On a discrete West Nile epidemic model

On a discrete West Nile epidemic model

Sophia R.-J. Jang

Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010 U.S.A. E-mail: [e-mail]

ABSTRACT

A West Nile epidemic model in discrete-time is proposed. The model consists of two interacting populations, the vector and the avian populations. The avian population is classified into susceptible, infective, and recovered classes while an individual vector is either susceptible or infective. The transmission of the disease is assumed only by mosquitoes bites and vertical transmission in the vector population. The model behavior depends on a lumpedparameter R₀. The disease-free equilibrium is locally asymptotically stable if R₀ < 1. The system is uniformly persistent and possesses a unique endemic equilibrium if R₀ > 1. Consequently, the disease can persist in the populations if R₀ > 1.

Mathematical subject classification: 92D25, 92D40, 39A11.

Key words: West Nile virus, uniform persistence, Liapunov function.

1 Introduction

West Nile virus (WNV) is a kind of arthropod-borne virus that are maintained in nature through biological transmission between susceptible vertebrate hosts and blood-feeding arthropods such as mosquitoes. Vertebrates can become infected when an infected arthropod bites them to take a blood meal. The susceptible vectors then become infected once feed on an infected host.

WNV was first isolated from a woman in the West Nile District of Uganda in 1937 and has emerged in recent years in many regions of the United States and Canada. The disease presents a threat and challenge to public and animal health. West Nile virus has been detected in dead birds of at least 138 species. Although birds, particularly crows and jays, infected with the virus can die or become ill, most infected birds do survive. We refer the reader to www.cdc.gov/ncidod/dvbid/westnile for more information about the virus history and its ecology.

Since data collected for the West Nile virus are usually discrete, we develop a discrete-time West-Nile model to investigate evolution of the disease between mosquitoes and bird reservoir hosts. Discrete time West Nile models have been studied in [15, 17]. However, our modeling assumptions are different from that given in [15, 17]. In [15] the vector population is partitioned into larval, susceptible, exposed, and infective classes, and all the newborns are in the larval class, while in [17] the vector population also has an exposed compartment and there is no vertical transmission. Moreover, our incidence rate is different from that studied in [15, 17]. Our model derivation is based on a recent continuous-time model proposed by Cruz-Pacheco et al. [5]. Although other vertebrates such as horses and humans do become infected, these populations are not modeled here.

The resulting epidemic model is a four-dimensional system of difference equations. Sufficient conditions for which solutions remain nonnegative are derived. It is shown that the disease-free equilibrium always exists. Its stability depends on a threshold R₀. The disease-free equilibrium is locally asymptotically stable if R₀ < 1 and unstable if R₀ > 1. There exists a unique endemic equilibrium and the system is uniformly persistent when R₀ > 1. Consequently, the WNV can persist if R₀ > 1. When there is no disease related death for the avian population, it can be shown that the disease-free equilibrium is globally asymptotically stable for R₀< 1.

In the following section, model derivation will be presented. Section 3 provides stability analysis of the model. Numerical simulations and a brief summary are given in the last section.

2 Model derivation

Our model consists of two interacting populations: birds and mosquitoes. The transmission of the disease is only by mosquitoes bites and vertical transmission in the vector population. Let N_a(t) and N_v(t) denote the avian and vector populations at time t, respectively for t = 0, 1, ¼. We assume that the mosquito population under the period of study is a constant, N_v, and the bird population has a constant recruitment rate L_a per unit time due to birth and immigration. However, the new arrival birds are all susceptible. The death rates of avian and vector populations are denoted by µ_a and µ_v, respectively.

For simplicity, the birth rate of the vector population is µ_v which is the same as its death rate. That is, µ_v is the number of births per individual per unit time for the mosquito population. It is also assumed that the bird population in the absence of the disease is governed by the difference equation N_a(t + 1) = L_a + (1 - µ_a)N_a(t). As a result, the bird population in the absence of the disease will always stabilize at the level (L_a)/(µ_a).

Similar to the idea used by Kermack and McKendrick [12] for modeling epidemics, the avian population at time t is separated into three compartments: susceptible S_a(t), the healthy susceptible individuals who can contract the disease, infectives I_a(t), the individuals who are infected and are infectious, and recovered R_a(t), who are cured. That is, N_a(t) = S_a(t) + I_a(t) + R_a(t) for t > 0. Since mosquitos have short life span, the vector population at any given time t is only classified into susceptible, S_v(t), and infectives, I_v(t). There is no recovered class for the vector population and S_v(t) + I_v(t) = N_v > 0 for t > 0.

Let b be the average number of bites per mosquito per unit time. The transmission probability from vectors to birds and from birds to vectors are constants and denoted by b_a and b_v, respectively. Hence a bird receives on average b bites per unit time. Therefore the infection rate per susceptible bird is

and the infection rate per susceptible mosquito is

We assume that the infected birds are recovered at a constant rate g_a, and let a_a be the disease related death rate for the avian population. From the data given in [13, 16] (cf. [5]) it is reasonable to assume that

(H1) a_a < g_a.

We remark that assumption (H1) may not be satisfied for some spices of birds such as American crow and blue jay. However, many other species of birds such as common grackle, house sparrow, European starling etc. do have small WNV mortality ([13]), and consequently (H1) will fit in to these particular species of birds.

Notice the average infectious period for an infected bird is

Furthermore, it is assumed that all these parameters L_a, µ_a, µ_v, b, b_a, b_v, N_v, g_a and a_a are positive. Since vertical transmission of the vector population has been found as an important mechanism in maintaining the virus in natural populations [3, 7, 10], we assume a constant fraction p, 0 < p < 1, of the offspring of the infectious vectors is infectious. Under these biological assumptions, the interaction between vector and avian populations are given below:

Notice as the birth and death rates of the vector population are the same and offsprings of susceptible mosquitoes are born susceptible, the equation for S_v has the above form.

Discrete time epidemic models have been studied by Allen [1, 2], Lewis et al. [15], Thomas and Urena [17], and more recently by Franke [9] on periodic epidemic models. In [1, 2], the models are expressed in terms of time unit Dt and nonnegativity of the solutions are derived using the quantity Dt and model parameters. Since data for the West Nile epidemics given in the literature [3, 5, 7, 10, 16] are in terms of days, our time unit is taken to be a day and model (2.1) does not involve time unit Dt as in [1, 2]. We now impose the following conditions on the parameters so that solutions of (2.1) will remain nonnegative as shown in Proposition 2.1.

(H2) bb_aN_v < L_a, g_a+ µ_a+ a_a< 1, bb_v< 1, and µ_v< 1.

We remark that the first three conditions in (H2) imposed on the parameters are reasonable restrictions. For example, since the time unit is taken to be one day, then according to the data given in [5, 16], we have b_a = 1, b = 0.75, and the maximum values of b_v is 0.68, of g_a is 0.36, of a_a is 0.19, of µ_a is 0.0004, and of µ_v is 0.06 for a variety of bird species such as blue jay, common grackle, American crow, house sparrow, American robin, rock dove etc. and different species of mosquitos. Therefore the first three conditions in (H2) are easily satisfied. However, we would need the total population of vector to be small or the new arrival of birds to be large in our study for the last inequality in (H2) to be true.

Proposition 2.1. Solutions of system (2.1) remain nonnegative and are bounded.

Proof. Let (S_a(t), I_a(t), R_a(t), N_a(t), S_v(t), I_v(t)) be a solution of (2.1) with S_a(0), I_a(0), R_a(0), S_v(0), I_v(0) > 0 and N_a(0) > 0. It is sufficient to prove nonnegativity for t = 1. Since S_a(0) + I_a(0) + R_a(0) = N_a(0) > 0 and S_v(0) + I_v(0) = N_v > 0, S_a(1) > L_a + (1 - µ_a)S_a(0) - bb_aI_v(0) > L_v + (1 - µ_a)S_a(0) - bb_aN_v > (1 - µ_a)S_a(0) > 0 by (H2) and (H3). It is clear that I_a(1), R_a(1) > 0 by (H3). Moreover, S_v(1) > S_v(0) + (1 - p)µ_vI_v(0) - bb_vS_v(0) > 0 by (H4). Similarly, N_a(1) > L_a + (1 - µ_a - a_a)N_a(0) > L_a and I_v(1) > 0 by (H3) and (H5), respectively. Therefore, solutions of (2.1) remain nonnegative by induction.

Notice N_a(t + 1) < L_a + (1 - µ_a)N_a(t) for t > 0 implies

As S_a(t), I_a(t), R_a(t) are nonnegative and satisfy S_a(t) + I_a(t) + R_a(t) = N_a(t) for t > 0, we have

Moreover, since S_v(t + 1) + I_v(t + 1) = N_v for t > 0 and solutions remain nonnegative, S_v(t), I_v(t) < N_v for t > 0. Therefore, solutions of (2.1) are bounded.

It follows from the proof of Proposition 2.1 that N_a(t) > L_a for t > 1 if N_a(0) > 0. Therefore system (2.1) is well-defined. Furthermore, since I_a(t) + S_a(t) + R_a(t) = N_a(t) and S_v(t) + I_v(t) = N_v for t > 0 from modeling assumptions, we are able to reduce the dimension of system (2.1) so that system (2.1) is equivalent to the following four-dimensional system of difference equations

3 Mathematical analysis

We first study the existence of steady state solutions of (2.2). Clearly there always exists a trivial steady state E₀ = , the disease-free equilibrium. The Jacobian matrix of (2.2) evaluated at E₀ has the following form

Let J₁ be the lower 3 × 3 submatrix of J(E₀). Then J₁ is similar to

Therefore eigenvalues of J(E₀) are 1 - µ_a of multiplicity 2 and eigenvalues of

Notice

tr J₂ = 2 - g_a - µ_a - a_a - (1 - p)µ_v

and

Jury conditions imply that eigenvalues l of J₂ satisfy |l| < 1 if and only if |tr J₂| < 1 + det J₂ < 2 [8]. It follows from (H2) that tr J₂ > 0 and thus we need to verify tr J₂ < 1 + det J₂ < 2 for the local stability of E₀.

Notice det J₂ < 1 is trivially true and thus 1 + det J₂ < 2 holds. To verify tr J₂ < 1 + det J₂ we shall separate our discussion into two cases: 0 < p < 1 and p = 1. When 0 < p < 1, a simple computation yields

Let

It follows that E₀ is locally asymptotically stable if R₀ < 1, and unstable if R₀ > 1. When p = 1,

Therefore tr J₂ < 1 + det J₂ if and only if

The above inequality is never valid. Therefore E₀ is always unstable when p = 1.

Observe that can be interpreted as the number of infections produced by a single infected bird during its infectious period in a susceptible mosquito population when the avian population is stabilized at the population level L_a/µ_a. Similarly, is the number of infections produced by a single infectious mosquito during its lifetime in a susceptible avian population. Therefore, , the geometric mean of these two quantities, may be regarded as the basic reproductive number of the disease.

We proceed to examine the existence of an interior steady state. An interior steady state (, , , ) must satisfy

Adding the first two equations of (3.5) resulting

L_a- µ_a - (g_a + µ_a + a_a) = 0.

Let

A = g_a + µ_a + a_a.

Then

and > 0 if and only if < . The third and fourth equilibrium equations imply

and

Substituting these into the second equilibrium equation we have for 0 < p < 1, > 0 must satisfy

where

Let

Notice

f(0) =

and since A = a_a + g_a + µ_a,

When R₀ < 1, < 0 and hence for f(x) to have at least one positive root in (0, L_a/A) it is necessary that

Notice that the last inequality is equivalent to , and

if and only if

bµ_ab_v - Aµ_v(1 - p)R₀ + 2(µ_a + g_a)µ_v(1 - p) > 0.

Substituting R₀ by the expression (3.4) and using (H2), one can see that the above inequality is trivially true. Therefore there exists no feasible solution for (3.6) in (0, L_a/A) if 0 < p < 1 and R₀ < 1. Consequently, system (2.2) has no interior steady state if 0 < p < 1 and R₀ < 1.

On the other hand if R₀ > 1 then since > 0 and f(L_a/A) < 0, it is clear that (3.6) has a unique solution Î (0, L_a/A). As a result, system (2.2) has a unique endemic equilibrium E₁ = (, , , ) if 0 < p < 1 and R₀ > 1. If 0 < p < 1 and R₀ = 1, then = 0 and f(x) = 0 has solutions 0 and -/Â. Notice = L_a(1 - p)µ_va_a - (1 - p)(µ_a + g_a)L_aµ_v - bL_aµ_ab_v < 0 by (H1). If  > 0 then it is straightforward to show that -/ > L_a/A, and if  < 0 then it is trivial that -/ < 0. Therefore if 0 < p < 1 and R₀ = 1, there is no feasible . We conclude that system (2.2) has no interior steady state if 0 < p < 1 and R₀< 1, and has a unique endemic equilibrium if R₀ > 1.

When p = 1, the threshold R₀ in (3.4) is not defined and the I_a-component, > 0, of an interior steady state must satisfy (3.6) with

and

= -bL_aµ_ab_v - b²b_ab_vµ_aN_v.

Since  > 0, > 0 and f(L_a/A) < 0, f(x) = 0 has a unique solution in (0, L_a/A). Consequently, (2.2) has a unique endemic equilibrium E₁ = (, , , ) when p = 1. Recall in this case that the disease-free equilibrium E₀ = (L_a/µ_a, 0, L_a/µ_a, 0) is unstable. The above discussion is summarized below.

Proposition 3.1. If 0 < p < 1 and R₀< 1, then E₀ = (L_a/µ_a, 0, L_a/µ_a, 0) is the only equilibrium and E₀ is locally asymptotically stable if R₀ < 1. If 0 < p < 1 and R₀ > 1, then E₀ is unstable and system (2.2) has a unique interior steady state E₁ = (, , , ). If p = 1, then E₀ is unstable and E₁ exists for (2.2).

Our next goal is to determine local stability of the steady state E₁. When p = 1, it follows from (3.5) that = N_v. Therefore the Jacobian matrix of system (2.2) evaluated at E₁ has the following form

Clearly 1 - bb_v/ is an eigenvalue of J(E₁) which is less than 1 but greater than zero by (H4). The upper 3 × 3 submatrix of J(E₁) - lI is similar to the following matrix

Using the third row expansion we see that 1 - µ_a is another eigenvalue and the rest of the two eigenvalues satisfy

l² + tr l + det = 0,

where

tr = bb_aN_v/ - 2 + µ_a + A

and

det = (1 - A)(1 - µ_a) - ba_ab_aN_v / - (1 - A)bb_aN_v/.

Since det < 1, applying the Jury conditions, we need to very that -1 -det < tr < 1 + det . A straightforward calculation yields tr < 1 + det if and only if

Since < and A > a_a, the above inequality is clearly true. Moreover, -1 - det < tr if and only if

This inequality also holds as A > a_a and < . We now summarize our discussion in the following proposition.

Proposition 3.2. System (2.2) has steady states E₀ = (L_a/µ_a, 0, L_a/µ_a, 0) and E₁ = (, , , ) when p = 1, where E₀ is unstable and E₁ is locally asymptotically stable.

It is not easy to verify whether the endemic-equilibrium E₁ is locally asymptotically stable when 0 < p < 1 and R₀ > 1. We show that the disease can persist by showing that the system is uniformly persistent. We first briefly discuss terminology used in Hofbauer and So [11] which will be adopted for our analysis. Let (X, d) be a metric space and h : X ® X be continuous with a closed subspace Y such that X\Y is forward invariant under h. It is assumed that X has a global attractor . Let M be the maximal compact invariant set in Y. Then h is uniformly persistent (with respect to Y) i.e., there exists m > 0 such that lim inf_{t ® ¥} d(h^t(x), Y) > m for all x Î X \ Y if and only if M is isolated in and W^s(M) = {x Î X : h^t(x) ® M as t ® ¥} Ì Y [11, Theorem 4.1].

Theorem 3.3. System (2.2) is uniformly persistent if either 0 <p < 1 and R₀ > 1 or if p = 1.

Proof. Let X = and Y = ¶, the boundary of X. Let H denote the map induced by system (2.2). It follows from the proof of Proposition 2.1 that S_a(t), I_a(t), N_a(t), I_v(t) > 0 for t > 1 if the initial condition is positive. Therefore X \ Y is positively invariant for system (2.2). Clearly system (2.2) has a global attractor and the only invariant set in Y is {E₀}, which is moreover isolated in {(S_a, I_a, N_a, I_v) Î : S_a + I_a < L_a/µ_a, N_a < L_a/µ_a, I_v < N_v}.

To show W^s({E₀}) Ì Y, suppose on the contrary that there exists a solution (S_a(t), I_a(t), N_a(t), I_v(t)) with S_a(0) > 0, I_a(0) > 0, N_a(0) > 0, and I_v(0) > 0 such that lim_{t ® ¥} S_a(t) = lim_{t ® ¥}N_a(t) = L_a/µ_a and lim_{t ® ¥} I_a(t) = lim_{t ® ¥} I_v(t) = 0. Then for any > 0 there exists t₀ > 0 such that

for t > t₀. We first consider the case when 0 < p < 1 and R₀ > 1. Since R₀ > 1, we can choose > 0 such that

We have by system (2.2) that

for t > t₀. Consider the following linear system

Let A denote the map induced by system (3.8). Notice each entry of A is positive and it follows from (3.7) that the spectral radius of A is larger than unity. Since x(t₀) = I_a(t₀) > 0 and y(t₀) = I_v(t₀) > 0, solutions of (3.8) are unbounded. As a result, I_a(t) and I_v(t) also become unbounded large as t ® ¥. We obtain a contradiction and conclude that W^s({E₀}) Ì Y. Therefore, system (2.2) is uniformly persistent with respect to Y by [11, Theorem 4.1], i.e., there exists m > 0 such that lim inf_{t ® ¥} S_a(t) > m, lim inf_{t ® ¥} I_a(t) > m, lim inf_{t ® ¥} N_a(t) > m and lim inf_{t ® ¥} I_v(t) > m for any solution (S_a(t), I_a(t), N_a(t), I_v(t)) with positive initial condition. The case when p = 1 can be shown similarly using instability of E₀.

Although it is known that the crow family of birds have very high WNV mortality rate, the mortality rate of some other species of birds such as house barrow and common grackle are usually very small. In particular, European starling, rock dove, American robin, and several other species of birds have zero WNV mortality rate as shown in an experimental study by Komar [13]. Therefore, it is reasonable to consider the special case when there is no WNV related mortality for the avian population. In this situation lim_{t ® ¥} N_a(t) = L_a/µ_a and (2.2) has the following three-dimensional limiting system

Notice R₀ becomes

We show that the disease-free equilibrium (L_a/µ_a, 0, L_a/µ_a, 0) is globally asymptotically stable for (2.2) if 0 < p < 1 and R₀< 1.

Theorem 3.4. The disease-free equilibrium E₀ = (L_a/µ_a, 0, L_a/µ_a, 0) is the only equilibrium which is moreover globally asymptotically stable for system (2.2) if a_a= 0, 0 < p < 1, and R₀< 1.

Proof. It is clear that (2.2) has only the disease-free equilibrium. Since S_a(t) + I_a(t) < L_a/µ_a and I_v(t) < N_v for t > 0, we let

We construct a Liapunov function V

where nonnegative Ã, and will be determined later. Let G denote the map induced by system (3.9). Then V > 0 on D and

We choose à = 0. Then and must satisfy

We now let

Then and clearly satisfy the above inequalities as R₀< 1. Hence V(G(S_a, I_a, I_v)) < V(S_a, I_a, I_v) and V is a Liapunov function on D.

Let = {(S_a, I_a, I_v) Î D : V(G(S_a, I_a, I_v)) = V(S_a, I_a, I_v)}. Then = {(S_a, I_a, I_v) Î D : I_a = I_v = 0} and the only invariant set in is (L_a/µ_a, 0, 0). Therefore, (L_a/µ_a, 0, 0) is globally asymptotically stable for system (3.9) by the LaSalle's invariance principle [8, 14]. Since the limiting system (3.9) has only one equilibrium which is moreover globally asymptotically stable when R₀< 1, applying [6], we conclude that the disease-free equilibrium is globally asymptotically stable for system (2.2) when a_a = 0, 0 < p < 1, and R₀< 1.

4 Discussion

It is showed in the previous section that the West Nile virus can be wiped out when R₀< 1 and µ_a = 0, and the disease can persist within the populations when R₀ > 1. Although it is proved that the disease-free equilibrium is globally asymptotically stable if µ_a = 0, 0 < p < 1 and R₀< 1, it is suspected that the disease-free equilibrium is globally asymptotically when R₀ < 1, µ_a > 0, and 0 < p < 1. Since stability analysis does not provide any information about the transient behavior of the model which may be very important in terms of eradication and management plans, we next use simple numerical methods to study (2.2).

To simulate model (2.2), we adopt the following parameter values: L_a = 140, g_a = 0.1, a_a = 0.1, µ_a = 0.02, b = 0.7, b_a = 1.0, b_v = 0.38, µ_v = 0.06, N_v = 200 and p = 0.2. Initial conditions are chosen to be S_a(0) = 1000, I_a(0) = 0, N_a(0) = 1000 and I_v(0) = 100 for all simulations presented.Notice in this case that R₀ = 0.5038 < 1 and system (2.2) has only the disease-free equilibrium. Simulations for this set of parameter values are plotted in Figure 1(a). Both infected populations go to a peak at approximately the same time before they are diminished. Therefore there is a serge of the disease for a short period of time even when R₀ < 1.

We next keep the same parameter values but change b_v from 0.38 to 0.78. Then R₀ = 1.0341 > 1 and system (2.2) has a unique endemic equilibrium according to Theorem 3.3. The time evolution of the infected populations are plotted in Figure 1(b). It can be seen that both infected populations also increase before they decrease to the equilibrium levels for initial conditions with I_v(0) > 10. When 0 < I_v(0) < 10, then both infected populations increase to equilibrium levels with increasing time. It is known that vertical transmission of virus in the vector population is an important factor for contributing the spread of the disease [3, 7, 10]. We shall investigate this factor using our built model. We vary the parameter value p with the above fixed parameter values so that R₀ > 1. When p = 0.2, it is calculated R₀ = 1.0341, when p = 0.3, R₀ = 1.1818, and R₀ = 1.3788 when p = 0.4. The resulting time series of the infected mosquitoes and birds are plotted in Figure 1 (c) and (d) respectively. We see from these two plots that increasing the vertical transmission rate p increases the equilibrium levels and hence increases severity of the epidemics as the peaks of infectives increase with increasing p. However, the time that these peaks occurred is approximately independent of p.

In this manuscript, a simple West Nile epidemic model in discrete-time is proposed and analyzed. Our modeling assumptions are based on a continuous-time model developed by Cruz-Pacheco et al. [5]. In particular, the avian population in the absence of the disease is stabilized in a constant population level and the transmission of the virus is either through infected mosquito bites or natural birth of infected vectors. The dynamics of the epidemics depend on a lumped parameter R₀. The disease-free equilibrium E₀ is the only equilibrium and is locally asymptotically stable if R₀ < 1. It is proved that E₀ is globally asymptotically stable when there is no disease related mortality for the avian population and R₀< 1. As a result, the disease can be wiped out in this special situation. However, the epidemic can persist if R₀ > 1. From the data given in [5, 16], it is very often that R₀ > 1 for many species of birds along with vertical transmission of the vector population. Therefore, very likely that the West Nile epidemic can persist in natural populations as it has been observed in recent years in the U.S.

It is demonstrated numerically via simulations that both infected populations increase initially even when R₀ < 1 and the transient behavior of the model depends on initial conditions when R₀ > 1. If the initial infected vector population is small, then both infected populations will increase until they reach the equilibrium levels. However, if the initial infected vector population is large, then both infected vector and host populations will reach a maximum number which is much larger than the equilibrium value in a short period of time before they decrease to the equilibrium levels as shown in Figure 1. Therefore in this situation there will be a severe outbreak of the disease in the beginning of the epidemic.

[4] http://www.cdc.gov/ncidod/dvbid/westnile

Received: 08/III/07. Accepted: 17/IV/07.

#714/07.

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Publication Dates

• Publication in this collection
  14 Nov 2007
• Date of issue
  2007

History

• Received
  08 Mar 2007
• Accepted
  17 Apr 2007