A New Susceptible-Infectious (SI) Model With Endemic Equilibrium
Ayse Peker-Dobie, Semra Ahmetolan, Ayse Humeyra Bilge, Ali Demirci
aa r X i v : . [ q - b i o . P E ] M a y A New Susceptible-Infectious (SI) Model With Endemic Equilibrium
Ayse Peker Dobie ∗ , Semra Ahmetolan † , Ayse Humeyra Bilge ‡ and Ali Demirci § Department of Mathematics, Faculty of Science and Letters, Istanbul Technical University, Istanbul,Turkey TDepartment of Industrial Engineering, Faculty of Engineering and Natural Sciences, Kadir HasUniversity, Istanbul, Turkey
June 1, 2020
Abstract
The focus of this article is on the dynamics of a new susceptible-infected model which consistsof a susceptible group ( S ) and two different infectious groups ( I and I ). Once infected, anindividual becomes a member of one of these infectious groups which have different clinical formsof infection. In addition, during the progress of the illness, an infected individual in group I maypass to the infectious group I which has a higher mortality rate. In this study, positiveness ofthe solutions for the model is proved. Stability analysis of species extinction, I -free equilibriumand endemic equilibrium as well as disease-free equilibrium is studied. Relation between thebasic reproduction number of the disease and the basic reproduction number of each infectiousstage is examined. The model is investigated under a specific condition and its exact solutionis obtained. Keywords: epidemic models; endemic equilibrium; disease free equilibrium; extinction;reproduction number, infectious diseases; dynamical systems; stability.
One major contribution of mathematics to epidemiology is the compartmental model introducedby Kermack and McKendrick in 1927 [1]. Since then, significant progress has been achieved andnumerous mathematical models have been developed to study various diseases [2, 3]. Major out-breaks such as SARS epidemic in 2002 [4, 5, 6], H5N1 influenza epidemic in 2005 [7, 8, 9], H1N1influenza pandemic in 2009 [10, 11, 12], Ebola in 2014 [13, 14, 15], and nowadays Covid-19 pan-demic [16, 17, 18] maintain the high interest in mathematical modelling and analysis of infectiousdiseases.The Susceptible-Infected-Removed (
SIR ), the Susceptible-Infected-Susceptible (
SIS ) and theSusceptible-Infected ( SI ) systems are the three fundamental epidemic models [19, 20]. In the SIR model, infected individuals gain permanent immunity after recovering from the disease whereas inthe
SIS model, they return to the susceptible group. On the other hand, in the SI model, infectedindividuals are lifetime infectious. Thus, the SI model consists of two different compartments;the susceptible group whose members are not yet infected by the pathogen and the infected groupwhose members are infected by the pathogen. Various infectious diseases such as AIDS caused byhuman immune deficiency virus (HIV) and Feline Infectious Peritonities (FIP) in cats caused by ∗ [email protected] Corresponding Author † [email protected] ‡ [email protected] § [email protected] eline Corona virus (FCoV) have more than one infectious stage. In this article, a new susceptible-infected model ( SI I ) which has two infectious stages is considered.The SI I model consists of three groups: susceptible population S , I -infected group and I -infected group. Each individual in the population is in one of these groups since individualsnever leave the infectious group once infected and each healthy newborn has no immunity. Oncesusceptible individuals are infected, they develop the disease in two different clinical forms, I and I . Each infected group contributes to its own infected group by transmitting the disease tosusceptible ones. In addition, I -infected individuals may develop the infection to a further stageand may become a member of the group I which has a higher mortality rate. Healthy newbornsare only seen in the susceptible population. The newborns from I and I -infected mothers aremembers of the same infected group as their mothers.The article consists of five sections. In Section 2, the mathematical model of the disease isdescribed. Theoretical results of the model about the basic reproduction number, disease freeequilibrium, endemic equilibrium, extinction, and the phase portraits are presented in Section 3.The relation between the basic reproduction number of each form of the infectious disease is alsogiven in this section. A model of the disease under specific conditions is presented in Section 4.Exact solutions of this reduced model are also obtained in this section. Concluding remarks andthe discussion of the results are given in Section 5. SI I Model
The susceptible-infected model considered in this work is described by the following system ofnonlinear ordinary differential equations S ′ = − β SI − β SI + f SI ′ = β SI − θI + f I I ′ = β SI + θI + f I (2.1)where f = µ − δ , f = µ − δ and f = µ − δ , and µ i and δ i are the birth and death ratesof each group. In this model denoted by SI I (figure 1), S represents the susceptible populationwhereas I and I are the two different groups of the population infected by the same virus butwhich develop the disease in different clinical forms. Each infected population comes into contactwith the susceptible population with different contact rates, β i . The parameter θ is the rate of theindividuals in the group I who become a member of the group I . The parameters β i , µ i , δ i and θ are positive and the nonconstant total population size N is equal to S + I + I . Note that it willbe necessary to choose θ − f > f < , otherwise I and I will be unbounded.The SI I model is based on the following assumptions: • Each individual in the population is in one of these three classes, S , I or I . • Infected individuals remain infective for the rest of their lives. • Newborns belong to the compartment of their mothers. • It is assumed that each infected group I i contributes to its own infected group by transmittingthe disease to susceptible individuals. This means that if individuals from the susceptible groupare infected by a member from the group I i they become a member of that infected group. • There is a flow from the group I to the group I since it is assumed that the members in thegroup I may develop the disease form of the group I . However, there is no flow from the group I to the group I . 2igure 1: Diagram of the SI I model. Parameters β i , µ i and δ i are the transmission, birth anddeath rates of the group I i , respectively whereas µ and δ are the birth and death rates of thegroup S . The parameter θ is the rate of the individuals in the group I who become a member ofthe group I . • The distinction between disease related and natural deaths is taken into account by differentmortality rates of each group.The SI I model admits solutions that are monotone, oscillatory or with decay in oscillatoryas shown in Figure 2. In this section, positiveness of the model is proved and the basic reproduction number is found.The disease free equilibrium at which the population remains in the absence of disease, and theendemic equilibrium are examined.
Proposition 1:
The susceptible group S and the infected groups I and I stay positive if theinitial conditions are chosen to be positive. Proof:
The first and the second equations in (2.1) can be expressed as follows S ′ /S = − β I − β I + f I ′ /I = β S − θ + f . Integration of the equations above gives S ( t ) = S exp (cid:18) Z t ( − β I − β I + f ) dτ (cid:19) I ( t ) = I , exp (cid:18) Z t ( β S − θ + f ) dτ (cid:19) (3.1)where S and I , are the initial conditions for the susceptible population S and the infected group I , respectively. As a consequence of the equations in (3.1), S ( t ) and I ( t ) remain positive if theinitial conditions are chosen to be positive.If the last equation in (2.1) is integrated, one finds I ( t ) = exp (cid:18) Z t ( β S + f ) dτ (cid:19)(cid:20) I , + Z t I ( τ ) exp (cid:18) − Z τ ( β S + f ) dσ (cid:19) dτ (cid:21) . (3.2)3 t S ,I ,I ( P e r c en t age s ) f =-0.1,f =0.9,f =1.5, β =2.5, β =1.5, θ =1.5 S I I t S ,I ,I ( P e r c en t age s ) f =0.5,f =0.5,f =1.1, β =1.5, β =1.7, θ =1.5 S I I t S ,I ,I ( P e r c en t age s ) f =1,f =0.5,f =1.1, β =1.3, β =2, θ =1.1 S I I Figure 2: Solutions of different types for the SI I model.The equation (3.2) yields that I ( t ) remains positive if the initial condition I , is chosen to bepositive. One of the key parameters in mathematical modelling of transmissible diseases is the basic repro-duction number ( R ) which is defined as the number of new infectious individuals produced by atypical infective individual in a fully susceptible population at a disease-free equilibrium. For mod-els with more than one infected compartments, the computation of R is based on the constructionof the next generation matrix whose ( i, j ) entry is the expected number of new infections in thecompartment i produced by the infected individual in the compartment j . For mutistrain models,the maximum of the eigenvalues of this matrix is the basic reproduction number [21, 22].For the SI I model, let I ∗ , = I ∗ , = 0 and S ∗ be the disease free equilibrium. If the system ofordinary differential equations in (2.1) is linearized about the disease free equilibrium one obtainsthe following linearized infection subsystem I ′ = β S ∗ I − θI + f I I ′ = β S ∗ I + θI + f I . (3.3)The matrices F = (cid:18) β S ∗ β S ∗ (cid:19) and V = (cid:18) − θ + f θ f (cid:19) are defined by using (3.3) suchthat K ′ = ( F + V ) K where K = (cid:18) I I (cid:19) . Note that the i th component of the matrix F K is therate of appearance of new infected individuals in the group I i whereas the i th component of thematrix V K is the rate of transfer of the individuals into and out of the group I i . Then the next4eneration matrix is used to derive the basic reproduction number L = − F · V − = β S ∗ θ − f − β S ∗ θf ( θ − f ) − β S ∗ f (3.4)where L ij is the number of secondary infections caused in the compartment i by an infected indi-vidual in the compartment j [21, 22]. Thus the eigenvalues of the next generation matrix are foundto be λ = β S ∗ θ − f , λ = − β S ∗ f . (3.5)Then the basic reproduction number which is the number of secondary cases produced by a singleinfective individual introduced into a population is the largest eigenvalue of the matrix L ; that is, R = max { λ , λ } . (3.6)Here, λ and λ are the basic reproduction numbers for each strain corresponding to the groups I and I , respectively. For the rest of the article, in order to distinguish the basic reproductionnumbers for each strain, λ and λ will be denoted by R and R , respectively. Proposition 2:
The disease free equilibrium is locally asymptotically stable if the following con-ditions are satisfied f < β S ∗ < θ − f β S ∗ < − f . Proof:
To determine the stability of the disease free equilibrium, the Jacobian matrix of thenonlinear system (2.1) is considered J = − β I ∗ − β I ∗ + f − β S ∗ − β S ∗ β I ∗ β S ∗ − θ + f β I ∗ θ β S ∗ + f . (3.7)The eigenvalues of the Jacobian matrix at the disease free equilibrium ( S ∗ , I ∗ , I ∗ ) = ( S ∗ , , λ ∗ = f , λ ∗ = β S ∗ − θ + f and λ ∗ = β S ∗ + f . Since all the eigenvalues are negative, thedisease free equilibrium is locally asymptotically stable.To express the conditions for asymptotic stability at the disease free equilibrium in terms of thebasic production number, one can express λ ∗ = ( θ − f )( R −
1) and λ ∗ = − f ( R − • Case 1: If R < R , then R = R by (3.6). Thus, if R < R = R <
1, it means that λ ∗ and hence λ ∗ are negative. • Case 2: Similarly, if R < R , then R = R . Thus, if R < R = R <
1, it means that λ ∗ and hence λ ∗ are negative.Since f is negative, the analysis of these two cases shows that the disease free equilibriumis unstable if R >
1; that is, the invasion of the disease is always possible. The disease freeequilibrium is locally asymptotically stable if R <
1. In other words, if R < R < R <
1, then solutions with initialvalues close to the disease free equilibrium remain close to this equilibrium and approach to thisequilibrium as t approaches infinity. 5 .4 Endemic Equilibrium Proposition 3:
The following statements hold for the system defined by the equations in (2.1).(i) Species extinction equilibrium at the point ( S ∗ , I ∗ , I ∗ ) = (0 , ,
0) is locally asymptoticallystable if f < I -free equilibrium at the point ( S ∗ , I ∗ , I ∗ ) = (cid:18) − f β , , f β (cid:19) is stable if f > − β f <β ( θ − f ).(iii) Endemic equilibrium at the point ( S ∗ , I ∗ , I ∗ ) = (cid:18) θ − f β , f β (cid:18) − β θβ f − β f (cid:19) , θf β f − β f (cid:19) is locally asymptotically stable if f > − β f > β ( θ − f ) and β > β . This equilibrium isstable if f > − β f > β ( θ − f ) and β = β . Proof:
To find the equilibrium points, the right hand side of each equation in (2.1) is set as 0.If ( S ∗ , I ∗ , I ∗ ) denote the ordered triple, then the following three critical points are found A (0 , , , B (cid:18) − f β , , f β (cid:19) , C (cid:18) θ − f β , f β (cid:18) − β θβ f − β f (cid:19) , θf β f − β f (cid:19) . (3.8)(i) The point A represents the extinction of population; that is, the final population size N iszero. If the Jacobian matrix in (3.7) is evaluated at the point A , one finds the matrix f − θ + f θ f whose eigenvalues are λ ∗ = f , λ ∗ = f − θ and λ ∗ = f . Note that λ ∗ = − β S /R and λ ∗ = − β S /R .Since all the eigenvalues are real, and λ ∗ as well as λ ∗ are already negative, the stability at thisequilibrium point depends entirely on the sign of the coefficient f . Therefore, the equilibrium at A which will lead to the species extinction is locally asymptotically stable if f is negative; that is,the death rate is greater than the birth rate in the susceptible population. This case is illustratedon the top left panel in Figure 2.(ii) The point B gives I -free equilibrium. Certainly, this equilibrium exists only if the com-ponents S ∗ and I ∗ of B are strictly positive; that is, f >
0. In other words, the births must begreater than the deaths in the susceptible population for the existence of I -free equilibrium.If the Jacobian matrix in (3.7) is evaluated at the point B , one finds the matrix β f /β f − ( β f /β ) − θ + f f θ whose eigenvalues are λ ∗ = − β f β − θ + f and λ ∗ , = ∓√− f f i .If two of the eigenvalues are pure imaginary complex conjugate numbers, then λ ∗ must benegative in order that the linear system is stable; that is, − β f < β ( θ − f ).To express the conditions for stability at the I -free equilibrium in terms of the basic productionnumber, one can express λ ∗ = β S (cid:18) R − R (cid:19) and λ ∗ , = ∓ r f β S R i . Thus, if R = R , thelinear system is stable at the point B .Stability at the I -free equilibrium is illustrated on the top right panel in Figure 2. Suitableparameters are chosen for such an equilibrium at which group I has gone extinct with time, whereas6 = f + Β Β ∆ f > Β >Β Θ= f Β Β ∆ - Β Β ∆ f Θ Figure 3: The shaded region gives ( f , θ ) pairs for which the endemic equilibrium at the point C isalways locally asymptotically stable if f > β > β . S and I are stable over time. Therefore, as can be observed in this figure, the groups S and I exhibit periodic behaviour.(iii) The point C represents endemic equilibrium, the existence of which requires f > − β f > β ( θ − f ).The determinant of the matrix J − λI at the point C is∆ = − λ ∗ + ( β S ∗ + f ) λ ∗ + S ∗ ( − β I ∗ − β I ∗ ) λ ∗ + β I ∗ S ∗ ( β β S ∗ − β S − β θ ) . (3.9) t S ,I ,I ( P e r c en t age s ) f =1,f =0.5,f =1.5, β =1.5, β =2, θ =1.5 S I I t S ,I ,I ( P e r c en t age s ) f =1,f =0.5,f =1.5, β =1.5, β =1.5, θ =1.5 S I I Figure 4: Asymptotic stability and stability at the endemic equilibrium on the left and right panels,respectively.Rather than finding the roots of this cubic equation in (3.9), the Routh-Hurwitz criterion isused to determine the character of the roots. This criterion states that all the roots of the cubicequation λ + aλ + bλ + c = 0 will have negative real parts if and only if a > c > ab > c . Itfollows immediately that the necessary and sufficient condition for the eigenvalues to have negativereal parts is β > β and − β f > β ( θ − f ). Therefore, the endemic equilibrium at the point C islocally asymptotically stable when f > β > β and − β f > β ( θ − f ). The shaded region inFigure 3 shows the ( f , θ ) pairs for which the endemic equilibrium is always locally asymptoticallystable.However, if two of the eigenvalues of the Jacobian matrix in (3.7) at the point C are pureimaginary complex conjugate and one is negative real, then the linear system at the endemicequilibrium is stable. To examine the conditions for stability, the eigenvalues at C are defined as λ ∗ = − a and λ ∗ , = ∓ bi where a, b >
0. Then, by equating the characteristic equation λ ∗ +7 λ ∗ + b λ ∗ + ab = 0 to the equation in (3.9) and using the restrictions on a , b and c , one finds − β f > β ( θ − f ) and β = β .The conditions for asymptotic stability and stability at the endemic equilibrium in terms of thebasic production number are as follows: • If f > R = R and β > β , then the endemic equilibrium is asymptotically stable. • If f > R = R and β = β , then the endemic equilibrium is stable.To demonstrate theoretical results for asymptotic stability at the endemic equilibrium, solutionof the system for suitable parameters is given on the left panel in Figure 4. The phase portraitcorresponding to this solution and its projection curves on SI and SI planes are given in Figure5 and 6, respectively. These figures show that such an equilibrium is asymptotically stable. Inaddition, phase portraits for the same parameter values but with different initial conditions aregiven in Figure 9. It is observed that different initial values do not change the asymptotic stabilitybut they affect the amplitude of solutions.To illustrate the theoretical results for stability at the endemic equilibrium, solution of thesystem for suitable parameters is given on the right panel in Figure 4. The phase portrait corre-sponding to this solution and its projection curves on SI and SI planes are given in Figures 7 and8, respectively. As can be seen in these figures, such an equilibrium is stable. Additionally, phaseportraits for the same parameter values but with different initial conditions are given in Figure 10.It is observed that different initial values do not change the stability but they affect the amplitudeof the solutions. In this section, a special case of the system defined by the relations in (2.1) is considered, and theexact solution of the new system is found.
The reduced system considered in this section is obtained when the system in (2.1) satisfies thefollowing conditions • the transmission rates are identical ( β = β ), • the birth and death rates of the susceptible population, I -infected group and I -infectedgroup are equal; that is, f = 0, f = 0 and f = 0.The corresponding system of nonlinear differential equations is defined by S ′ = − βS ( I + I ) I ′ = ( βS − θ ) I I ′ = βS I + θI (4.1)where S + I + I = N . To obtain the exact solution of this reduced case, a new variable T ′ = I + I (4.2)is defined. If S is considered as a function of T , one obtains the following by using the first equationin (4.1) and the equation in (4.2) together with the initial conditions S , I , , I , and T S = S exp( − β ( T − T )) . (4.3)8ubstituting (4.3) in the equation S + I + I = N gives I + I = N − S exp( − β ( T − T )) . (4.4)Substituting of (4.4) in (4.2) and then integrating the resulting equation yield T = T + 1 β ln (cid:20)(cid:18) − S N (cid:19) exp( βN t ) + S N (cid:21) . (4.5)In a similar manner, if I is also considered as a function of T , one gets the following by using thesecond equation in (4.1), the equations in (4.2) and (4.3)( I + I ) dI dT = ( βS exp( − β ( T − T )) − θ ) I . (4.6)Substituting of (4.4) in (4.6) and then integrating the resulting equation yield I = I , exp( − β ( T − T )) (cid:18) N exp( β ( T − T )) − S N − S (cid:19) − θβN . (4.7)If (4.7) is substituted in (4.4), one gets I = N − S exp( − β ( T − T )) − I , exp( − β ( T − T )) (cid:18) N exp( β ( T − T )) − S N − S (cid:19) − θβN . (4.8)If (4.5) is replaced in (4.3), (4.7) and (4.8), the exact solution of (4.1) is expressed as follows S = S exp ( − βN t )1 − ( S /N ) + ( S /N ) exp ( − βN t ) I = I , exp ( − θt )1 − ( S /N ) + ( S /N ) exp ( − βN t ) I = N − S exp ( − βN t ) + I , exp ( − θt )1 − ( S /N ) + ( S /N ) exp ( − βN t ) . (4.9)Graphs for the exact solution are given in Figure 11 for suitable parameters. As can be seen inFigure 11, the susceptible population disappears over time. A period of time after this disappear-ance, I -infected group also declines to zero. However, I -infected group survives. In this paper, a mathematical study describing a new susceptible-infected model is presented. The SI I epidemic model has two different infectious groups which have different clinical forms ofinfection. Members of one of the infectious group may become a member of the other infectiousgroup during the progress of the illness. It is assumed that the illness has no cure and thereforeindividuals who are infected will eventually die of the disease or some other unrelated cause.Initially, positiveness of the solutions of the SI I system is proved. In addition, the basic repro-duction number which has an important role in the epidemiology of a disease, and the equilibriumpoints are found. It is shown that the system may have three equilibrium points. Furthermore, thestability conditions of the equilibrium points are obtained in terms of the parameters. The phaseportraits for asymptotic stability and stability at the endemic equilibrium are given for suitableparameters and with different initial conditions.Stability analysis of the equilibrium points reveals the following aspects:91) The species will become extinct if the birth rate is smaller than the death rate in thesusceptible population in the presence of infection.(2) One form of the infection ( I ) may persist while the other form ( I ) dies out if the birth rateis greater than the death rate in the susceptible population, and if the basic reproduction numberof the system is equal to the basic reproduction number of the group I .(3) Endemic equilibrium may exist if the birth rate is greater than the death rate in the sus-ceptible population, and if the contact rate of I is greater than the contact rate of I , and if thebasic reproduction number of the system is equal to the basic reproduction number of the group I . A special case for specific parameters is investigated. The exact solution of this reduced systemis also obtained. Examination of this system reveals that I -infected group survives while othergroups disappear over time. ReferencesReferences [1] Kermack, WO, McKendrick AG. A contribution to the mathematical theory of epidemics.
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J MathBio. f =1,f =0.5,f =1.5, β =1.5, β =2, θ =1.5 S I I Figure 5: Phase portrait for asymptotic stability at the endemic equilibrium on SI I space.11 S I S I f =1,f =0.5,f =1.5, β =1.5, β =2, θ =1.5 Figure 6: Projection of the phase portrait for asymptotic stability at the endemic equilibrium on SI and SI plane. f =1,f =0.5,f =1.5, β =1.5, β =1.5, θ =1.5 S I I Figure 7: Phase portrait for stability at the endemic equilibrium on SI I space.12 S I S I f =1,f =0.5,f =1.5, β =1.5, β =1.5, θ =1.5 Figure 8: Projection of the phase portrait for stability at the endemic equilibrium on SI and SI plane. f =1,f =0.5,f =1.5, β =1.5, β =2, θ =1.5 I S I S(0)=0.7, I (0)=0.1, I (0)=0.2S(0)=0.7, I (0)=0.2, I (0)=0.1S(0)=0.2, I (0)=0.3, I (0)=0.5S(0)=0.2, I (0)=0.5, I (0)=0.3 Figure 9: Phase portrait for asymptotic stability at the endemic equilibrium for different initialconditions. 13 f =1,f =0.5,f =1.5, β =1.5, β =1.5, θ =1.5 I S I S(0)=0.7, I (0)=0.1, I (0)=0.2S(0)=0.7, I (0)=0.2, I (0)=0.1S(0)=0.2, I (0)=0.3, I (0)=0.5S(0)=0.2, I (0)=0.5, I (0)=0.3 Figure 10: Phase portrait for stability at the endemic equilibrium for different initial conditions. t S ,I ,I N =2, I (0)=10 -2 , I (0)=10 -3 , S(0)=N -I (0)-I (0), β =2.5, θ =1 S I I Figure 11: Graphs of S , I and I2