Modeling And Analysis Of An SVIRS Epidemic Model Involving External Sources Of Disease

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1 NNAONAL JOUNAL OF CHNOLOGY NHANCMN AND MGNG NGNNG ACH OL U 8 Modeling And Analysis Of An pidemic Model nvolving xternal ources Of Disease aid Kamel Naji Ahmed Ali Muhseen Department of mathematics College of science University of Baghdad. Baghdad -ra. mail: rnaji@scbaghdad.edu.i Assistant Lecturer Ministry of ducation usafa Baghdad-ra mail: aamuhseen@gmail.com. Abstract: n this paper a mathematical model that describes the flo of infectious disease in a population is proposed and studied. t is assumed that the disease divided the population into four classes: susceptible individuals vaccinated individuals infected individuals and recover individuals. he impact of immigrants vaccine and external sources of disease on the dynamics of epidemic model is investigated. he existence uniueness and boundedness of the solution of the model are discussed. he local and global stability of the model is studied. he occurrence of local bifurcation as ell as Hopf bifurcation in the model is investigated. Finally the global dynamics of the proposed model is studied numerically. Keyords: pidemic models tability accinated mmigrants external sources Local and Hopf bifurcation.. ntroduction Mathematical models have become important tools to study and analyze the spread and control of infectious disease. Most the proposed mathematical models those describe the transmission of infectious disease have been derived from the classical susceptible infective recover model hich is suggested originally by Kermac and Mcenderic []. n that model the susceptible individuals become infective by contact ith infected individuals and then the infected individuals may recover and transfer to removal individuals at a specific rate. Numbers of mathematical models ere developed to study and analyze the spread of infectious diseases in order to prevent or minimize the transmission of them through uarantine and other measures see for example [-5] and the references there in. On the other hand since the resistance against an infectious disease represents protection that reduces an individual s ris of contracting the disease therefore many epidemiological models involving vaccination have been proposed and studied see foe example [6-8] and the reference there in. Keeping the above in vie there are many infectious diseases spread ithin the population by direct contact beteen susceptible and infective individuals they may spread through external sources in the environment such as air ater insects etc. herefore recently Das et al. [9] have been proposed and studied a mathematical model consisting of eco-epidemiological model involving external sources of disease. n this paper e proposed and studied a mathematical model consisting of epidemic model involving immigrant individuals some of them may arrive infected ith the disease and vaccine in hich it is assumed that the disease transmitted by contact as ell as external sources in the environment.. he mathematical model: Consider a simple epidemiological model in hich the total population say Nt at time t is divided in to three sub classes the susceptible individuals t infected individuals t and recover individuals t. uch model can be represented as follos: d. Here is the recruitment rate of the population is the natural death rate of the population is the infected rate incidence rate of the susceptible individuals due to direct contact ith the infected individuals and is the natural recovery rate of the infected individuals. No since there are many infectious diseases Alanfelonzha bird s Anfelonzha and typhoid etc. spread in the environment by different factors including insects contact or other vectors therefore e assumed that the disease in the above model ill transmitted beteen the population individuals by contact as ell as external sources of disease in the environment ith an external incidence rate. Also it is assumed that the lifetime of removal individual s immunity may not continue forever and then the removal individuals return to be susceptible class ith a constant rate also non as losing removal individual s immunity rate. Further there is a constant flo say A of a ne members arriving into the population ith the fraction p of A arriving infected p. hen the above system. can be reritten in the form: d p A pa. Keeping the above in vie in order to study the effect of vaccination on the system. let t represented the vaccinated individuals in the population at time t and then the folloing assumptions are made: Copyright J.

2 NNAONAL JOUNAL OF CHNOLOGY NHANCMN AND MGNG NGNNG ACH OL U 9 he susceptible class is vaccinated at per capita rate. he infection can invade the susceptible class or vaccinated class depending on vaccine efficiency. he vaccine reduces the possibility of infection by a factor of hich is non as intensity vaccine immunity rate here. he vaccine may not give a permanent immunity for susceptible individuals so the vaccine may disappear and then the individuals loss the immunity ith rate. Accordingly the flo of disease in system. along ith the above assumptions can be representing in the folloing bloc diagram: Proof: Let t t t t be any solution of the system. ith non-negative initial condition since Nt= t+t+t+t then : hich gives dn d d dn N A No by solving the above linear differential euation e get that the total population is asymptotically constant by: A N t Hence all the solutions of system. that initiate in are confined in the region: Fig.. Bloc diagram of system.. herefore system can be modified to: d p A d pa. Clearly for the vaccine is completely affective. hile stand for the situation here the vaccine is totally ineffective. On the other hand denotes to the case hen immunity is life-long hile corresponds to the case here there is absolutely no vaccine induced immunity. herefore at any point of time t the total number of population becomes N t t t t. Obviously due to the biological meaning of the variables t t t and t system. has the domain: hich is positively invariant for system.. Clearly the interaction functions on the right hand side of system. are continuously differentiable. n fact they are Liptschizan function on. herefore the solution of system. exists and is uniue. Further all solutions of the system. ith non-negative initial conditions are uniformly bounded as shon in the folloing theorem. heorem.: All the solutions of system. hich are initiate in are uniformly bounded. A : N ; hich is complete the proof.. xistence of uilibrium points of system. n this section the existence of all possible euilibrium points of system. is discussed. Clearly if then the system. has an euilibrium point called a disease free euilibrium point and denoted by here: A. A Hoever if then the system. has an endemic euilibrium point denoted by here and represent the positive solution of the folloing set of euations: p A. pa traightforard computation to solve the above system of euations gives that: Copyright J.

3 NNAONAL JOUNAL OF CHNOLOGY NHANCMN AND MGNG NGNNG ACH OL U Copyright J. ] [ ] [ ] [ ] [ ] [ A p A p. hile is a positive root for the folloing third order euation: D D D D. here: ]] [ [ [ ] [ ] [ ] [ A p pa pa D pa A pa pa D A D D Clearly euation. has a uniue positive root given by and then exists uniuely in nt. if and only if at least one of the folloing to conditions hold. ] [ ] [ A.5a ] [ ] [ pa A pa.5b. Local stability analysis of system. n this section the local stability analysis of the euilibrium points and of the system. is studied as shon in the folloing theorems. heorem.: he disease free euilibrium point of system. is locally asymptotically stable if the folloing sufficient condition is satisfied:. Proof: he Jacobian matrix of system. at can be ritten as: a ij J Clearly J has the folloing eigenvalues: here represents the eigenvalue in - direction. Obviously and have negative real parts hile. herefore is locally asymptotically stable if and only if the eigenvalue hich is satisfied provided that condition. holds and hence the proof is complete. heorem.: Assume that the endemic euilibrium point of system. exists in the nt.. hen it is locally asymptotically stable if the folloing condition is satisfied:. Proof: he Jacobian matrix of system. at the endemic euilibrium point that denoted by J can be ritten: b ij No according to Gersgorin theorem if the folloing condition holds: j i i ii b ij b hen all eigenvalues of J exists in the region: : j i i b ii b ij U C U herefore according to the given condition. all the eigenvalues of J exists in the left half plane and hence is locally asymptotically stable.

4 NNAONAL JOUNAL OF CHNOLOGY NHANCMN AND MGNG NGNNG ACH OL U Copyright J. 5. Global stability analysis of system. n this section the global dynamics of system. is studied ith the help of Lyapunov function as shon in the folloing theorems. heorem 5.: Assume that the disease free euilibrium point of system. is locally asymptotically stable. hen the basin of attraction of say B it is globally asymptotically stable if satisfy the folloing conditions: 5.a pa 5.b Proof: Consider the folloing positive definite function: ln ln Clearly : is a continuously differentiable function such that and. Further e have: d d d By simplifying this euation e get: pa d herefore according to condition 5.a it is obtain that: pa d Obviously d for every initial points satisfying condition 5.b and then is a Lyapunov function provided that conditions 5.a-5.b hold. hus is globally asymptotically stable in the interior of B hich means that B is the basin of attraction and that complete the proof. heorem 5.: Let the endemic euilibrium point of system. is locally asymptotically stable. hen it is globally asymptotically stable provided that: 5.a 9 5.b 5.c 5.d 5.e 5.h Proof: Consider the folloing positive definite function: Clearly : is a continuously differentiable function such that and Further e have: d d d By simplifying this euation e get: d ith

5 NNAONAL JOUNAL OF CHNOLOGY NHANCMN AND MGNG NGNNG ACH OL U Copyright J. herefore according to the conditions 5.a-5.h e obtain that: d Clearly d and then is a Lyapunov function provided that the given conditions hold. herefore is globally asymptotically stable. 6. he local bifurcation analysis of system. n this section the occurrence of local bifurcations such as saddle-node transcritical and pitchfor near the euilibrium points of system. is studied in the folloing theorem. heorem 6.: ystem. has a transcritical bifurcation near the disease free euilibrium point but neither saddle-node bifurcation nor pitchfor bifurcation can accrue at the parameter 6. Proof: t is easy to verify that the Jacobian matrix of system. at can be ritten as: Df J here a a a. Clearly the third eigenvalue in -direction is zero further the eigenvector say K corresponding to satisfy the folloing: K K J then K J From hich e get that: 6.a 6.b 6.c o by solving the above system of euations e get: ; ; z y x here: z y x Here be any non zero real number. hus z y x K imilarly the eigenvector that corresponding to of J can be ritten: J his gives: Here is any non-zero real number. No rerite system. in a vector form as: X f dx here X and f f f f f ith i f i are given in system. and then determine f d df e get that: f then f herefore: f

6 NNAONAL JOUNAL OF CHNOLOGY NHANCMN AND MGNG NGNNG ACH OL U Conseuently according to otomayor theorem [] the system. has no saddle-node bifurcation near at. No in order to investigate the accruing of other types of bifurcation the derivative of f ith respect to vector X say Df is computed Df o Df K Again according to otomayor theorem if in addition to the above the folloing holds D f K K here Df is the Jacobian matrix at and then the system. possesses a transcritical bifurcation but no pitch-for bifurcation can occur. No since e have that: x D f K K y x y herefore: D f K K x y hen the system. has a transcritical bifurcation at hen the parameter passes through the bifurcation value. 7. epidemic model ithout vaccination Consider the proposed system. in case of absence of vaccine that is hen the parameter then system. ill be reduced to the folloing subsystem. d p A g pa g 7. g Clearly system 7. represents a simple epidemic model involving disease transmitted by contact and by external sources in the environment ith partially infective immigrant s individuals. Obviously the above subsystem is uniformly bounded and has to non negative euilibrium points: First the disease free euilibrium point that denoted by hich alays exists here: A 7. econd the endemic euilibrium point of system 7. that denoted by hich exists uniuely in the nt. here: here: p A 7.a 7.b D D D D 7.c D D D D D pa p A he local stability analysis of the above euilibrium points and of system 7. is studied as shon in the folloing theorems. heorem 7.: he disease free euilibrium point of system 7. is locally asymptotically stable provided that: 7.a hile it is a saddle point provided that: 7.b Proof: he Jacobian matrix of system 7. at can be ritten as: J Clearly J has the folloing eigenvalues: ; ; 7.5 Copyright J.

7 NNAONAL JOUNAL OF CHNOLOGY NHANCMN AND MGNG NGNNG ACH OL U here represents the eigenvalue in - direction. Obviously and are negative and then is locally asymptotically stable if and only if the eigenvalue hich is satisfied provided that condition 7.a holds hile it is saddle point provided that condition 7.b holds and hence the proof is complete. heorem 7.: he endemic euilibrium point of system 7. is locally asymptotically stable provided that: 7.6a 7.6b Proof: he Jacobian matrix of system 7. at the endemic euilibrium point can be ritten: J d ij hen the characteristic euation of J is given by: here: 7.7 d d d dd dd dd dd ddd d dd dd [ ] Further: dd d d dd d d dd d d dd d d ddd No according to outh-hurtiz criterion ill be locally asymptotically stable provided that ; and. Clearly: ; and provided that conditions 7.6a-7.6b are hold. Hence the proof is complete. he global dynamics of system 7. is studied ith the help of Lyapunov function as shon in the folloing theorems. heorem 7.: Assume that the disease free euilibrium point of system 7. is locally asymptotically stable. hen the basin of attraction of say B satisfy the folloing condition: pa 7.8 Proof: Consider the folloing positive definite function: F df L F Clearly L : is a continuously differentiable function such that L and L. Further e have: dl d By simplifying this euation e get: dl pa dl Obviously for every initial point satisfying condition 7.8 and then L is a Lyapunov function provided that condition 7.8 holds. hus is globally asymptotically stable in the interior of B hich means that B is the basin of attraction and this complete the proof. heorem 7.: Let the endemic euilibrium point of system 7. is locally asymptotically stable. hen it is globally asymptotically stable provided that the folloing conditions are satisfied: 7.9a 7.9b 7.9c Proof: Consider the folloing positive definite function: L Copyright J.

8 NNAONAL JOUNAL OF CHNOLOGY NHANCMN AND MGNG NGNNG ACH OL U 5 Clearly L : is a continuously differentiable function such that L and L. Further e have: dl d By simplifying this euation e get: dl ith ; ; ; ; ; herefore according to the conditions 7.9a-7.9c e obtain that: dl dl Clearly and then L is a Lyapunov function provided that the given conditions 7.9a-7.9c hold. herefore is globally asymptotically stable and hence the proof is complete. he occurrence of local bifurcations such as saddle-nod transcritical and pitchfor near the euilibrium point of system 7. is studied in the folloing theorem. heorem 7.5: ystem 7. has a transcritical bifurcation near the disease free euilibrium point but neither saddle-node bifurcation nor pitchfor bifurcation can accrue at the parameter 7. Proof: t is easy to chec that the Jacobian matrix of system 7. at can be ritten as: J J Clearly the second eigenvalue in -direction is zero hile and those are given in euation 7.7 are negative. Further the eigenvector say K corresponding to can be ritten as: z K y here z ; y be any non zero real number ith z and y. imilarly the eigenvector corresponding to of J can be ritten: Here is any non-zero real number. No rerite system 7. in a vector form as: dx gx here X and g g g g ith g i i are given in system 7. and then determine dg g e get that: d g Conseuently according to otomayor theorem [] the system has no saddle-node bifurcation near at. No in order to investigate the accruing of other types of bifurcation the derivative of g ith respect to vector X say Dg is computed and then e obtain that: Dg K Moreover since D g K K z Copyright J.

9 NNAONAL JOUNAL OF CHNOLOGY NHANCMN AND MGNG NGNNG ACH OL U 6 hen the system 7. has a trnscritical bifurcation but not pitch-for bifurcation at hen the parameter passes through the bifurcation value. he occurrence of Hopfbifurcation near the endemic euilibrium point of system 7. is also studied. Not that it is ell non that the necessary conditions of the three dimensional dynamical system 7. to have a Hopf bifurcation around at a specific parameter value say are given by and here and represent the coefficients of the characteristic euation of the dynamical system 7.. No since the conditions that guarantee the positivity of and are the same conditions that guarantee the positivity of. Hence there is no possibility of occurrence of Hopf bifurcation. Fig. ime series of the solution of system.. a trajectories of b trajectories of c trajectories of and d trajectories of. the solid line refers to the trajectory started at 59 hile the dotted line refers to the trajectory started at Numerical analysis of systems. and 7.: n this section the global dynamics of systems. and 7. is studied numerically. he objectives of this study are confirming our obtained analytical results and understand the effects of immigration existence of vaccine and existence of the external sources for disease on the dynamics of and epidemic models. Conseuently first system. is solved numerically for different sets of initial conditions and for different sets of parameters. t is observed that for the folloing set of hypothetical parameters that satisfies stability condition. of disease free euilibrium point system. has a globally asymptotically stable disease free euilibrium point as shon in folloing figure. A p Fig. ime series of the solution of system.. a trajectories of b trajectories of c trajectories of and d trajectories of. he solid line refers to the trajectory started at 555 hile dotted line refers to trajectory started at 559. Fig. ime series of the solution of system.. a for. b for. 5 c for. Copyright J.

10 NNAONAL JOUNAL OF CHNOLOGY NHANCMN AND MGNG NGNNG ACH OL U 7 Fig. 6 ime series of the solution of system.. a for. b for. c for. Fig. 5 ime series of the solution of system.. a for. b for. c for. 9. Copyright J.

11 NNAONAL JOUNAL OF CHNOLOGY NHANCMN AND MGNG NGNNG ACH OL U 8 Fig. 7 ime series of the solution of system.. a for. b for. c for. 6. Fig. 9 ime series of the solution of system. for the data given by 7. ith. and varying. a for. 5 b for.. n the folloing the global dynamics of system 7. is carried out. ystem 7. is solved numerically for the folloing set of parameters hich satisfies condition 7.a and then the trajectories are dran in Fig.. A p Fig. 8 ime series of the solution of system. for the data given by 7. ith varying. a for. 5 b for.. Fig. ime series of the solution of system 7.. a trajectories of b trajectories of and c trajectories of the solid starting at 5 and dotted starting at 55. Copyright J.

12 NNAONAL JOUNAL OF CHNOLOGY NHANCMN AND MGNG NGNNG ACH OL U 9 approaches to for. c he solution approaches to for.5. Fig. Phase plot of system 7. starting from three different initial points. Fig. Phase plot of system 7.. a he solution approaches to for. b he solution approaches to for. c he solution approaches to 5 for.9. Fig. Phase plot of system 7.. a he solution approaches to 5 for b he solution n Fig. shos that the solution of system. approaches asymptotically to the disease free euilibrium point has a globally asymptotically stable disease free starting from to different initial points and this is confirming our obtained analytical results. Fig. shos clearly the convergence of system. to the endemic euilibrium point asymptotically from to different initial points. his is indicates to occurrence of a transcritical bifurcation near the disease free euilibrium point at a specific value of.5. so became unstable and the solution of system. Copyright J.

13 NNAONAL JOUNAL OF CHNOLOGY NHANCMN AND MGNG NGNNG ACH OL U approaches to. n addition to that the above to figures refer to that increasing the contact rate beteen and causes destabilizing to disease free euilibrium point and the system approaches instead to the endemic point. Note that in the above figures -9 e ill use the folloing representations: olid line for describing trajectory of ; dashed line for describing trajectory of ; dash dot line for describing trajectory of ; dotted line for describing trajectory of and starting at 5 5. n Fig. as the incidence rate of disease resulting from external sources increases through increasing the disease free euilibrium point of system. becomes unstable point and the trajectory of system. approaches asymptotically to the endemic euilibrium point. n fact as increases it is observed that the number of susceptible and vaccinated individuals decrease and the number of recover and infected individuals increases. Fig. 5 it is clear that as the rate of vaccine coverage increases the endemic euilibrium point of system. becomes unstable point and the trajectory of the system approaches asymptotically to the disease free euilibrium point attendant that increasing in vaccined individuals and decreasing in susceptible individuals. Fig. 6 the increasing that is decreasing the lifetime of vaccine immunity destabilizes the disease free euilibrium point and then the solution of system. approaches to endemic euilibrium point attended that increasing in the susceptible infected and recover individuals hile the number of vaccinated individuals decreases. From Fig. 7 that as the recovery rate increases from. to.6 the endemic euilibrium point of system. becomes unstable point and the trajectory of system. approaches asymptotically to the disease free euilibrium point. But the number of susceptible and vaccinated individuals increases hile the number of the infected and recover individuals decreases. n Fig. 8 hoever increases the parameter more than. eeping other parameters fixed as in 7. ith. causes transferring in the stability of system. from endemic euilibrium point to disease free euilibrium point as shon in Fig. 9. herefore the death rate due to the disease plays a vital role as bifurcation parameter of system.. Fig. hos that the solution of system 7. approaches asymptotically to the disease free euilibrium point 5 from to different initial data. Fig. shos the existence of a uniue endemic euilibrium point of system 7. hich is globally asymptotically stable. Fig. the external incidence rate increases the disease free euilibrium point of system 7. becomes unstable point and the trajectory of system 7. approaches asymptotically to the endemic euilibrium point and then the number of susceptible individuals decrease hile the number of the infected and recover individuals increases. Fig. the recovery rate increases the endemic euilibrium point of system 7. becomes unstable point and the trajectory of system 7. approaches asymptotically to the disease free euilibrium point attendant that increasing the number of susceptible individuals and decreasing in the numbers of the infected and recover individuals. 9. Conclusion and discussion: n this paper to mathematical models have been proposed and analyzed. he objective is to study the effect of immigrants existence and nonexistence vaccine and then existence of external sources of the disease in the environment on the dynamical behavior of and epidemic models. he existence and the stability analysis of all possible euilibrium points are studied analytically as ell as numerically. t is observed that system. and system 7. have transcritical bifurcation near the disease free euilibrium point but neither saddle node nor pitchfor bifurcation can accrue. Further both the systems. and 7. do not have Hopf bifurcation near the endemic euilibrium point. Finally according to the numerically simulation the folloing results are obtained:. Both the systems. and 7. do not have periodic dynamic instead it they approach either to the disease free euilibrium point or else to endemic euilibrium point.. As the number of the infected immigrant individuals and the incidence rate of disease external incidence rate or contact incidence rate increase the asymptotic behavior of the systems. and 7. transfer from approaching to disease free euilibrium point to the endemic euilibrium point.. As the lifetime of vaccine immunity decreases the losing vaccine immunity rate increases then the disease free euilibrium point of system. becomes unstable and the solution ill approaches to the endemic euilibrium point. Further similar result is obtained in systems. and 7. hen the natural death rate decreases.. As the recovery rates in the systems. and 7. increase then the solution in both the systems ill be transfer from stability at endemic euilibrium point to stability at disease free euilibrium point. Further similar result is obtained in case of system. hen the vaccine coverage rate increases. 5. Finally changing the lifetime of removal individual's immunity in both the system. and 7. do not have vital effect on the dynamical behavior of each of them. eferences: [].O. Kermac A.G. Mcendric A contribution to the mathematical theory of epidemics Proc.. oc. London a [].M. Anderson. M. May Population Biology of nfectious Disease pringer erlag Ne Yor 98. [] F. Brauer C. Castillo-Chavez Mathematical Models in Population Biology and pidemiology pringer Ne Yor. [] O. Diemann J. A. P. Heesterbee Mathematical pidemiology of nfectious Disease: Model Building Analysis and nterpretation iley Ne Yor. Copyright J.

14 NNAONAL JOUNAL OF CHNOLOGY NHANCMN AND MGNG NGNNG ACH OL U [5] H.. Hethcote he mathematics of infectious disease AM ev [6] Kribs-Zaleta C.M. elasco-hern andez J.X.. A simple vaccination model ith multiple endemic states. Math. Biosci. 6: 8-. [7] Alexander M.. Boman C. Moghadas.M. ummers. Gumel A.B. ahai B.M. A vaccination model for transmission dynamics of influenza. AM J. Appl. Dyn. yst. :5-5. [8] huro. hafee he effect of treatment immigrants and vaccinated on the dynamic of epidemic model. M.c. thesis. Department of Mathematics College of cience University of Baghdad. Baghdad ra. [9] Krishna pada Das hovonlal oy and J. Chattopadhyay Biosystem [] otomayor J. Generic bifurcations of dynamical systems in dynamical systems M. M. Peixoto Ne Yor academic press 97. Copyright J.