# A Complete Model of Crimean-Congo Hemorrhagic Fever (CCHF) Transmission Cycle with Nonlocal Fractional Derivative

Authors: Hakimeh Mohammadi; Mohammed K. A. Kaabar; Jehad Alzabut; A. George Maria Selvam; Shahram Rezapour
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Crimean-Congo hemorrhagic fever is a common disease between humans and animals that is transmitted to humans through infected ticks, contact with infected animals, and infected humans. In this paper, we present a boxed model for the transmission of Crimean-Congo fever virus. With the help of the fixed-point theory, our proposed system model is investigated in detail to prove its unique solution. Given that the Caputo fractional-order derivative preserves the system’s historical memory, we use this fractional derivative in our modeling. The equilibrium points of the proposed system and their stability conditions are determined. Using the Euler method for the Caputo fractional-order derivative, we calculate the approximate solutions of the fractional system, and then, we present a numerical simulation for the transmission of Crimean-Congo hemorrhagic fever.


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## Body

## 1. Introduction

Crimean-Congo hemorrhagic fever is a common disease between human and livestock. The virus that causes this disease is one of the most important Arthropod-Borne viruses of the Bunyaviridae family, and it is a genus of Nairovirus that can cause severe and deadly disease in humans, but it is not associated with any specific clinical sign in livestock. The most common vector is a tick called Hyalomma, but it is also transmitted by other ticks [1]. The average mortality rate among infected people is 30 percent [2].The first known case of the disease was recorded in 1942 in the Crimean region of the former Soviet Union. The virus that caused the disease was also isolated from the blood of a feverish patient in 1956 in the Democratic Republic of the Congo. The relationship between these two reported places of disease and the attention to the main symptoms of the disease (fever and bleeding) has led to the choice of the current name of the disease (Crimean-Congo hemorrhagic fever) (see [3, 4]). The disease has been reported in more than 31 countries in Africa, Asia, and Eastern Europe [5].Numerous serological studies have confirmed infections in animals, especially domestic animals such as cattle, sheep, and goats that may occur as feverish reactions. Infection in animals occurs through the bite of ticks infected with the Crimean-Congo hemorrhagic fever virus [6]. Crimean-Congo hemorrhagic fever virus can also infect a wide range of wild animals. Among wild mammals, rabbits have been an important reservoir of the virus in the European part of the former Soviet Union and Bulgaria. In Asia, hedgehogs, rats, and particular species of rabbits are the reservoir of this virus [7].The most important ways of getting infected with the Crimean-Congo fever virus are as follows: the person is getting bitten by infected ticks, the contact of scratched or injured skin of a person’s body with the contents of infected crushed ticks, the contact of damaged skin or human mucosa with infected animal blood or secretions, and the contact with blood and other secretions of the infected person, as well as the contact with infected surgical instruments [8–10]. Because Crimean-Congo hemorrhagic fever is more likely to come from the contact with an infected animal or human or bite by infected ticks are transmitted to humans so hunters, farmers, ranchers, health personnel, and those contact with infected animals and humans due to occupations more likely to be infected.Clinical signs and the course of this disease include four stages:(i)
Incubation Period: After a tick bite, the incubation period usually lasts 1-3 days and reaches a maximum of 1 day. The incubation period following the contact with infected tissues or blood is usually 5-6 days, and the maximum time is 13 days [11](ii)
Prehaemorrhagy: In 80 percent of cases, Crimean-Congo hemorrhagic fever infections are asymptomatic. People in whom the disease has clinical manifestations, the onset of symptoms is sudden, and it lasts about 1 up to 7 days (average 3 days). The initial symptoms are severe headache, fever, chills, joint pain, muscle cramps, dizziness, pain and stiffness of the neck, eye pain, and fear of light. Nausea, vomiting, diarrhea, abdominal pain, loss of appetite, swelling and redness of the face, decreased heart rate, and low blood pressure have also been reported [12](iii)
Haemorrhagy: The bleeding phase is short and usually starts on days 3 to 5 and lasts 1 to 11 days (average 4 days). Bleeding in the mucosa, hematoma, bleeding gums and nose, bleeding from the uterus, bloody sputum, and bleeding from the conjunctiva and ears are the symptoms of the disease at this stage. Bleeding from various organs worsens the patient’s condition so that the patient may die in the second week of severe bleeding, intravascular coagulation, liver failure, and dehydration [13](iv)
Convalescence Period: Between days 7 and 20, the fever stops, and then the bleeding stops. From the tenth day, when the skin lesions fade, patients gradually recover. Most patients are discharged from the hospital in the third to sixth week after the onset of illness when blood and urine tests return to be normal [11]Biological and mathematical researchers have conducted research studies to model the transmission of Crimean-Congo fever. Kashkynbayev et al. have used an SI Model to study tick-borne diseases, including Crimean-Congo fever [14]. Ergena et al. have used an SIR Model to study the dynamic of tuberculosis and Crimean-Congo fever as epidemic diseases [15]. Switkes et al. have used the deterministic system of nonlinear differential equations to model the transmission of Crimean-Congo haemorrhagic fever with host immunity [16].In recent years, extensive studies [17–19] have been conducted on the mathematical analysis of fractional derivatives and integrals. The fractional-order derivative is nonlocal and includes the historical and long-term memory effect of the system, and this is one of its most important advantages over the integer-order derivative, which helps to model natural phenomena better [20–23].By the expansion of fractional differential calculus, researchers in many branches of science have turned to use the fractional differential equation system in their research. Mathematical modeling of the spread of viruses and the transmission of infectious diseases using systems of fractional differential equations are considered as one of the topics that has attracted the attention of researchers in recent decades [24]. Almeida et al. [25] proposed an epidemiologial MSEIR model formulated in the sense of Caputo fractional derivative. Baleanu et al. [26, 27] formulated new models of the HIV-1 infection of CD4+ T-cell and human liver via Caputo-Fabrizio fractional derivative. In addition, Rezapour et al. [28, 29] introduced new models for the spread of AH1N1 influenza and the transmission of Zika virus between humans and mosquitoes via Caputo-Fabrizio and Cuputo fractional derivatives, respectively. Singh analyzed the fractional blood alcohol model with a composite fractional derivative [30], and Singh et al. investigated the fractional fish farm model and fractional model of guava for biological pest control, [31, 32]. Also, Ghanbari et al. presented an efficient numerical method for the fractional model of allelopathic stimulatory phytoplankton species [33].In this article, we model the complete Crimean-Congo fever transmission cycle between humans, animals, and ticks, which in previous articles, researchers have only modeled a part of the cycle. Due to the effect of fractional derivative memory and good results obtained in recent years from fractional mathematical modeling, in this study, we use the fractional-order differential equation system to model the Crimean-Congo fever transmission.The structure of this paper is organized as follows: In Section2, some basic definitions and concepts of fractional calculus are recalled. A fractional-order mathematical model for the Crimean-Congo fever transmission cycle is formulated in Section 3. In Section 4, with the help of the fixed-point theory, our proposed system (10) is proven to have a unique solution. The approximate solution of the fractional differential equation system (10) is obtained numerically, and a numerical simulation for the transmission of the Crimean-Congo fever virus is also provided in Section 5. In Section 6, we conclude our research work.


## 2. Preliminary Results and Definitions

In the current section, we recall the two definitions of the fractional-order derivative and corresponding integral of each one. A concept of the Laplace transform of fractional derivative is also discussed.Definition 1 [34].
For an integrable functionw, the Caputo derivative of fractional order ϑ∈0,1 is given by(1)DCϑwt=1Γm−ϑ∫0twmυt−υϑ−m+1dυ,mϑ+1.The Gamma function, denoted byΓ., is defined as:(2)Γϑ=limm⟶∞m!mϑϑϑ+1ϑ+2⋯ϑ+m.Also, the corresponding fractional integral of orderϑ with Reϑ>0 is given by(3)ICϑwt=1Γϑ∫0tt−υϑ−1wυdυ.Definition 2 ([35, 36]).
Forw∈H1c,d and d>c, the Caputo-Fabrizio derivative of fractional order ϑ∈0,1 for w is given by(4)DCFϑwt=Mϑ1−ϑ∫ctexp−ϑ1−ϑt−υw′υdυ,where t≥0, Mϑ is a normalization function that depends on ϑ and M0=M1=1. If w∉H1c,d and 0<ϑ<1, this derivative for w∈L1−∞,d as given by(5)DCFϑwt=ϑMϑ1−ϑ∫−∞dwt−wυexp−ϑ1−ϑt−υdυ.Also, the correspondingCF fractional integral is presented by(6)ICFϑwt=21−ϑ2−ϑMϑwt+2ϑ2−ϑMϑ∫0twυdυ.The Laplace transform is one of the most important tools in solving differential equations, which has different definitions in fractional calculus. The next definition presents the Laplace transform of the Caputo fractional-order derivative.Definition 3 [34].
The Laplace transform of Caputo Fractional differential operator of orderϑ is given by(7)LDCϑwts=sϑLwt−∑i=0m−1sϑ−i−1wi0,m−1<ϑ≤m∈N.
This can also be obtained in the following form:(8)LDCϑwt=smLwt−sm−1w0−sm−1w′0−⋯−wm−1sm−ϑ.


## 3. Model Formulation

Mathematical models are considered as one of the most important tools in the study of disease transmission. In this section, we present a fractional-order mathematical model for the Crimean-Congo fever transmission cycle.Crimean-Congo haemorrhagic fever (CCHF) is a feverish hemorrhagic disease that is mostly transmitted by ticks. Although the virus is specific to animals, single infection, and epidemic cases of CCHF also occurred in humans. To model the transmission of this viral disease, we consider the population of transmitting ticksNk, the population of livestock and wild animals Nl, and the human population Nh. We divide the tick population into two groups and denote susceptible ticks with Sk and infected ticks with Ik. In the previous section, we have mentioned that livestock and some wild animals can also be infected with this disease and be a virus reservoir, which we divide into two groups, susceptible group Sl and infected group Il. Like the previous two populations, we divide the human population into two susceptible Sh and infected Ih groups. Susceptible ticks are infected through infected ticks at the effective contact rate β1 and through infected animals at the effective contact rate β2. Infected ticks transmit the virus to susceptible animals at the effective contact rate β3 when they feed on animal body. Crimean-Congo fever virus is transmitted to humans in three ways. The virus is transmitted to humans through infected ticks at the effective contact rate β4, through the blood and blood products of an infected animal at the effective contact rate β5, and through the blood and bloody mucosa of infected human at the effective contact rate β6. We also consider the recruitment rate of ticks, animals, and humans as Λk,Λl, and Λh, respectively. The natural mortality rates of ticks, animals, and humans are dk, dl, and dh, respectively.Based on the provided explanations, we present the Crimean-Congo fever transfer model with the system of differential equations as follows:(9)dSkdt=Λk−β1SktIkt−β2SktIlt−dkSkt,dIkdt=β1SktIkt+β2SktIlt−dkIkt,dSldt=Λl−β3SltIlt−dlSlt,dIldt=β3SltIlt−dlIlt,dShdt=Λh−β4ShtIkt−β5ShtIlt−β6ShtIht−dhSht,dIhdt=β4ShtIkt+β5ShtIlt+β6ShtIht−dhIht,where all of the initial conditions Sk0=S0k,Ik0=I0k,Sl0=S0l,Il0=I0l,Sh0=S0h, and Ih0=I0h are positive.The fractional-order system (FDEs) is related to systems with memory, history, or nonlocal effects which exist in the many biological systems that show the realistic biphasic decline behavior of infection or diseases but at a slower rate. In the above integer-order system, since the internal memory effects of the biological system of CCHF are not included, it is better that we extend the proposed ordinary model to a fractional model. In this alternative, the equality of the dimensions of both sides of the equation is disturbed, and we use an auxiliary parameterσ, with the dimension of sec., to solve this problem ([37]). Thus, the fractional-order model for the Crimean-Congo haemorrhagic fever (CCHF) is given as follows:(10)σϑ−1DCtϑSkt=Λk−β1SktIkt−β2SktIlt−dkSkt,σϑ−1DCtϑIkt=β1SktIkt+β2SktIlt−dkIkt,σϑ−1DCtϑSlt=Λl−β3SltIkt−dlSlt,σϑ−1DCtϑIlt=β3SltIkt−dlIlt,σϑ−1DCtϑSht=Λh−β4ShtIkt−β5ShtIlt−β6ShtIht−dhSht,σϑ−1DCtϑIht=β4ShtIkt+β5ShtIlt+β6ShtIht−dhIht,where t≥0 and 0<ϑ<1.


### 3.1. Nonnegative Solution

To show the nonnegativity of solutions, we claim thatM=Sk,Ik,Sl,Il,Sh,Ih∈R6+:Nkt≤Λk/dk,Nlt≤Λl/dl,Nht≤Λh/dh is the feasibility region of system (10). To prove this claim, we consider the following Lemma.Lemma 4.
The closed setM with respect to the fractional system (10) is positively invariant.Proof.
We first add two relations in the system (10) to obtain the fractional derivative of the total population of ticks. So,(11)σϑ−1DCtvNkt=Λk−dkNkt,where Nkt=Skt+Ikt. We apply the Laplace transform to the parties of the above relation, then(12)Nkt=Nk0Eϑ−dkσ1−ϑtϑ+∫0tΛkσ1−ϑηϑ−1Eϑ,ϑ−dkσ1−ηηϑdη.
In the above equation,Nk0 is the initial size of ticks population, and the terms Eϑ,Eϑ,ϑ are the Mittag-Leffler functions which are defined by(13)Eϑw=∑n=0∞wnΓ1+nϑ,Eϑ,ϑw=∑n=0∞wnΓϑ+nϑ,ϑ>0.
By simplifying the relations, we conclude that(14)Nkt=Nk0Eϑ−dkσ1−ϑtϑ+∫0tΛkσ1−ϑηϑ−1∑n=0∞−1ndknσn1−ϑηnϑΓnϑ+ϑdη=Λkσ1−ϑdkσ1−ϑ+Eϑ−dkσ1−ϑtϑNk0−Λkσ1−ϑdkσ1−ϑ,=Λkdk+Eϑ−dkσ1−ϑtϑNk0−Λkdk.
Now, ifNk0≤Λk/dk, then for t>0, Nkt≤Λk/dk. At the same way for Nl and Nh, we can prove that if Nl0≤Λl/dl and Nh0≤Λh/dh, then Nlt≤Λl/dl and Nht≤Λh/dh. Thus, the closed set M with respect to fractional model (2) is positively invariant.☐


### 3.2. Equilibrium Points

In the current section, we determine the equilibrium points of the system (10) and the basic reproduction number. We present the necessary conditions for the stability of the system at the equilibrium point. To determine the equilibrium points, we set the equations to zero in system (10),(15)DCϑSkt=DCϑIkt=DCϑSlt=DCϑIlt=DCϑSht=DCϑIht=0,We solve the resulting algebraic equations and determine the equilibrium point of the system. The disease-free equilibrium point, denoted byE0, is obtained as: E0=Λk/dk,0,Λl/dl,0,Λh/dh,0. The second equilibrium point, called the endemic equilibrium point, is obtained as E∗=Sk∗,Ik∗,Sl∗,Il∗,Sh∗,Ih∗,(16)Sk∗=Λkβ1Ik∗+β2Il∗+dk,Sl∗=Λlβ3Ik∗+dl,Sh∗=Λhβ4Ik∗+β5Il∗+β6Ih∗+dh.When the basic reproduction number is greater than one, and the spread of the disease continues, the endemic equilibrium point is defined. To obtain the basic reproduction number, we use the next generation method [38]. We consider the matrix form of the system (10) as follows:(17)DCϑvt=Fvt−Vvt,where(18)Fνt=σ1−ϑβ1SktIkt+β2SktIltβ3SltIktβ4ShtIkt+β5ShtIlt+β6ShtIht,Vνt=σ1−ϑdkIktdlIltdhIht.By calculating the Jacobian matrix forF and V at the disease-free equilibrium point, we obtain:(19)JFE0=σ1−ϑβ1Λkdkβ2Λkdk0β3Λldl00β4Λhdhβ5Λhdhβ6Λhdh,JvE0=σ1−ϑdk000dl000dh.The basic reproduction numberR0 is defined as the eigenvalue of next generation matrix of system (10), R0=ρFV−1. We obtain: R0=maxRh,Rkl,(20)Rh=β6Λhdh2,Rkl=β1Λkdl+β12Λk2dl2+4β2β3ΛkΛldk22dldk2.In commonly used infection models, whenR0>1, the infection will be able to start spreading in a population, but not if R0<1.


### 3.3. Stability of Equilibrium Points

To determine the necessary conditions for the stability of the disease-free equilibrium point, we investigate the roots of the characteristic equation of system (10). The Jacobian matrix of the system (10) is(21)J=σ1−ϑ×−β1Ik−β2Il−dk−β1Sk0−β2Sk00β1Ik+β2Ilβ1Sk−dk0β2Sk000−β3Sl−β3Ik−dl0000β3Slβ3Ik−dl000−β4Sh0−β5Sh−β4Ik−β5Il−β6Ih−dh−βhSh0β4Sh0β5Shβ4Ik+β5Il+β6Ihβ6Sh−dh.Then, the Jacobian matrix atE0 is obtained as:(22)JE0=σ1−ϑ−dk−β1Λkdk0−β2Λkdk000β1Λkdk−dk0β2Λkdk000−β3Λldl−dl0000β3Λldl0−dl000−β4Λhdh0−β5Λhdh−dh−βhΛhdh0β4Λhdh0β5Λhdh0β6Λhdh−dh.In the following theorem, we determine the necessary conditions for the stability of the disease-free equilibrium point.Theorem 5.
The disease-free equilibrium pointE0 is locally asymptotically stable if R0<1.Proof.
The characteristic equation of matrixJE0 is obtained as follows:(23)dl+λdk+λdh+λβ6Λhdh−dh−λdl+λβ1Λkdk−dk−λ−β2β3ΛkΛldkdl=0.
Therefore, the eigenvalues of the Jacobin matrix areλ1=−dl,λ2=−dk,λ3=−dh,λ4=β6Λh/dh−dh, and the roots of the following equation are:(24)λ2−λβ1Λkdk−dk−dl−β1Λkdldk+dldk+β2β3ΛkΛldkdl=0.
The three rootsλ1,λ2, and λ3 are negative. If R0<1, then Rh=β6Λh/dh2<1, we obtain λ4<0. It also follows from R0<1 that Rkl<1, then we conclude by simplifying β1Λk<dk2. In Equation (24), which is a quadratic equation, we have:(25)P=−β1Λkdldk+dldk+β2β3ΛkΛldkdl,S=β1Λkdk−dk−dl,since β1Λk<dk2 then P>0,S<0 so Equation (24) has 2 negative roots. Therefore, all of the eigenvalues are negative, and the disease-free equilibrium point is locally asymptotically stable.☐


## 3.1. Nonnegative Solution

To show the nonnegativity of solutions, we claim thatM=Sk,Ik,Sl,Il,Sh,Ih∈R6+:Nkt≤Λk/dk,Nlt≤Λl/dl,Nht≤Λh/dh is the feasibility region of system (10). To prove this claim, we consider the following Lemma.Lemma 4.
The closed setM with respect to the fractional system (10) is positively invariant.Proof.
We first add two relations in the system (10) to obtain the fractional derivative of the total population of ticks. So,(11)σϑ−1DCtvNkt=Λk−dkNkt,where Nkt=Skt+Ikt. We apply the Laplace transform to the parties of the above relation, then(12)Nkt=Nk0Eϑ−dkσ1−ϑtϑ+∫0tΛkσ1−ϑηϑ−1Eϑ,ϑ−dkσ1−ηηϑdη.
In the above equation,Nk0 is the initial size of ticks population, and the terms Eϑ,Eϑ,ϑ are the Mittag-Leffler functions which are defined by(13)Eϑw=∑n=0∞wnΓ1+nϑ,Eϑ,ϑw=∑n=0∞wnΓϑ+nϑ,ϑ>0.
By simplifying the relations, we conclude that(14)Nkt=Nk0Eϑ−dkσ1−ϑtϑ+∫0tΛkσ1−ϑηϑ−1∑n=0∞−1ndknσn1−ϑηnϑΓnϑ+ϑdη=Λkσ1−ϑdkσ1−ϑ+Eϑ−dkσ1−ϑtϑNk0−Λkσ1−ϑdkσ1−ϑ,=Λkdk+Eϑ−dkσ1−ϑtϑNk0−Λkdk.
Now, ifNk0≤Λk/dk, then for t>0, Nkt≤Λk/dk. At the same way for Nl and Nh, we can prove that if Nl0≤Λl/dl and Nh0≤Λh/dh, then Nlt≤Λl/dl and Nht≤Λh/dh. Thus, the closed set M with respect to fractional model (2) is positively invariant.☐


## 3.2. Equilibrium Points

In the current section, we determine the equilibrium points of the system (10) and the basic reproduction number. We present the necessary conditions for the stability of the system at the equilibrium point. To determine the equilibrium points, we set the equations to zero in system (10),(15)DCϑSkt=DCϑIkt=DCϑSlt=DCϑIlt=DCϑSht=DCϑIht=0,We solve the resulting algebraic equations and determine the equilibrium point of the system. The disease-free equilibrium point, denoted byE0, is obtained as: E0=Λk/dk,0,Λl/dl,0,Λh/dh,0. The second equilibrium point, called the endemic equilibrium point, is obtained as E∗=Sk∗,Ik∗,Sl∗,Il∗,Sh∗,Ih∗,(16)Sk∗=Λkβ1Ik∗+β2Il∗+dk,Sl∗=Λlβ3Ik∗+dl,Sh∗=Λhβ4Ik∗+β5Il∗+β6Ih∗+dh.When the basic reproduction number is greater than one, and the spread of the disease continues, the endemic equilibrium point is defined. To obtain the basic reproduction number, we use the next generation method [38]. We consider the matrix form of the system (10) as follows:(17)DCϑvt=Fvt−Vvt,where(18)Fνt=σ1−ϑβ1SktIkt+β2SktIltβ3SltIktβ4ShtIkt+β5ShtIlt+β6ShtIht,Vνt=σ1−ϑdkIktdlIltdhIht.By calculating the Jacobian matrix forF and V at the disease-free equilibrium point, we obtain:(19)JFE0=σ1−ϑβ1Λkdkβ2Λkdk0β3Λldl00β4Λhdhβ5Λhdhβ6Λhdh,JvE0=σ1−ϑdk000dl000dh.The basic reproduction numberR0 is defined as the eigenvalue of next generation matrix of system (10), R0=ρFV−1. We obtain: R0=maxRh,Rkl,(20)Rh=β6Λhdh2,Rkl=β1Λkdl+β12Λk2dl2+4β2β3ΛkΛldk22dldk2.In commonly used infection models, whenR0>1, the infection will be able to start spreading in a population, but not if R0<1.


## 3.3. Stability of Equilibrium Points

To determine the necessary conditions for the stability of the disease-free equilibrium point, we investigate the roots of the characteristic equation of system (10). The Jacobian matrix of the system (10) is(21)J=σ1−ϑ×−β1Ik−β2Il−dk−β1Sk0−β2Sk00β1Ik+β2Ilβ1Sk−dk0β2Sk000−β3Sl−β3Ik−dl0000β3Slβ3Ik−dl000−β4Sh0−β5Sh−β4Ik−β5Il−β6Ih−dh−βhSh0β4Sh0β5Shβ4Ik+β5Il+β6Ihβ6Sh−dh.Then, the Jacobian matrix atE0 is obtained as:(22)JE0=σ1−ϑ−dk−β1Λkdk0−β2Λkdk000β1Λkdk−dk0β2Λkdk000−β3Λldl−dl0000β3Λldl0−dl000−β4Λhdh0−β5Λhdh−dh−βhΛhdh0β4Λhdh0β5Λhdh0β6Λhdh−dh.In the following theorem, we determine the necessary conditions for the stability of the disease-free equilibrium point.Theorem 5.
The disease-free equilibrium pointE0 is locally asymptotically stable if R0<1.Proof.
The characteristic equation of matrixJE0 is obtained as follows:(23)dl+λdk+λdh+λβ6Λhdh−dh−λdl+λβ1Λkdk−dk−λ−β2β3ΛkΛldkdl=0.
Therefore, the eigenvalues of the Jacobin matrix areλ1=−dl,λ2=−dk,λ3=−dh,λ4=β6Λh/dh−dh, and the roots of the following equation are:(24)λ2−λβ1Λkdk−dk−dl−β1Λkdldk+dldk+β2β3ΛkΛldkdl=0.
The three rootsλ1,λ2, and λ3 are negative. If R0<1, then Rh=β6Λh/dh2<1, we obtain λ4<0. It also follows from R0<1 that Rkl<1, then we conclude by simplifying β1Λk<dk2. In Equation (24), which is a quadratic equation, we have:(25)P=−β1Λkdldk+dldk+β2β3ΛkΛldkdl,S=β1Λkdk−dk−dl,since β1Λk<dk2 then P>0,S<0 so Equation (24) has 2 negative roots. Therefore, all of the eigenvalues are negative, and the disease-free equilibrium point is locally asymptotically stable.☐


## 4. Existence of Unique Solution

In the current section, using the fixed-point theory, we prove that system (10) has a unique solution. Fixed-point theory is essential in proving the existence of a solution to the proposed system where adequate conditions are provided by fixed-point theorems such that a unique fixed point exists for a given function. To achieve this goal, we prove that kernels are satisfied under the Lipschitz condition, and they are contraction. Then, the existence of solution to the proposed system is constructed via fixed-point theorem. From the Lipschitz condition, the uniqueness of our obtained solution is proven when the obtained condition is satisfied.First, we consider system (10) in the following compact form:(26)σϑ−1DCtϑSkt=R1t,Skt,σϑ−1DCtϑIkt=R2t,Ikt,σϑ−1DCtϑSlt=R3t,Slt,σϑ−1DCtϑIlt=R4t,Ilt,σϑ−1DCtϑSht=R5t,Sht,σϑ−1DCtϑIht=R6t,Iht.We apply the fractional-order integral to the parties of the above equations, so(27)Skt−Sk0=σ1−ϑΓϑ∫0tR1μ,Skt−μϑ−1dμ,Ikt−Ik0=σ1−ϑΓϑ∫0tR2μ,Ikt−μϑ−1dμ,Slt−Sl0=σ1−ϑΓϑ∫0tR3μ,Slt−μϑ−1dμ,Ilt−Il0=σ1−ϑΓϑ∫0tR4μ,Ilt−μϑ−1dμ,Sht−Sh0=σ1−ϑΓϑ∫0tR5μ,Sht−μϑ−1dμ,Iht−Ih0=σ1−ϑΓϑ∫0tR6μ,Iht−μϑ−1dμ.In the following, we prove that the kernelsRj,j=1,2,3,4,5,6 are satisfied in the Lipschitz condition, and they are contraction.Theorem 6.
KernelR1 is satisfied in Lipschitz condition and contraction if we have:(28)0≤β1z1+β2z2+dk<1.Proof.
We can write forSk and S1k,(29)R1t,Sk−R1t,S1k=−β1IkSk−S1k−β2IlSk−S1k−dkSk−S1k,≤β1IkSk−Sk+β2IlSk−S1k+dkSk−S1k,≤β1Ik+β2Il+dkSk−S1k,≤β1z1+β2z2+dkSk−S1k.
Considere1=β1z1+β2z2+dk, where Ikt≤z1 and Il≤z2, are bounded functions. We get:(30)R1t,Sk−R1t,S1k≤e1Skt−S1kt,if 0≤β1z1+β2z2+dk<1, then the kernel R1 is satisfied in Lipschitz condition, and it is contraction.☐In a similar way, we can show that the kernelsRj,j=2,3,4,5,6 are satisfied in the Lipschitz condition as follows:(31)R2t,Ik−R2t,I1k≤e2Ikt−I1kt,R3t,Sl−R3t,S1l≤e3Slt−S1lt,R4t,Il−R4t,I1l≤e4Ilt−I1lt,R5t,Sh−R3t,S1h≤e5Sht−S1ht,R6t,Ih−R4t,I1h≤e6Iht−I1ht,so that e2=β1z4+dk,e3=β3z1+dl,e4=dl,e5=β4z1+β5z2+β6z3+dh,e6=β6z6+dh are bounded functions where Iht≤z3, Skt≤z4, Slt≤z5, and Sht≤z6. Also, if 0≤ej<1,j=2,3,4,5,6, then Rj are contraction for j=2,3,4,5,6.Based on system (27), we define:(32)A1nt=Snkt−Sn−1kt=σ1−ϑΓϑ∫0tR1μ,Sn−1k−R1μ,Sn−2kt−μϑ−1dμ,A2nt=Inkt−In−1kt=σ1−ϑΓϑ∫0tR2μ,In−1k−R2μ,In−2kt−μϑ−1dμ,A3nt=Snlt−Sn−1lt=σ1−ϑΓϑ∫0tR3μ,Sn−1l−R3μ,Sn−2lt−μϑ−1dμ,A4nt=Inlt−In−1lt=σ1−ϑΓϑ∫0tR4μ,In−1l−R4μ,In−2lt−μϑ−1dμ,A5nt=Snht−Sn−1ht=σ1−ϑΓϑ∫0tR5μ,Sn−1h−R5μ,Sn−2ht−μϑ−1dμ,A6nt=Inht−In−1ht=σ1−ϑΓϑ∫0tR6μ,In−1h−R6μ,In−2ht−μϑ−1dμ,where Sk0=S0k,Ik0=I0k,Sl0=S0l,Il0=I0l,Sh0=S0h, and Ih0=I0h are initial conditions. The norm of A1n in the above system is expressed as follows:(33)A1nt=Snkt−Sn−1kt≤σ1−ϑΓϑ∫0tR1μ,Sn−1k−R1μ,Sn−2kt−μϑ−1dμ.According to the Lipschitz condition (30), we conclude(34)A1nt≤σ1−ϑΓϑe1∫0tA1n−1μdμ.Similarly, we can prove that(35)A2nt≤σ1−ϑΓϑe2∫0tA2n−1μdμ,A3nt≤σ1−ϑΓϑe3∫0tA3n−1μdμ,A4nt≤σ1−ϑΓϑe4∫0tA4n−1μdμ,A5nt≤σ1−ϑΓϑe5∫0tA5n−1μdμ,A6nt≤σ1−ϑΓϑe6∫0tA6n−1μdμ.Therefore, we get(36)Snkt=∑i=1nA1it,Inkt=∑i=1nA2it,Snlt=∑i=1nA3it,Inlt=∑i=1nA4it,Snht=∑i=1nA5it,Inht=∑i=1nA6it.In the next theorem, we prove the existence of solution by the fixed-point theorem.Theorem 7.
The Crimean-Congo fever transmission fractional-order model (10) has a solution, if there exists tε such that(37)σ1−ϑΓϑtεei<1.Proof.
By Equation (34) and Equation (46), we obtain(38)A1nt≤Snk0σ1−ϑΓϑe1tn,A2nt≤Ink0σ1−ϑΓϑe2tn,A3nt≤Snl0σ1−ϑΓϑe3tn,A4nt≤Inl0σ1−ϑΓϑe4tn,A5nt≤Snh0σ1−ϑΓϑe5tn,A6nt≤Inh0σ1−ϑΓϑe6tn.
The above relations show that the system has a continuous solution. Now, it is sufficient to show that the above functions construct the solution for the fractional-order model (10). We consider the following relations:(39)Skt−Sk0=Snkt−U1nt,Ikt−Ik0=Inkt−U2nt,Slt−Sl0=Snlt−U3nt,Ilt−Il0=Inlt−U4nt,Sht−Sh0=Snht−U5nt,Iht−Ih0=Inht−U6nt.
The norm ofU1nt is obtained as follows:(40)U1nt=σ1−ϑΓϑ∫0tR1μ,Sk−R1μ,Sn−1kdμ≤σ1−ϑΓϑ∫0tR1μ,Sk−R1μ,Sn−1kdμ≤σ1−ϑΓϑe1Sk−Sn−1kt.
By continuing this repetitive method, we conclude:(41)U1nt≤σ1−ϑΓϑtn+1e1n+1k.
Attε, we have(42)U1nt≤σ1−ϑΓϑtεn+1e1n+1k.
If we take limit on the recent relation asn approaches to ∞, it results U1nt⟶0. Similarly, we conclude that Bjnt⟶0,j=2,3,4,5,6, and the proof is complete.☐To show the uniqueness of the solution of CCHF model, we consider that the fractional-order system (10) has another solution such as S1kt,I1kt,S1lt,I1lt,S1ht,I1ht, then for Sk,S1k can be written as:(43)Skt−S1kt=σ1−ϑΓϑ∫0tR1μ,Sk−R1μ,S1kdμ.We take the norm on the above equation, so(44)Skt−S1kt=σ1−ϑΓϑ∫0t∥R1μ,Sk−R1μ,S1k∥dμ.By Lipschitz condition (30), we obtain:(45)Skt−S1kt≤σ1−ϑΓϑe1tSkt−S1kt.Thus,(46)Skt−S1kt1−σ1−ϑΓϑe1t≤0.Theorem 8.
The solution of fractional-order system (10) is unique when the following condition is met:(47)1−σ1−ϑΓϑe1t>0.Proof.
Assume that the condition (46) holds, in which case we conclude from (46) and (47) that Skt−S1kt=0, and this shows that Sht=S1ht. In the same way, similar relationships can be reached for Ik,Sl,Il,Sh,Ih. This completes the proof.☐


## 5. Numerical Simulation and Discussion

In this section, we first obtain the approximate solution of the fractional differential equation system (10) by a numerical method, and then, we present a numerical simulation for the transmission of the Crimean-Congo fever virus.


### 5.1. Numerical Method

We use the fractional Euler method for Caputo derivative [39] to obtain the approximate solutions of the Crimean-Congo hemorrhagic fever virus transmission model. First, we consider the compact form of the system (10) as follows:(48)σϑ−1DCtϑφt=Qt,φt,φ0=φ0,0≤t≤T<∞,where φ=Sk,Ik,Sl,Il,Sh,Ih∈R+6, φ0=S0k,I0k,S0l,I0l,S0h,I0h, and Qt is a continuous real vector function that is satisfied in the Lipschitz condition as follows:(49)Qφ1t−Qφ2t≤mφ1t−φ2t,m>0.We apply the fractional-order integral operator corresponding to the Caputo fractional-order derivative on both sides of Equation (48), so(50)φt=σ1−ϑφ0+IϑQφt,0≤t≤T<∞.Setr=T−0/N and tn=nr, where t∈0,T and N is a natural number and n=0,1,2,⋯,N. Let φn be the approximation of φt at t=tn. By the fractional Euler method ([39]), we get:(51)φn+1=σ1−ϑφ0+rϑΓϑ+1∑p=0nωn+1,pQtp,φp,p=0,1,2,⋯,N−1,where(52)ωn+1,p=n+1−pϑ−n−pϑ,p=0,1,2,⋯,n.The obtained scheme is stable. Details of the stability analysis are given in Theorem (3.1) of [39]. According to the explanations provided, the answer of the system is obtained as follows:(53)Sn+1k=σ1−ϑS0k+rϑΓϑ+1∑p=0nωn+1,py1tp,φp,In+1k=σ1−ϑI0k+rϑΓϑ+1∑p=0nωn+1,py2tp,φp,Sn+1l=σ1−ϑS0l+rϑΓϑ+1∑p=0nωn+1,py3tp,φp,In+1l=σ1−ϑI0l+rϑΓϑ+1∑p=0nωn+1,py4tp,φp,Sn+1h=σ1−ϑS0h+rϑΓϑ+1∑p=0nωn+1,py5tp,φp,In+1h=σ1−ϑI0h+rϑΓϑ+1∑p=0nωn+1,py6tp,φp,so that ωn+1,p=n+1−pϑ−n−pϑ and the functions yj for j=0,1,⋯,6 are expressed as:(54)y1t,φt=Λk−β1SktIkt−β2SktIlt−dkSkt,y2t,φt=β1SktIkt+β2SktIlt−dkIkt,y3t,φt=Λl−β3SltIkt−dlSlt,y4t,φt=β3SltIkt−dlIlt,y5t,φt=Λh−β4ShtIkt−β5ShtIht−β6ShtIht−dhSht,y6t,φt=β4ShtIkt+β5ShtIlt+β6ShtIht−dhIht.


### 5.2. Simulation

In the present subsection, we present a numerical simulation to investigate the transmission of Crimean-Congo fever virus based on the amount of reproduction number. Also, we compare the results of the integer-order and fractional-order models.To perform the desired simulation, in two cases, we consider different values for the parameters. In the first case, we assume:β1=0.5×10−4,β2=0.3×10−4,β3=0.1×10−3,β4=0.03×10−4,β5=0.4×10−4,β6=0.7×10−4,Λk=0.6,Λl=0.3,Λh=0.6,dk=0.09,dl=0.07,dh=0.007,σ=0.99. We also consider the initial values as Sk=800,Ik=20,Sl=600,Il=30,Sh=1000,Ih=10.Using the above parameters, we obtainRh=0.0233,Rkl=0.00597; thus, R0=0.0233<1. Figure 1 shows the results of model (10) for all six groups for ϑ=0.98. In this case, R0<1, and Figure 1 shows that over time, the number of susceptible people is decreased, and the number of infected people is increased, but in less than 20 days, the number of infected people is decreased and eventually reaches zero, and the spread of the disease stops. In this case, St and It converge to the disease-free equilibrium point E0.Figure 1
Plots of the results of model (10) with R0<1 for ϑ=0.98.In the second case, we assume that the disease transmission rate increases from the susceptible group to the infected group, and the transmission rates are equal toβ1=0.5×10−3,β2=0.3×10−3,β3=0.1×10−2,β4=0.7×10−3,β5=0.4×10−3,β6=0.7×10−3. With these transfer rates, the value of the reproduction number is equal to R0=1.435>1. Figure 2 shows the results of model (10) for the six groups studied in this case. Over time, the population of susceptible groups decreases and the population of infected groups increases, and finally, after 100 days, the population of infected groups decreases and converges to the endemic equilibrium point. As the rate of disease transmission increases, the value of R0 increases, and we observe that the disease does not go away and its spread continues.Figure 2
Plots of the results of model (10) with R0>1 for ϑ=0.98.In this work, we have used the fractional-order derivative for modeling. In order to investigate the effect of derivation order, we have drawn the model results for infected groups with derivatives with integer-orderϑ=1 and fractional-order ϑ=0.98 in Figure 3. Figure 3 shows that the results of model (10) are similar for the integer-order and the Caputo fractional order, and a small change in the order of derivation has no effect on the overall trend of the results in terms of ascending and descending, but the resulting numerical values are different.Figure 3
Plots of the results of model (10) for infected groups with integer-order ϑ=1 and fractional-order ϑ=0.98 in the case R0>1.


### 5.3. The Reproduction Number Sensitivity Analysis

We investigate the effect of parameters in Crimean-Congo hemorrhagic fever fractional model (10) on reproduction number using the method introduced by [40]. For this simulation, we use the parameters in the first case of the previous subsection. Since R0 is defined as R0=maxRh,Rkl, therefore, we analyze the sensitivity of R0 in two cases.First, ifR0=Rh, by the mentioned method, we have Sβ6=∂R0/∂β6β6/R0=1>0,SΛh=∂R0/∂ΛhΛh/R0=1>0,Sdh=∂R0/∂dhdh/R0=−2β6Λh/dh3=−1.55<0. Figure 4 shows the sensitivity of R0 with respect to each of the parameters. As you can see, changing each of the parameters of model (10) that is involved in Rh changes the reproduction number. The reproduction number is directly related to parameters β6,Λh and inversely related to parameter dh. From an epidemiological point of view, whenever the reproduction number decreases, then the spread of the disease is controlled. Given that β6 has the most positive effect on the R0, so to control the spread of the disease, β6 should be reduced through the reduced communication of infected and susceptible humans.Figure 4
The graphs show the effect of model parameters on the reproduction number for the caseR0=Rh.In the latter case ifR0=Rkl, we obtain the same equations as above(55)SΛk=∂R0∂ΛkΛkR0=0.724>0,Sβ1=∂R0∂β1β1R0=0.4486>0,Sdl=∂R0∂dldlR0=−0.5513<0,Sβ2=∂R0∂β2β2R0=0.275>0,Sβ3=∂R0∂β3β3R0=0.276>0,Sdk=∂R0∂dkdkR0=−1.448<0.Figure5 shows the sensitivity of R0 with respect to each of the parameters. The reproduction number is directly related to parameters β1,β2,β3,andΛk and inversely related to parameter dl,dk. Among the mentioned parameters, parameters β1,β2, and β3 can be controlled, and all of which have a positive effect on causality, so to reduce the amount of reproduction number, it is enough to reduce the rate of disease transmission between ticks, animals, and humans.Figure 5
The graphs show the effect of model parameters on the reproduction number for the caseR0=Rkl.


## 5.1. Numerical Method

We use the fractional Euler method for Caputo derivative [39] to obtain the approximate solutions of the Crimean-Congo hemorrhagic fever virus transmission model. First, we consider the compact form of the system (10) as follows:(48)σϑ−1DCtϑφt=Qt,φt,φ0=φ0,0≤t≤T<∞,where φ=Sk,Ik,Sl,Il,Sh,Ih∈R+6, φ0=S0k,I0k,S0l,I0l,S0h,I0h, and Qt is a continuous real vector function that is satisfied in the Lipschitz condition as follows:(49)Qφ1t−Qφ2t≤mφ1t−φ2t,m>0.We apply the fractional-order integral operator corresponding to the Caputo fractional-order derivative on both sides of Equation (48), so(50)φt=σ1−ϑφ0+IϑQφt,0≤t≤T<∞.Setr=T−0/N and tn=nr, where t∈0,T and N is a natural number and n=0,1,2,⋯,N. Let φn be the approximation of φt at t=tn. By the fractional Euler method ([39]), we get:(51)φn+1=σ1−ϑφ0+rϑΓϑ+1∑p=0nωn+1,pQtp,φp,p=0,1,2,⋯,N−1,where(52)ωn+1,p=n+1−pϑ−n−pϑ,p=0,1,2,⋯,n.The obtained scheme is stable. Details of the stability analysis are given in Theorem (3.1) of [39]. According to the explanations provided, the answer of the system is obtained as follows:(53)Sn+1k=σ1−ϑS0k+rϑΓϑ+1∑p=0nωn+1,py1tp,φp,In+1k=σ1−ϑI0k+rϑΓϑ+1∑p=0nωn+1,py2tp,φp,Sn+1l=σ1−ϑS0l+rϑΓϑ+1∑p=0nωn+1,py3tp,φp,In+1l=σ1−ϑI0l+rϑΓϑ+1∑p=0nωn+1,py4tp,φp,Sn+1h=σ1−ϑS0h+rϑΓϑ+1∑p=0nωn+1,py5tp,φp,In+1h=σ1−ϑI0h+rϑΓϑ+1∑p=0nωn+1,py6tp,φp,so that ωn+1,p=n+1−pϑ−n−pϑ and the functions yj for j=0,1,⋯,6 are expressed as:(54)y1t,φt=Λk−β1SktIkt−β2SktIlt−dkSkt,y2t,φt=β1SktIkt+β2SktIlt−dkIkt,y3t,φt=Λl−β3SltIkt−dlSlt,y4t,φt=β3SltIkt−dlIlt,y5t,φt=Λh−β4ShtIkt−β5ShtIht−β6ShtIht−dhSht,y6t,φt=β4ShtIkt+β5ShtIlt+β6ShtIht−dhIht.


## 5.2. Simulation

In the present subsection, we present a numerical simulation to investigate the transmission of Crimean-Congo fever virus based on the amount of reproduction number. Also, we compare the results of the integer-order and fractional-order models.To perform the desired simulation, in two cases, we consider different values for the parameters. In the first case, we assume:β1=0.5×10−4,β2=0.3×10−4,β3=0.1×10−3,β4=0.03×10−4,β5=0.4×10−4,β6=0.7×10−4,Λk=0.6,Λl=0.3,Λh=0.6,dk=0.09,dl=0.07,dh=0.007,σ=0.99. We also consider the initial values as Sk=800,Ik=20,Sl=600,Il=30,Sh=1000,Ih=10.Using the above parameters, we obtainRh=0.0233,Rkl=0.00597; thus, R0=0.0233<1. Figure 1 shows the results of model (10) for all six groups for ϑ=0.98. In this case, R0<1, and Figure 1 shows that over time, the number of susceptible people is decreased, and the number of infected people is increased, but in less than 20 days, the number of infected people is decreased and eventually reaches zero, and the spread of the disease stops. In this case, St and It converge to the disease-free equilibrium point E0.Figure 1
Plots of the results of model (10) with R0<1 for ϑ=0.98.In the second case, we assume that the disease transmission rate increases from the susceptible group to the infected group, and the transmission rates are equal toβ1=0.5×10−3,β2=0.3×10−3,β3=0.1×10−2,β4=0.7×10−3,β5=0.4×10−3,β6=0.7×10−3. With these transfer rates, the value of the reproduction number is equal to R0=1.435>1. Figure 2 shows the results of model (10) for the six groups studied in this case. Over time, the population of susceptible groups decreases and the population of infected groups increases, and finally, after 100 days, the population of infected groups decreases and converges to the endemic equilibrium point. As the rate of disease transmission increases, the value of R0 increases, and we observe that the disease does not go away and its spread continues.Figure 2
Plots of the results of model (10) with R0>1 for ϑ=0.98.In this work, we have used the fractional-order derivative for modeling. In order to investigate the effect of derivation order, we have drawn the model results for infected groups with derivatives with integer-orderϑ=1 and fractional-order ϑ=0.98 in Figure 3. Figure 3 shows that the results of model (10) are similar for the integer-order and the Caputo fractional order, and a small change in the order of derivation has no effect on the overall trend of the results in terms of ascending and descending, but the resulting numerical values are different.Figure 3
Plots of the results of model (10) for infected groups with integer-order ϑ=1 and fractional-order ϑ=0.98 in the case R0>1.


## 5.3. The Reproduction Number Sensitivity Analysis

We investigate the effect of parameters in Crimean-Congo hemorrhagic fever fractional model (10) on reproduction number using the method introduced by [40]. For this simulation, we use the parameters in the first case of the previous subsection. Since R0 is defined as R0=maxRh,Rkl, therefore, we analyze the sensitivity of R0 in two cases.First, ifR0=Rh, by the mentioned method, we have Sβ6=∂R0/∂β6β6/R0=1>0,SΛh=∂R0/∂ΛhΛh/R0=1>0,Sdh=∂R0/∂dhdh/R0=−2β6Λh/dh3=−1.55<0. Figure 4 shows the sensitivity of R0 with respect to each of the parameters. As you can see, changing each of the parameters of model (10) that is involved in Rh changes the reproduction number. The reproduction number is directly related to parameters β6,Λh and inversely related to parameter dh. From an epidemiological point of view, whenever the reproduction number decreases, then the spread of the disease is controlled. Given that β6 has the most positive effect on the R0, so to control the spread of the disease, β6 should be reduced through the reduced communication of infected and susceptible humans.Figure 4
The graphs show the effect of model parameters on the reproduction number for the caseR0=Rh.In the latter case ifR0=Rkl, we obtain the same equations as above(55)SΛk=∂R0∂ΛkΛkR0=0.724>0,Sβ1=∂R0∂β1β1R0=0.4486>0,Sdl=∂R0∂dldlR0=−0.5513<0,Sβ2=∂R0∂β2β2R0=0.275>0,Sβ3=∂R0∂β3β3R0=0.276>0,Sdk=∂R0∂dkdkR0=−1.448<0.Figure5 shows the sensitivity of R0 with respect to each of the parameters. The reproduction number is directly related to parameters β1,β2,β3,andΛk and inversely related to parameter dl,dk. Among the mentioned parameters, parameters β1,β2, and β3 can be controlled, and all of which have a positive effect on causality, so to reduce the amount of reproduction number, it is enough to reduce the rate of disease transmission between ticks, animals, and humans.Figure 5
The graphs show the effect of model parameters on the reproduction number for the caseR0=Rkl.


## 6. Conclusion

In this work, we have presented a box model using the Caputo fractional-order derivative by taking into account the transmission of the Crimean-Congo hemorrhagic fever virus between ticks, animals (domestic and wild), and humans. We have calculated the feasible region and the equilibrium points of the system (10), and we have determined the necessary conditions for the stability of the equilibrium point. In the last section, using the Euler method for the Caputo fractional derivative, we have obtained the approximate solution of system (10), and then, we have provided a numerical simulation for the transmission of Crimean-Congo hemorrhagic fever virus. In two cases: R0<1 and R0>1, the results of the model have been plotted for the six groups in the model, which clearly show that in the case R0<1, the transmission of the disease stops after a while, and the results of the system converge to the disease-free equilibrium point. We have increased the rate of disease transmission among the groups, and in this case, the results for R0>1 show that the disease continues endemically, and also, the results converge to the endemic equilibrium point. The results of the model are compared with two types of derivatives of integer-order and fractional-order, and the result of comparison shows that changing the type of derivative with close order has no effect on the overall trend of the results but the obtained numerical values are different.Later, we have investigated the effect of each of the model parameters onR0, and the results show that the disease transmission rates among the groups have a positive effect on the value of R0; therefore, to control the spread of Crimean-Congo hemorrhagic fever, the disease transmission rate should be reduced by reducing contact between different groups.



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*Source: 1273405-2021-06-29.xml*