SQU Journal for Science, 2016, 21(2), 120-138 © 2016 Sultan Qaboos University 120 Global Dynamics and Sensitivity Analysis of a Vector-Host-Reservoir Model Ibrahim M. ELmojtaba1*, Santanu Biswas2 and Joydev Chattopadhyay2 1*Department of Mathematics and Statistics, College of Science, Sultan Qaboos University P.O. Box 36, PC 123, Al-Khoud, Muscat, Sultanate of Oman; 2Agricultural and Ecological Research Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700108, India. *Email: elmojtaba@squ.edu.om. ABSTRACT: The role of animal reservoir in the disease dynamics is not yet properly studied. In the present investigation a mathematical model of a vector-host-reservoir is proposed and analyzed to observe the global dynamics of the disease. We observe that the disease free equilibrium is globally asymptotically stable if the basic reproduction number ( 0 R ) is less than unity whereas unique positive equilibrium is globally asymptotically stable if 0 > 1R and transcritical bifurcation occurs at 0 = 1R . Our numerical result suggests that the biting rate plays an important role for the propagation of the disease and the recovery rate has not such important contrib ution towards eradication of the disease. We also perform sensitivity analysis of the model parameters and the results suggest that the death rate of reservoir may be used as a control parameter to eradicate the disease. Keywords: Vector-host-reservoir model; Basic reproduction number; Lyapunov function; Bifurcation analysis; Numerical simulation; Sensitivity analysis. المستودع والناقل-ذج العائلنمووتحليل الحساسية ل شاملةة الكيالدينامي جويديف شاتوباديو ، سانتانو بيسواس*إبراهيم محمد المجتبى ية يكيعرف المستودع للمرض بأنه نوع حيواني يعتبر وجوده ضرورياً النتشار المرض. لم تتم دراسة دور المستودع الحيواني في دينام :مستخلص مجتمعات ، العائل ، ة مرض ينتشر بين ثالثة كياألمراض بعد بصورة جيدة ، و لدراسة هذا الدور فقد قمنا باقتراح نموذج رياضي يصف دينامي إلعادة اإلنتاج المستودع والناقل. لوحظ من خالل التحليل الرياضي للنموذج أن نقطة اإلتزان الخالية من الوباء مستقرة عالمياً إذا كان العدد األساسي (R0( أقل من وحدة واحدة بينما هنالك نقطة إتزان موجبة وهي مستقرة عالمياً إذا كان )R0 > 1 ويحدث ، )( تشعب أمامي عندR0 = 1). نتائج لقضاء على المحاكاة العددية أوضحت أن معدل العض يلعب دوراً أساسياً في إنتشار المرض بينما أن معدل التماثل للشفاء ليس له مساهمة ملموسة في ا دة اإلنتاج يتأثر بشدة بمعدل وفيات المستودع ، مما المرض. قمنا أيضاً بإجراء تحليل للحساسية لمعلمات النموذج ، والذي أظهر أن العدد األساسي إلعا يعني أنه يمكن إستخدامه كمعلمة تحكم للقضاء على المرض. .، المحاكاة العددية وتحليل الحساسية، العدد األساسي إلعادة اإلنتاج، دالة ليبونوف، تحليل التشعبالمستودع والناقل-نموذج العائل كلمات مفتاحية: 1. Introduction enerally, a disease reservoir is defined as a species that is essential for the persistence and transmission of the disease [1]. There are several types of reservoirs depending on their role in the life cycle of the pathogen, some of which are not necessarily for the maintenance of the disease but they can get infected by the pathogen and transmit it [2]. Several studies showed that Lyme disease has many reservoir hosts; Salkeld et al. [3] observed an apparent statewide association between squirrel infection prevalence and Lyme disease incidence, which suggests that squirrels are an important reservoir host responsible for maintaining this zoonotic disease regionally through U.S.A., also Craine et al. [4] showed that gray squirrels are major reservoirs for Lyme disease in U.K., and Richter et al. [5] proved that American Robins act as reservoir hosts for Lyme disease Spirochetes across U.S.A. Diniz et al. [6] showed that there are several potential reservoir hosts for Leishmaniasis such as domestic dog and hamsters; Dantas-Torres [7] also proved that dogs act as a reservoir for Leishmainasis, and Faiman et al. [8] found that voles and rodents also act as major reservoirs for Leishmaniasis in Israel; Quinnell et al. [9] discovered that wide range of wild and domestic animals play the role of reservoir for Leishmaniasis such as the crab-eating fox, Cerdocyon thous, opossums, Didelphis spp., domestic cats, Felis cattus, and black rat, Rattus rattus. G GLOBAL DYNAMICS AND SENSITIVITY ANALYSIS OF A VECTOR-HOST-RESERVOIR MODEL 121 Melaun et al. [10] showed that pets are suspected to be potential reservoirs for many viruses like Bwamba virus, Kaeng Khoi virus, Rift Valley fever virus, Toscana virus, Western equine encephalitis, Sindbis virus, Chikungunya virus, Ross River virus, the Eastern equine encephalitis virus, the Venezuelan equine encephalitis virus, Yellow fever, Japanese Encephalitis, West Nile fever, Dengue fever, St. Louis encephalitis, Zika virus and Tacaribe virus. Besides viruses, some parasites are known, which occur in bats and humans, and can be transmitted through hemorrhagic insects. The first one is the Chagas disease and the coccidian genus Plasmodium, which is the pathogenic agent of malaria. Quite a good number of studies have been carried out to observe the disease dynamics with different settings and assumptions (for example, see [11, 12, 13]). As far our knowledge goes, no research has been done to describe the dynamics of a general vector-host-reservoir model. Keeping this factor in mind, we propose and analyze a non-fatal vector borne disease with reservoir. The basic aim of the present investigation is to observe the disease dynamics and to suggest some control strategies for eradication of the disease. In any epidemic model, the basic reproduction number plays an important role; we like to suggest the control strategies by sensitivity analysis of the model parameters related to basic reproduction number. The article is organized as follows: an introduction is given in Section 1, the model is formulated in Section 2, the model is fully mathematically analyzed in Section 3, sensitivity analysis for the parameters of the model is carried out in Section 4, some numerical simulation is givn in Section 5 and the paper ends with a conclusion. 2. Model formulation and equations To formulate this model, we will follow a model built by Elmojtaba et al. [14] to describe the dynamics of visceral leishmaniasis, see also [14, 15, 16]. Consider the transmission of a non fatal disease between our three different populations, human host population, ( ) H N t , reservoir host population, ( ) R N t , and vector population, ( ) V N t . Human host population will be divided into three categories, susceptible individuals ( ) H S t , infected individuals ( ), H I t and those who are recovered and have permanent immunity, ( ) H R t . This implies that ( ) = ( ) ( ) ( ) H H H H N t S t I t R t  . Similarly, the reservoir host population will be divided into two categories, susceptible reservoirs, ( ) R S t , and infected reservoirs, ( ) R I t , such that ( ) = ( ) ( ) R R R N t S t I t and the vector population have two categories, susceptible vectors ( ) V S t , and infected vectors ( ) V I t , such that ( ) = ( ) ( ) V V V N t S t I t It is assumed that susceptible individuals are recruited into the population at a constant rate h  and acquire infection with following contacts with infected vectors at a per capita rate V H I ab N , where a is the per capita biting rate of vectors on humans (or reservoirs), and b is the transmission probability per bite per human (as the case for malaria, [17, 18]). Infected humans recover and acquire permanent immunity at an average rate  . There is a per capita natural mortality rate h  in all human sub-population. Susceptible reservoirs are recruited into the population at a constant rate r  , and acquire infection following contacts with infected vectors at a rate V H I ab N where a and b as described above. A per capita natural mortality rate r  occurs in the reservoir population. Susceptible vectors are recruited at a constant rate v  , and acquire infection following contacts with infected humans or infected reservoirs at an average rate equal to H R H R I I ac ac N N  , where a is the per capita biting rate, and c is the transmission probability for vector infection. Vectors suffer natural mortality at a per capita rate v  regardless of their infection status. From the description of the terms, we get the following system of differential equations: = ' H H h H V h H H S S N abI S N    = ( ) ' H H V h H H S I abI I N    = ' H H h H R I R  (1) IBRAHIM M. ELMOJTABA ET AL 122 = ' R R r R V r R R S S N abI S N    = ' R R V r R R S I abI I N  = ' H R V v V V V v V H R I I S N acS acS S N N     = ' H R V V V V v V H R I I I acS acS S I N N   Invariant region All parameters of the model are assumed to be non-negative, furthermore since model (1) monitors living populations, it is assumed that all the state variables are non-negative at time 0t  . This shows that the biologically-feasible region: 7 = {( , , , , , , ) : , , , , , , 0} H H H R R V V H H H R R V V S I R S I S I R S I R S I S I     is positively-invariant domain, and thus, the model is epidemiologically and mathematically well posed, and it is sufficient to consider the dynamics of the flow generated by (1) in this positively-invariant domain  . 3. Analysis of the model In this Section, we analyze system (1) to obtain the steady states of the system and their stability. We consider the equations for the proportions by first scaling the sub-populations for H N , R N and V N using the following set of new variables = , = , = , = , = , = ,VH H H R R h h h r r v H H H R R V SS I R S I s i r s i s N N N N N N and = V v V I i N ; and let = V H N m N be the vector-human ratio defined as the number of vector per human host (see similar definition in malaria models, in [19, 20]). Note that the ratio m is taken as a constant because it is well known (see [21] pages 218- 220) that a vector takes a fixed number of blood meals per unit time independent of the population density of the host. Similarly, we let = V R N n N be the vector-reservoir ratio defined as the number of vector per reservoir host. Differentiating with respect to time t we get: =' vh h h h hs abmi s s   = ( ) ' vh h h h i abmi s i   =' h h h h r i r  (2) = ' r r v r r rs abni s s   = ' r v r r ri abni s i = ( ) ' v v r v v vh s ac i i s s    = ( ) ' v r v v vh i ac i i s i  with the feasible region (i.e. where the model makes biological sense) 7 = {( , , , , , , ) : h h h r r v v s i r s i s i R    0 , , ; 1; 0 , ; 1; 0 , ; 1} h h h h h h r r r r v v v v s i r s i r s i s i s i s i          . It can be shown that the above region is positively invariant with respect to the system (2), where 7 R  denotes the non-negative cone of 7 R including its lower dimensional faces. 3.1 Basic Reproduction Number of the Model To calculate the basic reproduction number we will use the next generation approach [22, 23], define F as the column-vector of rates of the appearance of new infections in each compartment; =V V V    , where V  is the column-vector of rates of transfer of individuals into the particular compartment; and V  is the column-vector of rates of transfer of individuals out of the particular compartment. Hence, from our model we have GLOBAL DYNAMICS AND SENSITIVITY ANALYSIS OF A VECTOR-HOST-RESERVOIR MODEL 123 = ( ) v h v r h r v abmi s F abni s ac i i s          and ( ) = . h h r r v v i V i i               then the matrices F and V from the partial derivatives of F and V with respect to the infected classes computed at the DFE are given by 0 0 ( ) 0 0 0 0 0 0 = , = 0 0 0 h r v abm abn F and V ac ac                             Then the basic reproduction number, 0 R defined as the spectral radius of matrix 1 FV  ; 2 2 1 0 ( ) = ( ) = ( ) h r r v h a bcn a bcm FV             R 3.2 Local Stability analysis of the disease-free equilibrium 0 E The disease-free equilibrium (DFE) of the system (2) is given by 0 0 0 0 0 0 0 0 = ( , , , , , , ) = (1, 0, 0, 0,1, 0,1, 0) h h h r r v v E s i r s i s i . using Theorem 2 of van den Driessche and Watmough [23], we have the following lemma: Lemma 3.1. The disease-free equilibrium is locally asymptotically stable if 0 < 1R and unstable if 0 > 1R . 3.3 Global Stability analysis of the disease-free equilibrium 0 E The following theorem shows that the DFE is globally asymptotically stable if 0 < 1R . Theorem 3.1. The disease-free equilibrium point is globally asymptotically stable if 0 < 1R Proof: Consider the following Lyapunov function = ( ) ( )r r r vh h hL ac i ac i i         with derivative = [ (1 ) ( ) ] ' r h h v h h L ac abm i r i i      ( )[ (1 ) ]h r v r rac abn i i i      ( )[ ( )(1 ) ]r h h r v v vac i i i i        2 2 2 2 1 [ ( ) ( )] ( (1 ) ) r h r v h v r h h v a bcm a bcn i a bcm i i                      2 2 1 1 2 ( (1 ) ) ( )( ) r h h v r h h h a bcm p i ac i                      2 1 2 1 2 ( )( ) ( )( ) r h h h h h r v ac p a bcn i i                      1 2 1 2 ( )( ) ( )( ) r h h r r h h h ac i ac i                       1 2 1 2 ( )( ) ( )( ) r h h h r h h r ac p ac i                       1 2 1 2 ( )( ) ( )( ) r h h h v r h h h v ac i i ac p i                       1 2 ( )( ) r h h r v ac i i           2 1 2 0 = ( )( )( 1) r v h h v i           R 2 2 2 1 2 1 ( (1 ) ) ( (1 ) ) r h h v r h h v a bcm i i a bcm p i                     IBRAHIM M. ELMOJTABA ET AL 124 2 1 2 1 2 ( )( ) ( )( ) h h r v r h h h v a bcn i i ac i i                      1 2 1 2 ( )( ) ( )( ) r h h h v r h h r v ac p i ac i i                       2 1 2 0 ( )( )( 1) r v h h v i            R and hence 0 ' L  if 0 < 1R . We observe that our system has the maximum invariant set for = 0 ' L if and only if 0 1R holds and = = = = = 0 h h h v r i p r i i . By Lyapunov-LaSallés Theorem [20], all the trajectories starting in the feasible region where the solutions have biological meaning approach the positively invariant subset of the set where = 0 ' L , so that as t  , ( ) 1 h s t  , ( ) 1 r s t  , and ( ) 1. v s t  This shows that all solutions in the set where = = = = = 0 h h h v r i p r i i , go to the disease-free equilibrium 0 E . Thus, 0 < 1R is the necessary and sufficient condition for the disease to be eliminated from the community. 3.4 Existence of the endemic equilibrium 1 E In order to prove the existence of 1 E we equate the right hand sides of system (2) to zeros, and substitute = 1 h h h s i r  , = 1 r r s i and = 1 v v s i , to obtain (1 ) = ( ) v h h h h abmi i r i    (3) = h h h i r  (4) (1 ) = v r r r abni i i (5) ( )(1 ) = h r v v v aci aci i i  (6) From equations (4) and (5) we have: = h h h r i     (7) = vr v r abni i abni      (8) Now substituting equations (7) and (8) in equations (3) and (6), respectively, then solving equations (3) and (6) we have either = 0 v i  , which gives the DFE, or v i  satisfies the following equation: 2( ) = 0 v v A i Bi C     (9) where 3 2 2 = ( )h h A a b cmn      2 0 = ( )( 1) ( )[ ( ) ] v r h v h r h r h ab B A m n                  R 2 0 = ( )( 1) v r h C     R We note that < 0A , and > 0C when 0 > 1R , hence we have one and only one positive solution for equation (9) when 0 > 1R , and then we have the following lemma: Lemma 3.2. The system (2) has precisely one positive endemic equilibrium 1 E given by 1 = ( , , , , , , ) h h h r r v v E s i r s i s i        and v i  satisfying equation (9), when 0 > 1R . 3.5 Local Stability analysis of the endemic equilibrium 1 E To investigate the local stability of the endemic equilibrium, we use the center manifold theorem, particularly, Theorem 5 in Castillo-Chavez and Song [25]. The Jacobian of the system (2) at the disease-free equilibrium 0 E is given by: GLOBAL DYNAMICS AND SENSITIVITY ANALYSIS OF A VECTOR-HOST-RESERVOIR MODEL 125 0 0 0 0 0 0 0 ( ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( ) = 0 0 0 0 0 0 0 0 0 0 0 0 0 h h h r r v v m m n E n ac ac ac ac                                               J Consider the case when 0 = 1R and suppose that = ab is chosen as a bifurcation parameter, then it can be shown that the jacobian of the system (2) has a right eigenvector given by 1 2 3 4 5 6 7 = ( , , , , , , ) T W w w w w w w w , where 1 7 = h m w w    2 7 = h m w w     3 7 = ( ) h h m w w       4 7 = r n w w    5 7 = r n w w   6 7 =w w and a left eigenvector given by 1 2 3 4 5 6 7 = ( , , , , , , )V v v v v v v v , where 1 3 4 6 = = = = 0v v v v 2 = h ac v   5 = r ac v  It can be shown that: 2 2 2 7 7 = [ ( 1)] ( ) h h h r r acm acm acn n a v w                   7 7 1 1 = [ ] h r b acm v w       it is clear that < 0a  and > 0b  , hence our System (2) undergoes a regular transcritical bifurcation at 0 = 1,R before the bifurcation the disease-free equilibrium is stable and there exists an unstable positive endemic equilibrium, and after the bifurcation the disease-free equilibrium became unstable the endemic equilibrium became stable. Then we have the following result: Lemma 3.3. The endemic equilibrium is locally asymptotically stable for 0 > 1R . 3.6 Global Stability of Endemic Equilibrium To prove the global stability of the endemic equilibrium we first start with some preliminaries. Lemma 1. (Lemma 2.1 in [26]) Let ( )A t be a continuous, cooperative, irreducible, and  -periodic n n matrix function. Let (.) ( ) A t be the fundamental matrix of the linear non-autonomous differential equation = ( )x A t x , where x is a 1n vector. Let (.) 1 = ln ( ( )) A r    , where, (.) ( ( )) A r  is the spectral radius of the monodromy matrix (.) ( ) A  . Then there exists a positive,  -periodic function ( )v t such that ( ) t e v t  is a solution of = ( )x A t x . IBRAHIM M. ELMOJTABA ET AL 126 Definition 1 System (2) is uniformly persistent if  an > 0 (depending only on parameter values not on initial condition) such that for any initial value ( (0), (0), (0), (0), (0), (0), (0)) h h h r r v v s i r s i s i   R int(  R )  int(  R )   R  int(  R )   R  int(  R ) such that every solution ( ( ), ( ), ( ), ( ), ( ), ( ), ( )) h h h r r v s t i t r t s t i t s t i v t of the system (2) satisfies inf ( )limt hs t  … , inf ( )limt hi t  … , inf ( )limt hr t  … , inf ( )limt rs t  … , inf ( )limt ri t  … , inf ( )limt vs t  … , inf ( )limt vi t  … . Definition 2 The system (2) is said to be permanent if there exists a compact region 0   int(  ) such that every solution of the system (2) with initial condition ( (0), (0), (0), (0), (0), (0), (0)) h h h r r v v s i r s i s i   R int(  R )  int(  R )   R  int(  R )   R  int(  R ) will eventually enter and remain in region 0  . Clearly, for a dissipative dynamical system proving permanence is equivalent to proving uniform persistence. Consider following sets: 7 =X  R , 0 =X  R int(  R )  int(  R )   R  int(  R )   R  int(  R ), 0 0 = \X X X . Let, :f X X be a continuous map and we define following set 0 0 = { : ( ) , 0} n M x X f x X n    … . Following lemma will be used to show uniform persistence of the system (2) Lemma 2. ([27]) Assume that 1. 0 0 ( )f X X and f has a global attractor A. 2. There exists a finite sequence 1 2 = { , ,...., } k M M MM of disjoint, compact, and isolated invariant sets in 0 X such that - ( )M = =1( ) k x M i i x M      ; - no subset of M forms a cycle in 0 X ; - i M is isolated in X ; - 0 ( ) = s i W M X  for each 1 i k„ „ Then there exists > 0 such that 0 lim ( ( ), )inf n n d f x X   … for all 0x X . We claim the following result Proposition 1 If 0 > 1R then the solutions of the system (2) is uniformly persistent. Proof. Consider the periodic semi-flow 7 7 :T   R R associated with the system (2) defined by: ( ) = ( , )T t x u t x , 7 x    R . Let 1 P be the associated Poincaré map defined as 1 := ( )P T  . We first show that 1 P is uniformly persistent with respect to 0 0 ( , )X X . It is clear that the set X and 0X are positively invariant for the system (2). Now the bounded set  ( =  ) attracts every solution of the system (2) and also  is compact. Thus the Poincaré map 1P is point dissipative and compact on X . Therefore, it follows from Theorem 1.1.3 in [27] there is a global attractor A of 1P that attracts each bounded set in X . For the system (2) the set M  is defined as 0 1 0 = {( (0), (0), (0), (0), (0), (0), (0)) : ( (0), (0), (0), (0), (0), (0), (0)) , 0} n h h h r r v v h h h r r v v M s i r s i s i X P s i r s i s i X n     … . We claim that, = {( , 0, 0, , 0, , 0) : 0, 0, 0} h r v h r v M s s s s s s  … … … . It is clear that {( , 0, 0, , 0, , 0) : 0, 0, 0} h r v h r v s s s s s s M  … … … . Now, let 0 ( (0), (0), (0), (0), (0), (0), (0)) \ {( , 0, 0, , 0, , 0) : 0, 0, 0} h h h r r v v h r v h r v s i r s i s i X s s s s s s … … … . If (0) = 0, (0) = 0, (0) = 0, > 0 h h r v i r i i , then we get (0) > 0, (0) > 0, (0) > 0, (0) > 0. h r v v s s s i From second equation of the system (2) we have, (0) > (0) (0) > 0 ' h v h i abmi s . (0) > (0) > 0 ' h h r i . (0) > (0) (0) > 0 ' r v r i abni s . GLOBAL DYNAMICS AND SENSITIVITY ANALYSIS OF A VECTOR-HOST-RESERVOIR MODEL 127 and (0) > (0)( (0) (0)) > 0 ' v v h r i acs i i . Similarly, for other cases also. Therefore, 0 ( (0), (0), (0), (0), (0), (0), (0)) h h h r r v v s i r s i s i X for all 0 < 1t = . This implies that = {( , 0, 0, , 0, , 0) : 0, 0, 0} h r v h r v M s s s s s s  … … … . Now, 1 P has a unique fixed point 0 ( , 0, 0, , 0, , 0) h r v E s s s in M  . It is easy to show that 0 { }E is isolated in X and as 0 E is global attracting in M  therefore we have, 0 ( ) = ( ) { } x M M x E       , where, ( )x is the omega limit set of x . It is clear that no subset of 0 { }E can forms a cycle in 0 X . Now, we shall show that, 0 0 ( ) = s W E X  , where, 0 ( ) s W E is the stable set of 0 E . Let 0 0 0 = ( (0), (0), (0), (0), (0), (0), (0)) ( ) s h h h r r v v x s i r s i s i W E X  . Since 0 0 x X , therefore by continuity of solution with respect to initial conditions we have, for any (0,1)  , there exists a > 0 such that 0 0 x X  satisfying 0 0 || || 0 sufficiently small so that (.) (.) (.) ( ( )) > 1 F V M r       . Thus by lemma 1 and standard comparison theorem [28]   -periodic function ( )f t such that 1( ) ( ) s t x t f t e… , where, ( ) = [ , , ] h r v x t i i i and 1 (.) (.) (.) 1 = ln ( ( )) > 0. F V M s r        This implies as t  , ( ) h i t   , ( ) r i t   and ( ) v i t   . This is a contradiction as 0 1 0 sup || ( ) ||< .lim m t P x E   Therefore, we have, 0 1 0 sup || ( ) ||lim m t P x E   … 0 0 x X  . Which is again impossible as 0 0 ( ) s x W E (as 0 ( ) s x W E implies 0 1 0 || ( ) ||= 0lim m t P x E  ). Thus we have, 0 0( ) = s W E X  . Therefore, by lemma 2 we have 1 P is uniformly persistent with respect to 0 0 ( , )X X . Therefore, by Theorem 3.1.1 in [27] the periodic semi-flow T is uniformly persistent in X . Thus, if 0 > 1R then solution of the system (2) is uniformly persistent. Next, we claim the following result Proposition 2. If 0 > 1R then the endemic equilibrium 1 E is globally asymptotically stable. IBRAHIM M. ELMOJTABA ET AL 128 Proof. Let, 1 = ( ( ), ( ), ( ), ( ), ( ), ( ), ( )) h h h r r v v E s t i t r t s t i t s t i t . We shall first show that ( ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( )) h h h h r r v v s t i t p t r t s t i t s t i t is globally asymptotically stable We construct following Lyapunov function ( ) =| ( ) ( ) | | ( ) ( ) | | ( ) ( ) | | ( ) ( ) | | ( ) ( ) | | ( ) ( ) | | ( ) ( ) | . h h h h h h r r r r v v v v L t s t s t i t i t r t r t s t s t i t i t s t s t i t i t             We use following formula, | | = ( ) , ' ' x sgn x x to calculate the upper right-hand derivative (Dini's derivative) of ( )L t . Therefore, we have ( ) | | | | | | | | | | | | | | h h h h h h h h h r r r r r r v v v v v v D L t s s i i r r s s i i s s i i                       „ (11) Let, = min{ , , } r v h K    . Therefore, > 0K . Now, ( ) (| | | | | | | | | | | | | |) h h h h h h r r r r v v v v D L t K s s i i r r s s i i s s i i               „ . Which implies L is non-increasing on [0, ) . Thus we have, ( ) = 0limt L t . Therefore it follows that | |= 0limt h hs s  , | |= 0limt h hi i  , | |= 0limt h hr r  , | |= 0limt r rs s  , | |= 0lim t r ri i  , | |= 0limt v vs s  , | |= 0lim t v vi i  . Thus, ( ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( )) h h h h r r v v s t i t p t r t s t i t s t i t is globally asymptotically stable. 4. Sensitivity Analysis of 0 R 0 R is considered one of the most important quantities in epidemic theory [29], therefore studying the sensitivity of 0 R to the other parameters will give some more insight ideas about the best way of interventions to reduce 0 R below unity. There are many ways of conducting sensitivity analysis, all resulting in a slightly different sensitivity ranking [30]. Following [31, 32, 33] we used the normalized forward sensitivity index also called elasticity as it is the backbone of nearly all other sensitivity analysis techniques [30] and are computationally efficient [31]. The normalized forward sensitivity index of the basic reproduction number, 0 R with respect to a parameter value, P is given by: 00 0 = p P S p    R R R (12) because our 0 R contains square root, then it is convenient to use this version of equation 12: 2 00 2 0 1 = 2 p P S p    R R R (13) Using equation 13 together with parameter values given in the Table 1, we have our sensitivity indices for 0 R with respect to the other model parameters, which is presented in Table 2. Table 1. Parameter values for sensitivity analysis. Parameter Parameter description Value Source h  Natural mortality rate of humans 0.00004 1day [34] r Natural mortality rate of reservoirs 0.000274 1 day  [14] v  Natural mortality rate of vectors 0.189 1day [35] a Biting rate of vectors 0.285 [36] b Progression rate of the disease in vectors 0.22 [36] c Progression rate of the disease in human and reservoir 0.0714 [37]  Recovery rate 0.1,0.5,0.9 Assumed m Vector-human ratio 5 Assumed n Vector-reservoir ratio 10 Assumed GLOBAL DYNAMICS AND SENSITIVITY ANALYSIS OF A VECTOR-HOST-RESERVOIR MODEL 129 It can be seen from Table 2 that the sensitivity index of a , b , c , n and v  are fixed for all values of  , which means that the effect of these parameters on 0 R is not affected by the recovery rate (i.e. different values of  ), where a , b , c and n has positive effect on 0 R , for example if the biting rate is increased by 10%, then 0 R will increase by 10%, and if c is decreased by 10% then 0 R will decrease by 5%, while v  has a negative impact on 0 R , therefor if for example v  is decreased (increased) by 10% then 0 R will increase (decrease) by 5%. Table 2. Sensitivity indices of 0 .R Parameter Sensitivity Index = 0.1 = 0.5 = 0.9 a +1 +1 +1 b +0.5 +0.5 +0.5 c +0.5 +0.5 +0.5 n +0.499 +0.499 +0.499 m +0.0006 +0.0001 +0.00001  -0.0006 -0.0001 -0.00001 h  -2.7e-7 -1.1e-8 -3.4e-9 r  -4.99 -0.99 -0.55 v  -0.5 -0.5 -0.5 It is also clear that m ,  and h  have a very small effect on 0 R because their sensitivity indexes are very small (less than 0.001). However when the recovery rate,  , is small, then 0 R became very sensitive to r  , for example if r  is decreased (increased) by 10% then 0 R will increase (decrease) by 49.9%, this is also can be seen from Figure 1, for example when = 0.1 , then 0 R ranges between 5.6 and 15.5 for different values of r  , and when = 0.5 0 R ranges between 1.1 and 3.9 for the same values of r  and when = 0.9 0 R ranges between 0.83 and 2.1 for the same values of r  , which shows the effect of r  on 0 R , and that effect is really clear for small values of  . Different Scenarios Regarding Disease Control: • If the animal reservoir is kept out of the system (assuming that there is no transmission between reservoir and vector, or if the animal reservoir is kept away from humans so vectors can't transmit the disease from reservoir to human), then the threshold for the disease to invade the human population can be kept less than 1 easily, i.e. the disease can be eliminated. • If the human population is kept out of the system (assuming that all vectors stay near to the animal reservoir population to the their blood meal) then the threshold cannot be kept smaller than one which means that the disease will always persist in the reservoir and vector populations. • Considering the full system, then the threshold can be kept less than one using one of the following control strategies:  Applying human treatment at a very high rate, which is not cost-effective.  Decrease the animal reservoir population throw culling and apply human treatment at a medium rate, which is not ethical and also not so cost-effective.  Keep the vector biting rate in a low level either by using pesticides or changing the human behavior, and applying human treatment at a low rate. IBRAHIM M. ELMOJTABA ET AL 130 Figure 1. Values of 0 R for different values of r  with: (a) = 0.1 , (b) = 0.5 , (c) = 0.9. 5. Numerical Simulation of the Model In this section we solve our model numerically with initial conditions: 0 0 0 0 0 0 0 = 0.9, = 0.1, = 0, = 0.9, = 0.1, = 0.8, = 0.2 h h h r r v v s i r s i s i . Some of the parameter's values were obtained from literature, and some of them were assumed or made varying in order to study their role. The parameter values used are in Table 3. Table 3. Parameter values for numerical simulation. Parameter Parameter description Value Source h  Natural mortality rate of humans 0.00004 1day [34] r  Natural mortality rate of reservoirs 0.000274 1day [14] v  Natural mortality rate of vectors 0.189 1day [35] a Biting rate of vectors Variable Variable b Progression rate of the disease in vectors Variable Variable c Progression rate of the disease in human and reservoir Variable Variable  Recovery rate Variable Variable m Vector-human ratio Variable Variable n Vector-reservoir ratio Variable Variable Varying the values of a, the biting rate of vectors Simulation results show that when the biting rate of vectors is small that leads to some delay in the time needed for the epidemic curve to reach the peak in the infected human population; however, the peak itself remains the same for all values of the biting rate, therefore reducing the biting rate of vectors helps in reducing the prevalence of the disease in the human population; however, to reduce the epidemic's peak other intervention is needed. Nonetheless, it seems to have less effect on other population as can be seen from Figure 2. GLOBAL DYNAMICS AND SENSITIVITY ANALYSIS OF A VECTOR-HOST-RESERVOIR MODEL 131 Figure 2. Simulation results for different values of a. Varying the values of b, the progression rate of the disease in vectors Simulation results show that the epidemic curve remains the same for small and big values of b and it just shifted to the right, which means that when the time needed for the pathogen to progress within vectors is big, the disease will need more time to hit the population which gives a window for some interventions, as seen from Figure 3. However, as we know this parameter is out of control. IBRAHIM M. ELMOJTABA ET AL 132 Figure 3. Simulation results for different values of b. Varying the values of c, the progression rate of the disease in humans and reservoirs It can be seen from Figure 4 that the effect of changing the values of c is almost the same as changing the values of b ; but the differences between the curves of the small value and big value of c is less than the differences between the curves of the small value and big value of b in infected human and infected reservoir populations, and the curves of different values of b and c are the same for the population of infected vectors, which means that the progression time of the pathogen within the vector has more effect than the progression time of the pathogen within the host or within reservoir. This is due to the fact that the vector population is much bigger than the other populations and the recruitment rate of the vector population is big compared to the recruitment rate of the other populations. GLOBAL DYNAMICS AND SENSITIVITY ANALYSIS OF A VECTOR-HOST-RESERVOIR MODEL 133 Figure 4. Simulation results for different values of c. Varying the values of m, the vector-human ratio, and n, the vector-reservoir ratio Results show that there is a positive relationship between m and the fraction of infected humans, and there is almost no relationship between m and the fraction of infected reservoir, which is something predictable, as seen from Figure 5. Also, there is a positive relationship between n and the fraction of infected reservoirs, and there is almost no relationship between n and the fraction of infected humans, as seen from Figure 6. From Figures 5 and 6 it can be shown that the effect of different values of m on the population of vectors is the same as the effect of different values of n on the population of vectors, and that is because it is assumed that vectors take a fixed amount of blood despite the available number of hosts or reservoirs, and also because it is assumed that vectors do not prefer humans over reservoirs. IBRAHIM M. ELMOJTABA ET AL 134 Figure 5. Simulation results for different values of m. GLOBAL DYNAMICS AND SENSITIVITY ANALYSIS OF A VECTOR-HOST-RESERVOIR MODEL 135 Figure 6. Simulation results for different values of n. Varying the values of  (the recovery rate) Simulation results show that recovery rate has a strong impact on the population of infected humans, and it has no effect on the reservoir population nor on the vector population; which shows that although the recovery rate is an important factor in the fight against the disease, it is not enough for eradication, and it should be accompanying other interventions on the other populations, as seen from Figure 7. IBRAHIM M. ELMOJTABA ET AL 136 Figure 7. Simulation results for different values of . 6. Conclusion We have developed a general model for the dynamics of a vector-host-reservoir model. Our analysis of the model showed that the disease-free equilibrium is globally asymptotically stable when 0 R is less than unity; and unstable when 0 R is greater than unity, and our system posses only one endemic equilibrium which is globally asymptotically stable when 0 R is greater than unity. Our sensitivity analysis shows that 0 R is sensitive to all of the parameters of the model either positively or negatively, and the most influential has been the natural death of the reservoir, r  which indicates that the best control strategy is culling the infected animals, and the second one is the biting rate, a which indicates that reducing the biting rate by using, for example, bed-nets is the second best control strategy against the disease. Numerical simulations were used to examine the effect of all of the parameters of the model, and the results showed that reducing the biting rate of vectors helps in reducing the prevalence of the disease in the human population. 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