A Fractional-Order Study for Bicomplex Haemorrhagic Infection in Several Populations Conditions

1  Introduction

LF is an acute viral illness that leads to hemorrhagic fever in humans. The virus responsible belongs to the Adenoviridae family and is caused by a single-stranded RNA virus [1,2]. The primary reservoir host of this virus is Mastomys natalensis, commonly known as the multimammate mouse, a rodent species that is widely distributed throughout the sub-Saharan region of Africa [3]. LF was first identified in northern Nigeria in the early 1950s and again in 1969 [4]. The disease is named after the town of Lassa in Borno State, Nigeria, where it was first recognized. Unfortunately, it has since become a significant health concern in West Africa [5]. The Centers for Disease Control and Prevention (CDC) and the World Health Organization (WHO) report that there are approximately 100 to 30,000 cases and 5000 deaths each year in West Africa [6]. Because the eastern and, even more so, western parts of West Africa continuously record frequent and expansive LF outbreaks, they are more often described as the Lassa belt. Such countries are Liberia, Guinea, Sierra Leone, Nigeria, and a number of others [7]. Due to protracted civil conflict, Lassa fever’s endemicity has spread to the south coast of Liberia and West Africa, continuing to be a serious public health concern [8]. Human contamination with the Lassa virus is generally attributed to the consumption of foods or products containing rat urine or feces [9]. Moreover, direct contact with the patients and laboratory conditions increases the chances of second viral transmission [10]. Clinical symptoms of an LF appear 6–21 days after infection, with 80% of cases being mild and manifesting as fever, headaches, coughing, muscle pains, sore throat, and exhaustion. Side symptoms like nose bleeding, facial swelling, breathing issues, and low blood pressure might occur in severe situations [11]. Some of the outbreaks have occurred in these countries. In 2018, one of the biggest pandemics occurred in Nigeria. It is used as an example for this article. The Lassa Browning fever during this outbreak has affected 18 out of 36 states in Nigeria, with over 400 confirmed cases. It was the worst outbreak that has ever occurred in the country to date. Subsequently, after the first case was identified in 2018, the number of LF cases confirmed and deaths surged in Nigeria [12]. The increase in the trend of LF is directly and mostly linked to the host reservoir’s increase in LF as influenced by environmental changes, namely, rainfall. Other important causes of this seasonal pattern are congestion of health facilities, pollution of the environment, and lack of proper hygiene. Because Mastomys rats translocate from their natural ecological niche to human settings during the rainy season, the scarcity of LF occurrence primarily depends on the possibilities Africans would employ in constraining the disease’s spread [13]. Mathematical models have emerged as the core components of the disease dynamics in populations. Recent mathematical modeling activities have ranged from highly focused to general. Different models have been developed to address common questions and improve our understanding of disease distributions. Research has been conducted on the transmissibility of LF in [14–16]. Some authors have used various methods to mathematically model the transmission of Lassa fever disease in [17,18], and fractional order application on the epidemic model is studied [19].

Fractional calculus is an emerging and rapidly developing branch of mathematics that has found extensive applications across various fields of science and engineering. It has grown to contain complicated real-world dynamics as new concepts are applied and tested on actual data (see [20–23]). Fractional calculus is useful in many disciplines because of its precision and versatility, as well as its capacity to preserve historical and current records. Abdullahi [24] developed a new Lassa fever model by culling rodent populations. This lessens the number of rats that are affected, but it doesn’t totally eliminate the sickness in people. Since it is practically impossible to eradicate entire rodent populations, this conclusion is pertinent to ecological research. Using an Atangana Baleanu Caputo (ABC) fractional derivative, Ali et al. [25] investigated the behavior of COVID-19 using a nonlinear susceptible, infected, quarantine, and recovered (SIQR) pandemic model. The approximate solution was calculated using the Adams-Bashforth numerical technique. For fractional and fractal-order fractions, improved converging effects of dynamics were validated by analytical and numerical computations. In [26], an etiological model was constructed using fractal-fractional operators, which built three consecutive approximation approaches based on various kernels. Chaos is demonstrated in the study by Ghanbari and Kumar [27], which explores the dynamics of a fractional predator-prey-pathogen model. They used a numerical scheme to apply the Atangana-Baleanu fractional operator to the model, which produced intriguing chaotic behaviors of the prey and predator populations and some new findings. Rashid et al. [28] used fractional differential equations and nonlinear solutions to establish an analytical framework for Lassa fever outbreaks. They discovered the impact of ecological emergence and aerosol pathways, among other propagation processes, on the development of infected people and rodents. The results emphasize the significance of both immediate and future infectiousness through different kernels. Zhang et al. [29] examined a fractional dynamical system of tuberculosis (TB) transmission and co-infection in the Human Immunodeficiency Virus HIV community, including ten classes. According to the study, TB will probably spread throughout society if unprotected or HIV-positive people come into contact with another person who has the disease. In [30], the Caputo operator is applied to develop a fractional-order model for the transmission dynamics of susceptible, exposed, infected, and recovered (SEIR) type Lassa fever, examining its spread from rodents to humans, person-to-person, and within societal infection areas. When all transmission pathways are considered, the model reveals an increase in the negative impacts of Lassa fever. However, fractional-order data suggest that the adverse effects on specific transmission routes are less severe. The study by Kumar et al. [31] used arbitrary-order derivative operators with three different kernels to better approximate a variable-order COVID-19 outbreak system. Abidemi and Owolabi [32] created a compartmental model of LF dynamics, considering the routes of transmission from hospitalized people to infected rats and symptomatic humans. The model shows that because of the cumulative number of new cases, hospitalized persons contribute to the community’s LF burden. The study [33] presented a fractional-order dynamical system of the host-parasitoid population, investigated the prospect of generating new chaotic behaviors with the Caputo operator, and demonstrated chaotic behavior at various fractional orders. Using an effective numerical technique, researchers in [34] investigated the nonlinear structure of the multi-strain TB model under the fractal-fractional operator.

This study employs non-linear fractional differential equations to analyze a fractional-order model of Lassa fever dynamics in human and rodent populations. The model is explored by replacing the standard first-order derivative with the Caputo fractional derivative. This approach reveals self-similar patterns and long-term memory effects in the transmission of Lassa fever, distinguishing it from models using integer-order derivatives. Introducing fractional operators incorporates memory effects and more complex dynamics into the system, making them especially suitable for modeling epidemic processes. The aim of this study is to demonstrate that Caputo fractional derivatives are better suited for understanding biological systems in mathematical epidemiology. The structure of the paper is organized as follows: Section 1 introduces the topic, while Section 2 defines the fractional operators used in this research. Section 3 presents a fractional-order mathematical model employing the Caputo derivative to analyze the dynamics of Lassa fever transmission among both rodents and humans. Section 4 provides a qualitative analysis of the Lassa fever model, followed by a computational study in Section 5. Section 6 explores the stability of the proposed model, and Sections 7 and 8 present the results, discussion, and conclusions.

2  Preliminaries

This section provides a review of fundamental concepts related to fractional calculus and its key findings. To capture the drawbacks which are mentioned in Table 1, we incorporated the Caputo fractional derivative into our Lassa Fever model to enhance its accuracy and better capture the complexities of the disease dynamics, and key contributions with limitations of work are also discussed in Table 1.

Definition 1 [35]. Let H be a continuous function in ℒ1([0,𝒯],R). The Riemann-Liouville fractional integral of order ν∈(0,1) is defined as:

Itν0RLH(t)=1γ(ν)∫0t(t−x)ν−1H(x)dx.(1)

Definition 2 [35]. If H is continuous on [0,T], then its Caputo fractional derivative is defined as:

Dtν0CH(t)=1γ(n−ν)∫0t(t−x)n−ν−1dndxnH(x)dx,(2)

where n is the smallest integer greater than ν. At n=1, Eq. (2) simplifies to:

Dtν0CH(t)=1γ(1−ν)∫0t(t−x)−νddxH(x)dx.(3)

Lemma 1 [36]. Let κ∗ represent the equilibrium point of a system governed by the Caputo derivative. Then, the following expression holds if and only if H(t,κ∗)=0:

Dtν0CH(t)=H(t,κ(t)),ν∈(0,1).(4)

Remark 1. Let Φ(t)∈C[0,b] and the Caputo fractional derivative Dtν0CH(t)∈(0,b] for 0<ν≤1. Following from the Definition (1), if Dtν0CH(t)≥0 then Φ(t) is a non-decreasing function, and if Dtν0CH(t)≤0 then Φ(t) is a non-increasing function ∀t∈[0,b].

3  Mathematical Fractional Order of Lassa Fever Model

In past years, the dynamics of mathematical modeling of infectious diseases have emerged as a significant influence. This study looks at a fractional-order compartmental model for the new virus LF in Nigeria, focusing on how to keep the disease under control and prevent outbreaks, unlike traditional models that split populations into separate groups of different sizes over time. The paper presents a six-compartment LF model, where M(t) represents total active cells as follows:

M(t)=Sh(t)+Eh(t)+Ih(t)+Rh(t)+Sr(t)+Ir(t)(5)

The compartments in the model are described as follows:

•   Susceptible Humans Sh(t): Individuals who are at risk of contracting the infection at time t.

•   Exposed Humans Eh(t): Individuals who have come into contact with the infection but are not yet infectious.

•   Infectious Humans Ih(t): Individuals who are infected and capable of transmitting the disease to others.

•   Recovered Humans Rh(t): Individuals who have recovered from the infection and are considered immune.

•   Susceptible Rodents Sr(t): Rodents that can potentially become infected.

•   Infectious Rodents Ir(t): Rodents that are infected and able to transmit the disease.

The use of a fractional-order model utilizing the Caputo derivative stems from its effectiveness in accounting for memory effects and long-range interactions that are characteristics of many epidemiological systems, including the transmission of Lassa Fever. The Caputo derivative is used to construct a fractional-order derivative that accommodates non-local effects and long-term dependence associated with the transmission of disease through complicated landscapes. This model is an enhancement over traditional models with the object of supplying better continuance predictions by capturing real-world dynamics. Ojo & Goufo [37] posited the model for LF transmission as follows:

{dShdt=ψh+ζhRh−δhSh−μhSh,dEhdt=δhSh−(σh+μh)Eh,dIhdt=σhEh−(Th+μh+λh)Ih,dRhdt=ThIh−(μh+ζh)Rh,dSrdt=ψr−δrSr−μrSr,dIrdt=δrSr−μrIr.(6)

Lassa Fever Model with Fractional Derivative

Epidemics generally have long-term effects, & fractional-order models which can be used to explore these aspects. Fractional-order ‘derivative’, which permits derivatives between 0 and 1, which are a different class of systems, very similar to most real-world systems, and will provide insight into the recurrence of outbreaks or the effect of interventions. This makes it particularly suitable for applications in engineering and the applied sciences. Maintaining consistency with standard differential equations enhances the precision of disease modeling. The LF infectious disease transmission model (6) is expanded and altered using an auxiliary parameter ϕ, which reduces the model’s dimension to a time−1. Thus, the model (6) can be expressed using the Caputo derivative of order ν∈(0,1], represented by DnuC, as follows:

{1ϕν−1D0,tνCSh(t)=ψh+ζhRh−δhSh−μhSh,1ϕν−1D0,tνCEh(t)=δhSh−(σh+μh)Eh,1ϕν−1D0,tνCIh(t)=σhEh−(Th+μh+λh)Ih,1ϕν−1D0,tνCRh(t)=ThIh−(μh+ζh)Rh,1ϕν−1D0,tνCSr(t)=ψr−δrSr−μrSr,1ϕν−1D0,tνCIr(t)=δrSr−μrIr,(7)

with initial conditions:

Sh(0)≥0,Eh(0)≥0,Ih(0)≥0,Rh(0)≥0,Sr(0)≥0,Ir(0)≥0.(8)

Fig. 1 presents the flow diagram of the dynamical system. The susceptible human population increases at a rate of ψh, which shows the recruitment from birth or immigration. Additionally, the susceptible population is augmented by the loss of immunity from recovered people from LF at a rate of ζh. On the other hand, the number of susceptible humans declines due to Lassa fever infections, which result from effective interactions with the LF-infected humans or rodents at a specific rate. δh=βrhIr+βhIhNh. Natural death reduces the susceptible human population at a rate of μh. An infection in the susceptible population creates the exposed human population. Natural deaths and disease progression to the LF infectious individuals decrease the exposed human population at μh and δh, respectively. The transition rate from the exposed population to the infectious human compartment is governed by a specific rate. Once individuals enter the infectious compartment, they are subsequently reduced due to recovery through therapy at a rate of Th, as well as natural mortality and disease-induced death at a rate of λh. When Lassa fever is treated early, people recover, and the recovered human population grows. Nevertheless, the recovered population may be diminished if recovered individuals get re-infected as a result of immunity loss (ζh). Additionally, the recovered humans are reduced by natural death (μh). At a rate ψr, rats are recruited through birth, forming a susceptible rodent population. This population is lowered by natural death at a rate of μr and Lassa virus infection as a result of direct contact with an infected person or rodent at a rate of δr=βhrIhNh+βrIrNr. Infecting the susceptible rodent population creates the infectious rodent population, which is thereafter reduced by natural mortality at a rate of μr. Every model parameter is considered to be positive. Table 2 describes model parameters.

Figure 1: Lassa fever flow sheet

4  Analysis of Proposed Lassa Model

4.1 Positive-Ness and Bounded-Ness of Solutions

A major purpose of this study is to examine the conditions under which LF’s model, whose solution sets must be limited and positive, can be implemented in real-world scenarios. We are aware of the following facts about classical derivatives. The system’s positivity and boundedness are demonstrated through deduced results [38]. We have

1ϕν−1D0,tνCSh(t)=ψh+ζhRh−δhSh−μhSh,≥−(δh+μh)Sh,⇒ Sh(t)=Sh(0) Eν(−(δh+μh)tν),∀t≥0.(9)

we obtain,

Eh(t)=Eh(0) Eν(−(σh+μh)tν),Ih(t)=Ih(0) Eν(−(Th+μh+λh)tν),Rh(t)=Rh(0) Eν(−(ζh+μh)tν),Sr(t)=Sr(0) Eν(−(δr+μh)tν),Ir(t)=Ir(0) Eν(−(μr)tν).(10)

Theorem 1. The suggested LF fractional-order model (7) has a unique and bounded solution along the initial conditions (8) in R6+.

Proof: Based on model (7), we discover ◻

{D0,tνCSh(t)|Sh=0 = ψh+ζhRh≥0,D0,tνCEh(t)|Eh=0 = δhSh≥0,D0,tνCIh(t)|Ih=0 = σhEh≥0,D0,tνCRh(t)|Rh=0 = ThIh≥0,D0,tνCSr(t)|Sr=0 = ψr≥0,D0,tνCIr(t)|Ir=0 = δrSr≥0.(11)

4.2 Uniqueness and Existence Findings

The fixed point theory approach is utilized to identify the uniqueness of a mathematical model’s solution, revealing the existence of model (7) under specific conditions. The Caputo model (7) acts as the mathematical basis for analyzing the dynamics of the LF disease model, along with feasibility and stability. The sub-sections of the model computed the Lipchitz condition, including existence, uniqueness, and stability of the system.

Let X(J) denote the Banach space of real-valued functions defined on the interval J=[0,T], with continuous Sup norm. Space M is defined in Cartesian product below:

M=X(J)×X(J)×X(J)×X(J)×X(J)×X(J),

‖Sh,Eh,Ih,Rh,Sr,Ir‖=‖Sh‖+‖Eh‖+‖Rh‖+‖Ih‖+‖Sr‖+‖Ir‖,

where

‖Sh‖=supt∈J|Sh(t)|,‖Eh‖=supt∈J|Eh(t)|,‖Ih‖=supt∈J|Ih(t)|,‖Rh‖=supt∈J|Rh(t)|,‖Sr‖=supt∈J|Sr(t)|,‖Ir‖=supt∈J|Ir(t)|.

Taking ∫0t on the LF model (7) yields:

{Sh(t)−Sh(0)=∫0t[ψh+ζhRh−δhSh−μhSh]dϖ,Eh(t)−Eh(0)=∫0t[δhSh−(σh+μh)Eh]dϖ,Ih(t)−Ih(0)=∫0t[σhEh−(Th+μh+λh)Ih]dϖ,Rh(t)−Rh(0)=∫0t[ThIh−(μh+ζh)Rh]dϖ,Sr(t)−Sr(0)=∫0t[ψr−δrSr−μrSr]dϖIr(t)−Ir(0)=∫0t[δrSr−μrIr]dϖ.(12)

Let

{Λ1(t,Sh)=ψh+ζhRh−δhSh−μhSh,Λ2(t,Eh)=δhSh−(σh+μh)Eh,Λ3(t,Ih)=σhEh−(Th+μh+λh)Ih],Λ4(t,Rh)=ThIh−(μh+ζh)Rh]dϖ,,Λ5(t,Sr)=ψr−δrSr−μrSr,Λ6(t,Ir)=δrSr−μrIr,(13)

Λj, j=1,2,3,…,6, satisfies the Lipschitz condition when model (7) has upper bounds. Assume S and S1 are two functions; then

‖Λ−Λ1‖≤(μh+δh)‖S−S1‖.(14)

Setting κ1=μh+δh gives:

‖Λ−Λ1‖≤κ1‖Sh−Sh1‖.(15)

For Λi and 0≤κi<1 for i=1,2,3,…,6, the Lipschitz condition is satisfied. Using the same approach again for the remaining compartments, we get:

‖Λ−Λ2‖≤κ2‖Eh−Eh1‖,‖Λ−Λ3‖≤κ3‖Ih−Ih1‖,‖Λ−Λ4‖≤κ4‖Rh−Rh1‖,‖Λ−Λ5‖≤κ5‖Sr−Sr1‖,‖Λ−Λ6‖≤κ6‖Ir−Ir1‖,(16)

where κ2=σh+μh, κ3=(Th+μh+δh), κ4=(μh+ζh), κ5=δr+μr and κ6=μr. Following from (13), the system is represented as follows:

{Sh(t)=Sh(0)+1Γ(ν)∫0t(t−x)ν−1Λ1(x,Sh(x))dx,Eh(t)=Eh(0)+1Γ(ν)∫0t(t−x)ν−1Λ2(x,Eh(x))dx,Ih(t)=Ih(0)+1Γ(ν)∫0t(t−x)ν−1Λ(x,Ih(x))dx,Rh(t)=Ih(0)+1Γ(ν)∫0t(t−x)ν−1Λ(x,Rh(x))dx,Sr(t)=Sr(0)+1Γ(ν)∫0t(t−x)ν−1Λ(x,Sr(x))dx,Ir(t)=Ir(0)+1Γ(ν)∫0t(t−x)ν−1Λ(x,Ir(x))dx.(17)

Subsequently Eq. (17) transforms into:

{Snh=1Γ(ν)∫0t(t−x)ν−1Λ1(x,Snh(x))dx,Enh=1Γ(ν)∫0t(t−x)ν−1Λ2(x,Enh(x))dx,Inh=1Γ(ν)∫0t(t−x)ν−1Λ(x,Inh(x))dx,Rnh=1Γ(ν)∫0t(t−x)ν−1Λ(x,Rnh(x))dx,Snr=1Γ(ν)∫0t(t−x)ν−1Λ(x,Snr(x))dx,Inr=1Γ(ν)∫0t(t−x)ν−1Λ(x,Inr(x))dx.(18)

Based on the initial conditions in (8), the difference between successive terms is given as follows:

{ΨSh,n(t)=1Γ(ν)∫0t(t−x)ν−1(Λ1(x,Sh,n−1(x))−Λ1(x,Sh,n−1(x)))dx,ΨEh,n(t)=1Γ(ν)∫0t(t−x)ν−1(Λ1(x,En−1(x))−Λ1(x,En−2(x)))dx,ΨIh,n(t)=1Γ(ν)∫0t(t−x)ν−1(Λ1(x,Is,n−1(x))−Λ1(x,Is,n−2(x)))dx,ΨRh,n(t)=1Γ(ν)∫0t(t−x)ν−1(Λ1(x,Rs,n−1(x))−Λ1(x,Rs,n−2(x)))dx,ΨSr,n(t)=1Γ(ν)∫0t(t−x)ν−1(Λ1(x,Sr,n−1(x))−Λ1(x,Sr,n−2(x)))dx,ΨIr,n(t)=1Γ(ν)∫0t(t−x)ν−1(Λ1(x,Ir,n−1(x))−Λ1(x,Ir,n−2(x)))dx.(19)

Let us consider:

Sh,n(t)=∑k=0nΨSh,n,k(t),Eh,n(t)=∑k=0nΨEh,n,k(t),Ih,nn(t)=∑k=0nΨIh,n,k(t),Rh,nn(t)=∑k=0nΨRh,n,k(t),Sr,nn(t)=∑k=0nΨSr,n,k(t),Ir,nn(t)=∑k=0nΨIr,n,k(t).(20)

Hence, we have the following expressions: S1=Sh,n−1(t)−Sh,n−2(t),E1=Eh,n−1(t)−Eh,n−2(t)I1=Ih,n−1(t)−Ih,n−2(t),R1=Rh,n−1(t)−Rh,n−2(t)S1∗=Sr,n−1(t)−Sr,n−2(t)&I1∗=Ir,n−1(t)−Ir,n−2(t)

ΨSh,n−1(t)=S1,ΨEh,n−1(t)=E1,ΨIh,n−1(t)=I1,ΨRh,n−1(t)=R1,ΨSr,n−1(t)=S1∗,ΨIr,n−1(t)=I1∗.(21)

and we obtain the desired relations as follows:

{‖Sh,n(t)‖=U1∫0t‖ΨSh,n−1‖Sdx,‖Eh,n(t)‖=U2∫0t‖ΨEh,n−1‖Sdx,‖Ih,n(t)‖=U3∫0t‖ΨIh,n−1‖Sdx,‖Rh,n(t)‖=U4∫0t‖ΨRh,n−1‖Sdx,‖Sr,n(t)‖=U5∫0t‖ΨSr,n−1‖Sdx,‖Ir,n(t)‖=U6∫0t‖ΨIr,n−1‖Sdx,(22)

where

S=(t−x)ν−1,Ui=KiΓ(ν),fori=1,2,⋯,6.

Building on the previous analysis, we present and prove the following theorem:

Theorem 2. Assume that the functions Λj:[0,T]×R6→R, (j=1,2,3,…,6) with initial conditions in 8 meet the Lipschitz and contraction requirement for 0<ν<1. The Caputo fractional-order Lassa model in (7) possesses a unique solution if

TνΓ(ν+1)κj<1,j=1,2,3,…6.

is true for all t∈[0,T].

Proof. As Sh,Eh,Ih,Rh,Sr,Ir are bounded and χj, j=1,2,3,…6, satisfies the Lipschitz condition, then

‖ΨSh,n−1(t)‖≤‖ΨSh,0(t)‖(TνΓ(ν+1)κ1)n,‖ΨEh,n−1(t)‖≤‖ΨEh,0(t)‖(TνΓ(ν+1)κ2)n,‖ΨIh,n−1(t)‖≤‖ΨIh,0(t)‖(TνΓ(ν+1)κ3)n,‖ΨRh,n−1(t)‖≤‖ΨRh,0(t)‖(TνΓ(ν+1)κ4)n,‖ΨSr,n−1(t)‖≤‖ΨSr,0(t)‖(TνΓ(ν+1)κ5)n,‖ΨIr,n−1(t)‖≤‖ΨIr,0(t)‖(TνΓ(ν+1)κ6)n.(23)

after simplification We obtain the following:

‖ΨSh,n(t)‖≤‖ΨSh,0(t)‖(TνΓ(ν+1)κ1)n+1,‖ΨEh,(t)‖≤‖ΨEh,0(t)‖(TνΓ(ν+1)κ1)n+1,‖ΨIh,(t)‖≤‖ΨIh,0(t)‖(TνΓ(ν+1)κ1)n+1,‖ΨRh,n(t)‖≤‖ΨRh,0(t)‖(TνΓ(ν+1)κ1)n+1,‖ΨSr,n(t)‖≤‖ΨSr,0(t)‖(TνΓ(ν+1)κ1)n+1,‖ΨIr,n(t)‖≤‖ΨIr,0(t)‖(TνΓ(ν+1)κ1)n+1.(24)

As n→∞,

‖ΨSh,n(t)‖→0,‖ΨEh,n(t)‖→0,‖ΨRh,n(t)‖→0,‖ΨIh,n(t)‖→0,‖ΨSr,n(t)‖→0,‖ΨIr,n(t)‖→0.

This guarantees that the suggested model has a solution. To demonstrate that this is a unique solution, we will follow the steps outlined below.

Let a system of solutions to (7) exists, say Sh,1(t), Eh,1(t), Ih,1(t), Rh,1(t), Sr,1(t) and Ir,1(t), then

‖Sh−Sh,1‖=1Γ(ν)∫0t(t−τ)ν−1[Υ1(τ,Sh(τ))−Υ1(τ,Sh,1(τ))]dτ≤(TηΓ(η+1)κ1)‖Sh−Sh,1‖,

where

‖Sh−Sh,1‖(1−TνΓ(ν+1)κ1)≤0.(25)

This leads to the conclusion that

‖Sh−Sh,1‖=0asSh→Sh,1.

Following a similar approach, similar results can be derived for the remaining compartments. ◻

4.3 Equilibrium Points Analysis

This section aims to provide a detailed review of equilibrium points. To establish the existence of the endemic equilibrium, we need to solve for the steady-state solutions of the system, i.e., the points where the derivatives of all compartments of (7) are zero. We have the following system of equations,

{0=ψh+ζhRh−δhSh−μhSh,0=δhSh−(σh+μh)Eh,0=σhEh−(Th+μh+λh)Ih,0=ThIh−(μh+ζh)Rh,0=ψr−δrSr−μrSr,0=δrSr−μrIr.(26)

The Lassa fever-free equilibrium point, denoted by υ0, is calculated as:

υ0=(Sr0,Er0,Ir0,Rr0,Sh0,Ih0)=(ψhμh,0,0,0,ψrμr,0).(27)

Firstly we consider the system of Sr from (26):

0=ψr−δrSr−μrSr.

Rearrange to solve for Sr:

δrSr+μrSr=ψr.

Factor out Sr:

(δr+μr)Sr=ψr.

Finally, solve for Sr:

Sr=ψrδr+μr.(28)

Now we consider the system of Ir from (26)

0=δrSr−μrIr.

Rearrange to solve for Ir:

Ir=δrSrμr.

Substitute the expression for Sr from Step 5:

Ir=δrμr⋅ψrδr+μr.(29)

The rest of the endemic equilibrium points are computed as follows, which are denoted by υ∗, and are given by

υ∗=(Sr∗,Er∗,Ir∗,Rr∗,Sh∗,Ih∗).(30)

Sh∗=ψhK1K2K3K1K2K3δh+K1K2K3μh−σhThζhδh,Eh∗=δhψhK2K3K1K2K3δh+K1K2K3μh−σhThζhδh,Ih∗=δhψhK3σhK1K2K3δh+K1K2K3μh−σhThζhλh,Rh∗=δhψhK3K1K2K3δh+K1K2K3μh−σhThζhδh,Sr∗=ψrδr+μr,Ir∗=δhψrμr(δr+μr).(31)

where K1=σh+μh, K2=Th+μh+λh, and K3=μh+ζh. An equilibrium point with asymptotically stable behavior for all delays is categorized as absolutely stable; one with asymptotically stable behavior for some intervals, but not all, is then classified as conditionally stable.

4.4 Reproduction Number

In disease epidemiology, the basic reproduction number is a threshold value that determines whether a disease can be controlled. Researchers often use the next-generation matrix approach [37] to calculate this number for biological models. At the disease-free equilibrium (DFE) point, two matrices are considered: F for new infections and V for individual transfer. ℛ0 is calculated based on the spectral radius of FV−1. This approach provides a comprehensive understanding of disease dynamics and management strategies. Using this technique, the suggested model (7) basic reproductive number was determined to be:

ℛ0=12{βhσhK1K2+βrμr+(βhσhK1K2−βrμr)2+4(βhrSr0σhβrhSh0K1K2μr)2}.(32)

LF spreads in populations where ℛ0>1, but not when ℛ0<1.

4.5 Stability Analysis of Lassa Model

Here, we concentrate on the stability analysis of the proposed model’s solutions, demonstrating how important system stability is for achieving the expected output and carrying out the proposed system’s intended function.

4.5.1 Local Stability

Theorem 3. If ℛ0<1, the disease-free equilibrium point (υ0) is locally asymptotically stable; otherwise, it is unstable.

Proof. We obtained the Jacobian matrix of (7) as follows:

𝒥=(−βrhIrNh−βhIhNh−μh00ζh00βrhIrNh+βhIhNh−μh−σh00000σh−Th−μh−λh00000Th−μh−ζh000000−βhrIhNh−βrIrNr−μr00000βhrIhNh−βrIrNr−μr).(33)