Global Dynamics and Sensitivity Analysis of a Vector-Host-Reservoir Model

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 > 1 R and transcritical bifurcation occurs at 0 = 1 R . 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 contribution 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.


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.

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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.

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, ()

N t S t I t 
and the vector population have two categories, susceptible vectors ()

V
St, and infected vectors ()

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 , 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.From the description of the terms, we get the following system of differential equations:

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 0 t  .This shows that the biologically-feasible region: , , , , , ) : , , , , , , 0} 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  .

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 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 with the feasible region (i.e.where the model makes biological sense) 7 = {( , , , , , , ) : 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.

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  is the column-vector of rates of transfer of individuals into the particular compartment; and V  is the column-vector of rates 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 R defined as the spectral radius of matrix 1

Local Stability analysis of the disease-free equilibrium 0 E
The disease-free equilibrium (DFE) of the system (2) is given by 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 <1 R and unstable if 0 >1 R .

Global Stability analysis of the disease-free equilibrium 0 E
The following theorem shows that the DFE is globally asymptotically stable if 0 <1 R .
Theorem 3.1.The disease-free equilibrium point is globally asymptotically stable if 0 <1 R Proof: ] ) ) ) R .We observe that our system has the maximum invariant set for  , go to the disease-free equilibrium 0 E .Thus, 0 <1 R is the necessary and sufficient condition for the disease to be eliminated from the community.

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 From equations ( 4) and ( 5) we have: Now substituting equations ( 7) and (8) in equations ( 3) and ( 6), respectively, then solving equations ( 3) and ( 6) we have either =0 We note that <0 A , and >0 C when 0 >1 R , hence we have one and only one positive solution for equation ( 9) when 0 >1 R , and then we have the following lemma: Lemma 3.2.The system (2) has precisely one positive endemic equilibrium 1

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: = ( , , , , , , )


It can be shown that:

] ()
, 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 >1 R .
Thus, if 0 >1 R then solution of the system (2) is uniformly persistent.Next, we claim the following result Proposition 2. If 0 >1 R then the endemic equilibrium 1 E is globally asymptotically stable.

Sensitivity Analysis of
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: because our 0 R contains square root, then it is convenient to use this version of equation 12: 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.  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

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. .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.

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.

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.

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.

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.

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.

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.However, to reduce the epidemic's peak other interventions are needed; nonetheless, it seems to have less effect on other populations.

Acknowledgment
The authors acknowledge with thanks the comments and suggestions of anonymous reviewers which enhanced the quality and the readability of the paper.
-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 st , ( ) 1 r st , and ( ) 1.

vst
This shows that all solutions in the set where = = = = = 0 v i , which gives the DFE, or v i  satisfies the following equation: a bifurcation parameter, then it can be shown that the jacobian of the system (2) has a right eigenvector given by

1 
, for example when = 0.the same values of r  , which shows the effect of r  on 0 R , and that effect is really clear for small values of  .

Figure 2 .
Figure 2. Simulation results for different values of a.

Figure 3 .
Figure 3. Simulation results for different values of b.

Figure 4 .
Figure 4. Simulation results for different values of c.

Figure 5 .
Figure 5. Simulation results for different values of m.

Figure 6 .
Figure 6.Simulation results for different values of n.

Figure 7 .
Figure 7. Simulation results for different values of . System (2) is uniformly persistent if  an >0  A r   is the spectral radius of the monodromy matrix (.) () A   .Then there exists a positive,  t e v t  is a solution of = ( ) x A t x .Definition 1 a continuous map and we define following set

Table 1 .
Parameter values for sensitivity analysis.It can be seen from Table2that 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%,

Table 3 .
Parameter values for numerical simulation.
r  v 