control model

We now integrate time-dependent controls , , and into model (5). These control functions are bounded and Lebesgue integrable. Specifically, denotes the treatment for COVID-19, refers to the treatment effort for Typhoid fever, and indicates the combined treatment effort for both diseases. The controls , , and are constrained such that . The control system is defined by

27

28

The optimal control system examines scenarios where the number of co-infected cases and the cost of implementing the controls , , and are minimized subject to the state system (5). For this, we consider the objective functional:

29

is the final time, and we seek to find an optimal control such that

30

where such that are measurable with for is the control set. The Pontryagin’s maximum principle³³ gives the necessary conditions that an optimal control pair must satisfy. This principle transforms (27), (29), and (30) into a problem of minimizing a Hamiltonian, , pointwise with respect to the control functions .

31

where , are the co-state variables associated with . The adjoint equations are obtained by

32

with transversality

33

From Eq. (30), we obtain the following adjoint equations

34

The optimality of the control problem is obtained by where . The solutions of , , and are presented in compact form as

35

Numerical simulation of the model

In this section, we evaluated how treatment control can influence co-infections of COVID-19 and Typhoid fever. The parameter values are provided in Table 3. To address the optimal control problem, we employed the forward-backward sweep method to solve the optimality system of both the state and adjoint systems. The adjoint system was solved using the fourth-order Runge-Kutta method with initial conditions , , , , , , , , and .

Table 3.

Model parameter values.

Parameter	Value	Source
	447	Assumed
	0.0000207	27
		24
	0.001	Assumed
	0.05	Assumed
	0.3	25
	0.014	13
	0.033	26
	0.119	Assumed
	0.052	27
	0.05	Assumed
	0.0447	26
	0.002	27
	0.94	14
	0.13978	28
	1000	Assumed
	1000	Assumed
	10	Assumed
	100	Assumed
	0.2	Assumed
	0.4	Assumed
	0.2	Assumed
	0.43	Assumed
	0.25	Assumed
	0.2	Assumed
	0.5	3
	0.5	Assumed
	0.043	29
	0.005	26
	0.04	26
	0.014	24
	0.014	24
	1,000,000	Assumed
	1,000,000	Assumed

From Fig. 5, it is observed that the number of susceptible individuals declines rapidly within the first five days due to infections from Salmonella Typhi and SARS-CoV-2, as well as natural deaths. This decline does not reach zero; instead, it stabilizes at a non-negative value due to continuous recruitment and recovered individuals losing protective immunity. Additionally, the number of exposed individuals rises within eight days as many susceptible people are infected by Salmonella Typhi, SARS-CoV-2, or both, moving into the exposed class. The number of infected individuals increases as exposed individuals are screened and identified with either disease, and through the natural recovery of co-infected individuals. However, infections decrease due to disease-induced deaths, natural deaths, and effective treatments. Ultimately, the number of recovered individuals grows with the increase in treatment rates. This suggests that to minimize or eliminate the transmission of Typhoid fever and COVID-19 in the human population, effective control measures are necessary.

Figure 5.

Human population.

Figure 6 shows the effects of treatment control on infected humans. In Fig. 6a, treatment control reduces typhoid fever infections, while its absence leads to an increase. Figure 6b indicates that COVID-19 infections also decline with treatment control but rise without it. Figure 6c demonstrates a decrease in co-infected individuals with treatment controls, contrasting with an increase without them. Figure 6d,e reveal that treatment control for typhoid fever, COVID-19, or both reduces the presence of Salmonella typhi and SARS-CoV-2. Figure 6f shows the control profile remains high for about 28 days before dropping. Thus, implementing treatment control during outbreaks is crucial to minimize transmission in a population.

Figure 6.

Simulation of the model showing the effect of treatment control on the infected human.

Figure 7 illustrates the dynamics of Salmonella typhi and SARS-CoV-2 concentrations in the contaminated environments and , respectively. The graphs show that Salmonella typhi concentration increases over the first 10 days, while SARS-CoV-2 concentration rises for 18 days before both decline and stabilize at non-negative constant values. This initial increase is attributed to the growth rates of the pathogens and shedding rates from infected individuals. Subsequently, natural decay rates and the introduction of treatment as a control measure contribute to the decline of pathogens in compartments and .

Figure 7.

Pathogens population.

Effect of varying the treatment rates of infectious humans , and

Figure 8 illustrates the dynamics of Typhoid fever and COVID-19 transmission in the community, focusing on the impact of treatment rates for Typhoid fever, denoted as . When , the number of Typhoid fever infections rises within 5–9 days. However, as increases (i.e., ), the infection rate decreases. Thus, we conclude that the infection’s endemicity increases when treatment is not implemented.

Figure 8.

Typhoid fever infected humans.

In Fig. 9, the co-infected population increases as the treatment rate for co-infection decreases. Conversely, an increase in the treatment rate leads to a decrease in the number of people infected with Typhoid fever and COVID-19, indicating effective treatment and recovery. When , the number of co-infected individuals rises, but when , it decreases. Therefore, it is concluded that implementing treatment during the Typhoid fever and COVID-19 pandemic is essential to reduce disease transmission in the community.

Figure 9.

Co-infected humans.

Figure 10 shows how the human population infected by COVID-19 changes with the treatment rate . The number of infected individuals increases as the treatment rate decreases over 12 days. Conversely, as increases, the number of infections decreases. Specifically, when , infections rise, but when , infections decline. This suggests that higher treatment levels lead to more recoveries among infected individuals.

Figure 10.

COVID-19 infected humans.

Impact of varying the screening rates for infectious humans: m, n, and (m+n)

The results in Figs. 11, 12, and 13 show the impact of screening on the transmission of diseases. Higher screening rates lead to the early detection of COVID-19, typhoid fever, and co-infection cases, allowing for the timely isolation and treatment of infected individuals. As a result, this reduces the number of individuals infected with COVID-19, typhoid fever, and co-infection. Conversely, lower screening rates lead to undetected cases and an increase in the number of individuals infected with these diseases, contributing to higher transmission within a community. When , , and , the number of individuals infected with typhoid fever, COVID-19, and co-infection rises. However, when , , and , the number of individuals infected with these diseases decreases. Therefore, it is recommended that implementing screening during the typhoid fever and COVID-19 pandemic is essential to reduce disease transmission in society.

Figure 11.

Typhoid fever infected humans.

Figure 12.

COVID-19 infected humans.

Figure 13.

Co-infected humans.

Discussion and conclusion

COVID-19 and typhoid fever are pandemics that affect lives and economies worldwide. In this paper, a deterministic model was developed and analyzed to investigate the effects of treatment as a control measure on the transmission dynamics of co-infection with COVID-19 and typhoid fever. The model considered both human and pathogen populations. An optimal control was formulated from the basic model, incorporating three control variables: treatment effort for COVID-19 , treatment effort for typhoid fever , and treatment effort for co-infection .

The next-generation matrix method was applied to compute the effective reproductive number , and the local asymptotic stability at the disease-free equilibrium and global asymptotic stability at the disease-free equilibrium and endemic equilibrium point was assessed using the Lyapunov function method. Some limitations of the next-generation matrix method include population mixing, sensitivity to parameter estimation, complexity in multi-pathogen systems, temporary dynamics, applicability of non-exponential growth models, and data quality requirements.

Sensitivity analysis of the model was carried out using the partial Rank Correlation Coefficient (PRCC) method. The results indicate that , , , , and are directly proportional to , meaning that they are responsible for increasing the number of infected humans whenever they increase. On the other hand, parameters such as , , and are inversely proportional to , meaning that they lower the number of infected humans .

Additionally, parameters such as , , , , , and are positively correlated to . These parameters are responsible for increasing the number of SARS-CoV-2 infections in the human population when they increase. Furthermore, parameters such as , , , , and are directly proportional to the number of . The increase or decrease of these parameters at specific times influences the increase or decrease in the number of co-infection cases. On the other hand, , , , , and are inversely proportional to . This implies that an increase or decrease in these parameters results in an increase or decrease in co-infection cases.

The optimality system for the co-infection model is established using Pontryagin’s maximum principle. Numerical simulations were carried out, and the results suggest that treatment control and screening should be introduced during the infection period to combat the transmission of COVID-19 and typhoid fever in the community. These results align with the findings in^29,30, which indicate that treatments as control measures should be implemented to reduce the number of individuals infected by COVID-19 or typhoid fever.

Based on the findings, it is recommended that during disease outbreaks, policymakers should invest in treatments and screening as control measures to minimize the transmission of COVID-19 and typhoid fever within the population. The findings from this study can also guide public health and infectious disease management in understanding disease interactions and optimizing treatment strategies. For example, they can be useful in identifying the most effective treatment approaches for managing co-infections, such as the timing and combination of interventions for COVID-19 and typhoid fever. In addition, the findings can inform public health policies, including vaccination programs, quarantine measures, and resource allocation. In a future study, we expect to investigate how SARS-CoV-2 affects the immune response to Salmonella Typhi and vice versa. Finally, future studies can be conducted to investigate drug interactions, optimal dosing regimens, and the timing of antibiotics.

Author contributions

D.S.M. Developed and created the model; analyzed and interpreted the data; and wrote the manuscript, A.H. Developed and created the model, Manuscript - review & editing, and Insightful suggestions, A.H. Developed and created the model, Manuscript - review & editing, and Insightful ideas, and O.D.M. Developed and created the model; and made critical feedback.

Data availability

All data generated or analysed during this study are included in this published article and its supplementary information files.

Declarations

Competing interests

The authors declare no competing interests.