Dynamics and Optimal Control of a West Nile Virus Model with Seasonality Ahmed Abdelrazec1 , Suzanne Lenhart2∗and Huaiping Zhu3† The Canadian Applied Mathematics Quarterly, October 2015. 1 School
of Mathematics and Statistics, Arizona State University, Tempe, AZ, USA
2 Department 3 Department
of Mathematics, University of Tennessee, Knoxville, TN, USA
of Mathematics and Statistics, York University, Toronto, ON, Canada
Abstract West Nile virus (WNv) is a mosquito-borne zoonotic arbovirus which arrived in Canada in 2001. It has kept spreading across the country and still remains a threat to public health. In this study, we establish compartment models to investigate the dynamics of the transmission of WNv in the mosquito-bird cycle and humans. Firstly, we establish and study the model without seasonality and prove the existence of the backward bifurcation of the model. Secondly, we expand the model to include the seasonal variations to study the impact of seasonal changes on the transmission of the virus. We prove the existence of periodic solutions, in the seasonal model, under specific conditions. We also introduce and calculate the basic reproduction number for this seasonal forced model. Furthermore, we examine the dynamics of the model when the seasonal variation becomes stronger. Then, we extend the first and second models to assess the impact of some anti-WNv control measures, by re-formulating the models as an optimal control problem. This entails the use of three control functions: adulticide, larvicide and human protection. The numerical simulations of this optimal control problem lead to the following outcomes: (1) Larvicide is the most effective strategy to control an ongoing epidemic in reducing disease cost. (2) The importance of considering the other animals that could be infected in any region and its effect ∗
Lenhart’s work was partially supported by the National Institute for Mathematical and Biological Synthesis through the National Science Foundation award # EF-0832858. † This work was supported by an Early Researcher Award of Ontario, the Pilot Infectious Disease Impact and Response Systems Program of Public Health Agency of Canada, and Natural Sciences and Engineering Research Council of Canada. Corresponding author:
[email protected].
1
Figure 1: Reported human cases of WNv in Ontario and Canada, from 2002 to 2012. Data from Public Health Agency of Canada (PHAC) website.
regarding the cost of control. (3) Identifying the time of applying the control to achieve the best control strategy.
Keywords: West Nile virus; equilibria; stability; bifurcation; seasonality; periodic solutions; optimal control.
1
Introduction
West Nile virus (WNv) is a mosquito-borne zoonotic arbovirus belonging to the genus Flavivirus in the family Flaviviridae that can cause swelling and inflammation of the brain and spinal cord in birds, humans and many other species of animals (e.g. horses, cats, bats, and squirrels) [7]. When an infected mosquito bites a bird or some other mammal including a human, the virus can be transmitted. A mosquito can be infected when it bites an infected bird. Also, the virus can be vertically transmitted from a mosquito to its offspring [37]. The virus is named after the West Nile region of Uganda, where the virus was first identified in 1937 [36]. Since then, WNv has dispersed in Africa, southern Europe, the Middle East and Asia [35]. The virus was first identified in North American in New York in 1999, and has since then become established across the North American continent [7]. The United States experienced one of its worst epidemics to date in 2012 [39]. The WNv activity was first reported in Canada in 2001, when the virus was found in dead birds and mosquito pools in southern Ontario [7]. In Fig.1, we present the reported WNv positive human cases in Ontario and Canada from 2002 to 2012 [39]. From the figure, one can see that the number of WNv infected humans in Canada was remarkably decreasing 2
during the years of 2007-2010. However, it started increasing once again in Ontario in 20102012 despite the immense efforts by the specialized agencies to control the virus. There are no indications that the spread of the virus has stopped. This fact reveals that the disease is evolving towards an endemic situation where the infected proportion is rather small. Studies of such epidemics often suggest that weather related factors could have been responsible for this recrudescence. Seasonal variations in temperature, rainfall and resource availability are ubiquitous and can exert strong pressures on density of vector mosquitoes. Three different mechanisms are responsible for seasonality: host behavior changes, climate and environmental changes, and pathogen appearance and disappearance [13]. In Canada, there are three species that carry the WNv: Culex Pipiens, Culex Quinquefasciatus and Culex Restuans. Because WNv mosquitoes culex are sensitive to temperature change, WNv shows very clear seasonal variation in any given year in Southern Ontario and other regions in Canada. This variation would not necessarily be labeled as an outbreak. The incidence during any part of the year should be compared to the situation in the previous years to demonstrate a clear increase to be declared as an epidemic. Mathematical models played an important role in gaining some insights into the transmission dynamics of WNv [4, 5, 10, 19, 22, 40, 43]. Those models considered different aspects of transmission of WNv and determined threshold conditions regarding control strategies but all the previous models deal with birds as only one species. In North America, the virus has been found in more than 300 bird species. The authors in [1, 6, 16, 30] categorized the birds into two groups and studied the effects in transmission of the virus. Likewise, the article [12] studied the effect of the interaction between different species of birds and mosquitoes living in the same locality on the emergence and prevalence of the disease. A common feature of all previous WNv models is that they are annual models with constant parameters, therefore, the seasonal impact on WNv was ignored. There have been some epidemiological models using a time-varying rate of some parameters [24, 25]. The authors in [41] proposed statistical relationships between environmental parameters and WNv by using the mosquito surveillance data, and temperature and precipitation records. Paper [11] is the only work that tackles the seasonal effects on new outbreaks in WNv starting from an endemic situation. It numerically concluded that the frequency of the new outbreaks depends on the relationship between the intrinsic and seasonal frequencies. By assuming that the birth rate of mosquitoes follows a periodic pattern, we study the impact of seasonal variations of the mosquito population on the dynamics of WNv. We also prove the existence of periodic solutions under specific conditions. Moreover, we introduce and calculate the basic reproduction number for this seasonal forced model. Furthermore, we numerically study the effect of seasonality and the dynamics of the model when the seasonal 3
variation becomes stronger. Control efforts are carried out to limit the spread of the disease, and in some cases, to prevent the emergence of drug resistance. Optimal control theory can be applied to models of many infectious diseases. The authors in [3] used a time dependent model to study the effects of prevention and treatment on malaria. Similarly, the authors in [32] used a time dependent model to study the impact of a possible vaccination with treatment strategies in controlling the spread of malaria in a model that includes treatment and vaccination with waning immunity. Optimal control theory has been applied to models with vector-borne diseases [3, 29, 34]. Time dependent control strategies have been applied for the studies of HIV/AIDS, Tuberculosis, Influenza and SARS [2, 8, 23, 44]. Although studies are underway, there is no vaccine currently available for WNv. The methods used to reduce the risk of WNv infection are based on mosquito reduction strategies (such as larvaciding, adulticiding, and elimination of breeding sites) and personal protection (based on the use of appropriate insect repellents). These measures are intensified during mosquito seasons. The only work on optimal control for a WNv model is [4], which determines cost-effective strategies for combating the spread of WNv in a given population. We are interested in the importance of carefully taking into account the impact of the seasonal variation when applying the control, and in identifying the best control strategy. The current article is organized as follows. We formulate our mosquito model connecting to past work [1, 4, 40], and then we construct our new full model in Section 2. In Section 3, we find and study the stability of the equilibrium points of the model and then existence of the backward bifurcation. The impact of seasonal variation, including proof of existence of periodic solutions, is demonstrated in Section 4. The model is then extended in Section 5 to incorporate anti-WNv control measures. Moreover, in Section 5 the necessary conditions for an optimal control and the corresponding states are derived, along with the presentation of numerical simulations and discussions.
2
Formulation of our models
There are two types of models that can help describe the effects of WNv, epidemic and endemic. Epidemic models describe outbreaks that occur in less than a single year while endemic models are used in studying diseases over a long period of time. Modeling mosquito population dynamics has become an important part of understanding the transmission of mosquito-borne arboviruses.
4
2.1
Modeling of mosquito population; connecting to past work
Suppose that the female mosquito lays eggs with a rate denoted by γm . Mosquito maturation rate denoted by mL . The natural death rates of larva and adult mosquitoes are denoted by dL and dm respectively. Thus, the general model for larval L and adult M mosquito populations, without the presence of the virus, is given by the following: dL dt = γm − (mL + dL )L, dM dt = mL L − dm M.
(2.1)
In some WNv models γm is considered as a constant term (see [4, 5, 10, 14, 34]), or as logistic growth (see [1]). In this paper we follow [9, 15, 27, 31, 40, 43] by assuming γm = rm M where rm denoted to the mosquitoes per capita birth rate. In this case, the parameters must satisfy rm mL = dm (mL + dL ) so as to guarantee the existence of a equilibria.
2.2
The basic model
Usually the host population remains unchanged over years if there is not host disease or other environmental changes. The recruitment rate of of host population is denoted by γs , and the natural death rate is denoted by ds . The general compartmental model for host population Ns , irrespective of disease infection, is given by the following: dNs = γs − ds Ns , dt
(2.2)
Based on the discussion on modeling the population of mosquitoes and hosts, and to extend the modeling for the WNv [1, 4, 27, 43], we propose to study a new model:
5
dLs = rm (Ms + (1 − q)Mi ) − (dL + mL )Ls , dt dLi = qrm Mi − (dL + mL )Li , dt dMs B1i + B2i = mL Ls − βm bm Ms − dm Ms , dt N dMi B1i + B2i Ms − dm Mi , dt = mL Li + βm bm N dBjs Bjs = γbj − db Bjs − βb bm Mi , dt N dBji Bjs = −(db + νj + µj )Bji + βb bm Mi , j = 1, 2 dt N (2.3) dBjr dt = −db Bjr + νj Bji , dS S = γ Mi − dh S, h − β h bm dt N dE S = βh bm Mi − αE − dh E, dt N dI = αE − (γ + µl + r + dh )I, dt dH = γI − (µh + τ + dh )H, dt dR = τ H + rI − dh R, dt where j = 1, 2, correspond to different avian populations, 1 for corvids and 2 for non-corvids. The definitions and values of the parameters used in the model (2.3) are summarized in Table 1. In the model (2.3) the total human population denoted by Nh , is split into the populations of susceptible S, exposed E, infectious I, hospitalized H and recovered R humans. Susceptible, infected and recovered corvid birds are denoted by B1s , B1i and B1r respectively. Similarly, susceptible, infected and recovered non-corvid birds are denoted by B2s , B2i and B2r . The total number of bird population is denoted by Nb . The parameter A denotes the number of other living organisms that mosquitoes will bite, and N = Nb + Nh + A represents the total of all organisms that mosquitoes will bite. As we mentioned in the previous section, the female mosquitoes can transmit WNv vertically [37], and the fraction of progeny of infectious mosquitoes that is infectious is denoted by q, with 0 ≤ q < 1. Then the L population is split into the populations of susceptible larval Ls and infectious larval Li . Similarly the M population splits into susceptible adults
6
Par. rm dm dL βm bm mL q γbj db βb µj νj γh βh α γ µl µh τ r dh
Value variable (0.02 − 0.07) (0.1 − 1.5) (0.018 − 0.24) (0.2 − 0.75) (0.07 − 0.1) (0 − 1) (50 − 500) (10−4 − 10−3 ) (0.088 − 0.9) (0.02 − 0.3) (0 − 0.2) 0.05 0.01 0.1 0.0009 0.015 0.0005 0.05 0.0002 0.00008
Meaning mosquitoes per capita birth rate natural death rate of adult mosquitoes natural death rate of larva mosquitoes WNv transmission probability from birds to mosquitoes biting rate of mosquitoes mosquito maturation rate vertical transmission rate in mosquitoes recruitment rate of corvid j = 1 and noncorvid birds j = 2 nature death rate of birds WNv transmission probability from mosquitoes to birds death rate of corvid and noncorvid birds due to the infection recovery rate of corvid and noncorvid birds the recruitment rate of humans WNv transmission probability from mosquitoes to humans the rate of development of clinical symptoms of WNv the hospitalization rate of infected humans the WNv-induced death rate of humans the death rate of hospitalized humans the treatment-induced recovery rate the natural recovery rate the natural death rate for humans
Ref. [43] [43] [43] [43] [43] [43] [26] [26] [43] [43] [26] [26] [4] [4] [4] [4] [4] [4] [4] [4] [4]
Table 1: Description of parameters of the model (2.3). Ms and infectious adults Mi . Thus, Nm = L + M = Ls + Li + Ms + Mi is the total number of mosquitoes. Due to its short life, a mosquito never recovers from the infection, so we do not consider the recovered class in the mosquitoes [20]. Let h ∈ [0, 1] be the percentage of corvids in new recruitment of birds. If γb is the recruitment rate, then in the model (2.3) we have γb1 = hγb and γb2 = (1 − h)γb . If h = 0, then all birds are non-corvid, and if h = 1, all birds are corvids. Each of the total subpopulations Nm , Nb and Nh is assumed to be positive for t = 0. Let us denote γb1 , B˜1 = db
γb2 B˜2 = , db
γh S˜ = , dh
7
˜ = B˜1 + B˜2 + S˜ + A. N
(2.4)
To better organize the analysis, we denote δj = db + µj + νj , (j = 1, 2), for all adjusted infectious period taking into account of mortality rates of birds [1].
1 δj
is the
We begin analysis by making two assumptions; the first is the parameter constraint dm (mL + dL ) so as to guarantee the existence of a disease-free equilibrium. The second rm = m L ˜ and L(0) = L ˜ = dm M ˜. is that the adult and larval mosquito populations satisfy: M (0) = M mL Then in a given period of time the mosquito population has constant size equal to Nm (t) = dm ˜ )M . (1 + m L Next, we will determine the equilibrium points and assess their stability, and we will also prove the existence of backward bifurcation
3
Equilibrium and dynamics of the model
The model (2.3) has a disease-free equilibrium E0 , obtained by setting the right hand sides ˜ 0, M ˜ , 0, B˜1 , 0, 0, B˜2 , 0, 0, S, ˜ 0, 0, 0, 0). The local stability of (2.3) to zero, resulting in E0 = (L, of E0 is governed by the basic reproduction number R0 . The basic reproduction number is obtained by using the next-generation matrix [38]: v u u ˜ M R0 = tq + βm βb b2m ˜2 dm N
! ˜2 ˜1 B B , + δ1 δ2
(3.1)
˜ are defined in (2.4). where B˜1 , B˜2 , N Theorem 2 of van den Driessche and Watmough [38] gives the following stability result with R0 given by (3.1). dm Proposition 3.1. For system (2.3), under the assumption rm = m (mL + dL ), the diseaseL free equilibrium E0 is locally asymptotically stable if R0 < 1 and unstable if R0 > 1.
An endemic equilibrium is given by the solution of the algebraic system obtained by ˜ − Mi and Ls = L ˜ − Li . setting the derivatives of model (2.3) equal to zero with Ms = M
8
qrm Mi − (dL + mL )Li B1i + B2i ˜ mL Li + βm bm (M − Mi ) − dm Mi N Bjs Mi γbj − db Bjs − βb bm N Bjs −δj Bji + βb bm Mi Nb + A −db Bjr + ν2 Bji S γh − βh bm Mi − dh S N S βh bm Mi − αE − dh E N αE − (γ + µl + r + dh )I
= 0,
(3.2)
= 0,
(3.3)
= 0,
(3.4)
= 0,
(3.5)
= 0,
(3.6)
= 0,
(3.7)
= 0,
(3.8)
= 0,
(3.9)
γI − (µh + τ + dh )H = 0,
(3.10)
τ H + rI − dh R = 0.
(3.11)
First we write the susceptible and recovered birds variables in terms of B1i and B2i δj Bjs = B˜j − Bji , db
Bjr =
νj Bji , db
j = 1, 2.
(3.12)
By combining (3.5) and (3.12) one can verify that B2i =
δ1 B˜2 B1i . δ2 B˜1
(3.13)
From (3.2) we have qdm Mi = mL Li . As assumed in the previous section we know that at ˜ and L = L. ˜ Then from equation (3.3), we have any positive equilibrium, we have M = M ˜ − Mi ) B1i +B2i . As a result we get the following: (1 − q)dm Mi = βm bm (M N Mi =
˜ (B1i + B2i ) βm bm M . (1 − q)dm N + βm bm (B1i + B2i )
(3.14)
It follows from (3.5) that B1i + B2i =
βb bm Mi N
B˜1 B˜2 B1i B2i + − − δ1 δ2 db db
! .
(3.15)
Eliminating Mi from equation (3.14) and (3.15), a straight forward calculation yields that if an endemic equilibrium exists, its B1i and B2i coordinates should satisfy the following quadratic equation: 2 2 c20 B1i + c11 B1i B2i + c02 B2i + c10 B1i + c01 B2i + c00 = 0,
9
(3.16)
where
µ1 − βm bm µdb1 , db 2(1 − q)dm µdb1 µdb2 − βm bm ( µdb1 + µdb2 ), 2 (1 − q)dm µdb2 − βm bm µdb2 , ˜ − 2(1 − q)dm N ˜ µ1 + βm βb b2m M˜ , βm bm N db db ˜ − 2(1 − q)dm N ˜ µ2 + βm βb b2m M˜ , βm bm N db db ˜2 − M ˜ βm βb b2m B˜1 + B˜2 . (1 − q)dm N δ1 δ2
c20 = (1 − q)dm c11 = c02 = c10 = c01 = c00 =
2
˜ βb βm b2m NM ˜2
Using the expression for R0 in (3.1) we can write can rewrite c00 in (3.17) as ˜ 2 dm 1 − R2 . c00 = N 0
B˜1 δ1
+
B˜2 δ2
(3.17)
= dm (R02 − q), so we (3.18)
To obtain the positive equilibrium points, we have to find the intersection of the line (3.13) with the quadratic curve (3.16). This is similar to the results illustrated in our previous work [1]. Theorem 3.2. If we set ˜ k1 ˜ − βb bm M ∆ = (βm bm )2 N k2 db ˜
˜
!2 − 4βm bm ˜
˜ M k1 k1 ˜ − k2 ), ((1 − q)dm − βm bm )βb bm (N k2 k2 db
˜
with k1 = dbδB1 1 + dbδB2 2 and k2 = µ1δB1 1 + µ2δB2 2 , then under assumption (1 − q)dm > βm bm kk12 the system (2.3) can have up to two positive equilibrium. More precisely, ∗ 1. If R0 > 1, there exists a unique positive stable equilibrium E2 = (L∗s , L∗i , Ms∗ , Mi∗ , B1s , ∗ ∗ ∗ ∗ ∗ B1i , B1r , B2s , B2i , B2r , S ∗ , E ∗ , I ∗ , H ∗ , R∗ ). Moreover, √ ˜ βm bm k1 db B˜1 M ˜ 2(1 − q)d − β b + N + ∆ m b m db δ1 k2 ∗ , B1i = 2 ((1 − q)dm k2 − βm bm k1 )
∗ B2i =
δ1 B˜2 ∗ B1i , δ2 B˜1
Mi∗ = L∗i =
δ1 ∗ ∗ B1s = B˜1 − B1i , db
δ2 ∗ ∗ B2s = B˜2 − B2i , db
∗ B1r =
∗ ∗ ˜ (B1i βm bm M + B2i ) , ˜ − µ1 B ∗ − µ2 B ∗ ) + βm bm (B ∗ + B ∗ ) (1 − q)dm (N 1i 2i db 1i db 2i
qdm ∗ M , mL i I∗ =
˜ − L∗ , L∗s = L i α E ∗, γ + r + dh
S ∗ = S˜ H∗ =
∗ B1s , ∗ ∗ δ1 βh B1s + B1i dh βb
γ I ∗, τ + dh
2. If R0 < 1, then 10
R∗ =
ν1 ∗ B , db 1i
∗ B2r =
ν2 ∗ B , db 2i
˜ − M∗ Ms∗ = M i
E∗ =
dh (S˜ − S ∗ ), dh + α
1 (τ H ∗ + rI ∗ ). dh
˜ ((1−q)dm k2 −βm bm k1 ) db , and ∆ > 0, we have two positive equi(a) If βbdbbm < M < ˜ βm bm k1 βb bm N ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ , , B2r , B2i , B2s , B1r , B1i librium points, E1 = (Ls , Li , Ms , Mi , B1s ∗ ∗ , , B1i S ∗∗ , E ∗∗ , I ∗∗ , H ∗∗ , R∗∗ ) unstable point and E2 = (L∗s , L∗i , Ms∗ , Mi∗ , B1s ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ B1r , B2s , B2i , B2r , S , E , I , H , R ) stable point. Moreover, √ ˜ βm bm k1 db B˜1 M ˜ 2(1 − q)dm − βb bm db + N − ∆ δ1 k2 ∗∗ B1i = . 2 ((1 − q)dm k2 − βm bm k1 ) By just replacing ”*” with ”**”, we can obtain the other values of the coordinates of E1 by the same relations between the coordinates in E2 . These two equilibria coalesce if and only if ∆ = 0. (b) Otherwise there is no positive equilibrium. 3. If R0 = 1, then ˜ ((1−q)dm k2 −βm bm k1 ) db (a) If M < , there exists a unique endemic equilibrium E2 . ˜ βm bm k1 βb bm N (b) Otherwise, there is no endemic equilibrium. In general from the parameter assumptions, the system (2.3) has infinitely many degendm erate stationary points satisfying L(t) = m M (t). From 2(a) in the Theorem 3.2 suggests L the possibility of backward bifurcation at any given initial larval and adult mosquito popdm ˜ ˜ ) (where the locally-asymptotically stable DFE co-exists with a locallyulation ( m M, M L asymptotically stable endemic equilibrium) when R0 is nearly below unity. To check this, let ∆ = 0 and solve for the critical value of R0 , denoted by R1 : v u ˜ u ((2(1 − q)dm − βm bm kk12 ) − βm bm kk12 βb bm dM˜N )2 t b R1 = 1 − . (3.19) 4dm ((1 − q)dm − βm bm kk12 ) Thus, the backward bifurcation scenario involves the existence of a subcritical transcritical bifurcation at R0 = 1 and of a saddle-node bifurcation at R0 = R1 . It should be mentioned that the proofs of stability and existence of the backward bifurcation are similar to the proofs in our earlier work [1]. Similar to Theorem 4.1 in [1] we can summarize and prove the next theorem. Note that the proof of the next theorem is based on the center manifold theory similar to the proof in [1]. Theorem 3.3. Consider model (2.3) with positive parameters. If ˜ ˜ B1 βm B2 βm ˜ )) + µ2 − (ν2 + db (1 + )) , A + S < µ1 − (ν1 + db (1 + (1 − q)dm δ1 (1 − q)dm δ2 then system (2.3) undergoes a backward bifurcation when R0 = 1. 11
From Theorem 3.3 we can conclude that the backward bifurcation occurs at R0 = 1. If we assume that all the birds as one family and A = 0, then the condition of the for occurrence of the backward bifurcation in Theorem 3.3 can be simplified as !! ˜ ˜ βm B N ν + db 1 + , (3.20) µ> ˜ − (S˜ + A) ˜ B (1 − q)dm N and is similar to one of the conditions in [4]; it is also considered a generalization of the same form in [1, 40]. With reference to equation (3.20) we notice that the existence of another important condition that is required for occurrence the backward bifurcation and that is, the ratio between total number of birds and the other mammals that can be infected by mosquitoes is greater than unity. When forward bifurcation occurs, the condition R0 < 1 is a necessary and sufficient condition for disease eradication, whereas it is no longer sufficient when a backward bifurcation occurs.
(a) Infected mosquitoes.
(b) Infected humans.
Figure 2: Time series of model (2.3) of (a) infected mosquitoes, (b) infected humans when R0 = 0.9908 > R1 = 0.9846, all the parameters are as in Table 1.
Fig.2 shows convergence to both the disease free equilibrium and the endemic equilibrium for system (2.3) when βb = 0.3, βm = 0.05 and (µ1 , µ2 ) = (0.27, 0.07), (in this case R0 = 0.9908 > R1 = 0.9846). The profiles can converge to either the disease free equilibrium or an endemic equilibrium point for the trajectories of system (2.3), depending on the initial sizes of the population of the model. The epidemiological significance of these is that the usual requirement of R0 < 1 is, although necessary, no longer sufficient for disease elimination. In other words, for R0 < 1, a stable disease-free equilibrium coexists with two endemic equilibria: a smaller equilibrium (i.e., with a smaller number of infective individuals) which is unstable and a larger one (i.e., with a larger number of infective individuals) which is stable. In such a scenario, disease 12
elimination would depend on the initial sizes of the sub-populations (state variables) of the model. That is, the presence of backward bifurcation in model (2.3) suggests that the feasibility of controlling WNv when R0 is nearly below unity, may depend on the initial sizes of the sub-populations. On the other hand, reducing R0 below the saddle-node bifurcation value R1 , may result in disease eradication. It follows from (3.20) that one can see, in order for backward bifurcation to occur the virus induced death rate must be high enough and the total number of initial bird population should be greater than the sum of total number of initial other mammals.
4
The impact of seasonal variations
(a) Percentage egg hatching in Culex quinquefasciatus in Toronto during the year 2011.
(b) Temperature and percentage egg hatching in Culex quinquefasciatus in Toronto
Figure 3: Impact of the temperature in birth rate of mosquitoes in Toronto
Strong pressure on population dynamics can be exerted by seasonal variations in temperature. Field observations show that the strength and mechanisms of seasonality can alter the spread and persistence of WNv. Hatching of the Culex mosquito eggs varies during the year; being low in the winter and high in the summer [21]. Fig.3 shows the relation between temperature and percentage of eggs hatching from Culex quinquefasciatus. The figure also demonstrates the percentage of hatching eggs from Culex quinquefasciatus in Toronto, ON, Canada in 2011. From the data (see [21]) in Fig.3, we can assume that the birth rate of mosquitoes follows a periodic pattern. Therefore, we propose that: rm = r1 (1 − cos(ωt)),
13
(4.1)
where r1 is the mean of birth rate of mosquitoes, 0 ≤ ≤ 1 is a measure of the influence of 2Π day −1 is the frequency. the seasonality on the birth process and ω = 365
4.1
Existence of periodic solutions
By replacing rm given in (4.1) into system (2.3), we can conclude that the total number of larval and adult mosquitoes satisfy the following equations: dL = (r1 (1 − cos(ωt))M − (dL + mL )L, dt (4.2) dM = −dm M + mL L. dt Using the trajectories of (4.2) the equation for total number of adult mosquitoes M can be written as dM d2 M + d − (δ − κ cos(ωt)) M = 0, (4.3) dt2 dt for all d = dm + dL + mL , δ = (mL r1 − dm (dL + mL )) and κ = mL r1 , where d2 + 4δ > 4κ with 0 ≤ < 1. From equation (4.3) we can conclude that the parametric resonance appears. Parametric resonance comes from changes in the parameters of the system as opposed to the classical resonance which originates from external forcing. The fundamental property of parametric resonance is that resonance peaks are expected at integer fractions of the natural period, once a control parameter has exceeded a certain threshold, with each parametric resonance peak having its own threshold value. In general, the solutions of equation (4.3) are not periodic. However, for a given δ, periodic solutions exist for special value of κ. The most general method to analyze equation (4.3) is the classical Floquet method [18], which is based on the calculation of the monodromy matrix and an analysis of its eigenvalues. However, this method requires a large number of numerical integrations, which restricts its possibilities, especially if the coefficients of the equations depend on some parameters. Noted that the stability analysis of differential equations with periodic coefficients is rather cumbersome, but it can be successfully made with computer software such as the Maple program. According to the general theory of linear differential equations with periodic coefficients, the behavior of solutions of (4.3) is determined by its characteristic multipliers ρ, which are the eigenvalues of the monodromy matrix X(T ) for (4.3), where X(t) is the principal fundamental matrix which is defined ! M1 (t) M2 (t) X(t) = , dM1 dM2 dt
14
dt
with M1 (t) and M2 (t) are two linearly independent solutions of (4.3) satisfying the initial 2 1 (0) = 0 and dM (0) = 1. conditions M1 (0) = 1, M2 (0) = 0, dM dt dt The characteristic equation det(X(T ) − ρI2 ) = 0 can be rewritten as ρ2 − 2Dρ + B = 0, 2 2 1 where D = 21 (M1 (T ) + dM (T )) and B = M1 (T ) × dM (T ) − M2 (T ) × dM (T ). Thus, the dt dt dt characteristic roots ρ1,2 are functions of two parameters D and B which are given by the formula √ (4.4) ρ1,2 = D ± D2 − B.
The parameter B can be found without solving equation (4.3). Indeed, since the functions M1 (t) and M2 (t) satisfy (4.3), we can write d2 Mj dMj + d − (δ − κ cos(ωt)) Mj = 0, dt2 dt
(j = 1, 2).
By solving these two equations together, we can conclude that the function y(t) = M1 (t) × dM2 1 (t) − M2 (t) × dM (t) satisfies the following differential equation: dt dt dy = −dy(t). dt Thus, the parameter B is given by the formula B = e−dT ,
(4.5)
Note that the parameter B can be found by Liouvilles formula directly. Hence, the system (4.3) is asymptotically stable for |D| < 12 (B + 1), stable for |D| = 12 (B + 1), and unstable for |D| > 12 (B + 1), for all 0 < B < 1. Next we use the the Poincare-Lyapunov theorem to calculate D. The general solution of (4.3) can be represented as a power series M (t) =
n X
Nj (t)κj ,
(4.6)
j=0
where Nj (t) are continuous functions, and sufficiently small κ. In order to obtain differential equations determining the functions Nj (t), we substitute expansion (4.6) into (4.3). Next, by equating the coefficients of κj , j = 0, 1, .. on both sides of the equation, we obtain the following system of differential equations. dN0 d2 N0 +d − δN0 = 0, 2 dt dt 15
and
d2 Nj dNj − δNj = − cos(ωt)Nj−1 (t), j = 1, 2, ... +d 2 dt dt Accordingly, we have two linearly independent solutions of N0 (t), satisfying the initial condition of the fundamental matrix, N0 (t) = e
−d t 2
(cosh(w0 t) +
N0 (t) = where w0 =
q
d2 4
d sinh(w0 t)), 2w0
1 −d t e 2 sinh(w0 t), w0
+ δ. Then the initial conditions for the functions Nj , j = 1, 2, .... can be dN
written as Nj (0) = dtj (0) = 0. Using these initial conditions we can obtain the following expression for the functions Nj (t), j = 1, 2, .... as Z t Z t 1 ( d2 +w0 )(s−t) ( d2 −w0 )(s−t) Nj (t) = cos(ωs)Nj−1 (s)e ds − cos(ωs)Nj−1 (s)e ds . (4.7) 2w0 0 0 It should be noted that the solution N0 (t) is increases unboundedly as t → ∞ (unstable) L) when δ > 0 (i.e r1 > dm (dmL +m ), decreases to 0 as t → ∞ (asymptotically stable) when L dm (dL +mL ) L) δ < 0 (i.e r1 < ), and stable when δ = 0 (i.e r1 = dm (dmL +m ). This is achieved mL L with previous results when = 0 (κ = 0). Using the recurrence relation (4.7), we can successively calculate the coefficients Nj in expansion (4.6). However, as j grows, the calculations become more and more cumbersome. Therefore, this method can be reasonably realized with computer software. With accuracy of κ2 , we have found the parameter D as a power series in :
D =
2d −d e 2 T (sinh(w0 T ) w0
+ mL ω 2 + 2 ((4w02 − ω 2 )((w02 − d2 ) sinh(w0 T ) (4.8)
+ 2dω cosh(w0 T )) + (w02 − d2 (3w02 − d4 )) sinh(w0 T ) + 2w0 d cosh(w0 T ))). From the above, we are able to state the principal results about the existence of the periodic solutions. 2
r1 ml ml 2 Theorem 4.1. Let < = dm (m − wr21+d 2 2 + o( ) and (L, M ) be the solution of system L +dL ) 2 (4.2) through (L(0), M (0)) ∈ R+ . Then the following statements are valid:
1. If < < 1, then lim [L(t), M (t)] = (0, 0), t→∞
2. If < > 1, then lim [L(t), M (t)] = (∞, ∞), t→∞
3. The periodic solutions exists only if < = 1. 16
Proof. By studying the dynamics of equation (4.3), we can conclude that the domain of stability of equation (4.3) is inside the triangle bounded by the lines B = 1 and B = −1±2D in the D − B plane. The points that lie on the boundary of the triangle determine the stable behavior of its solutions, while the domain outside the triangle is the domain of instability. Thus, from (4.8) and (4.5) we can conclude that D > 0 and 0 < B < 1. Furthermore, the line B = −1 + 2D is periodical condition in the (D, B) plane. The periodic condition can be written in the form of the relation between (r1 , ) as shown below r1 ∼ =
dm (mL + dL )(w2 + d2 ) 2 . mL ((w2 + d2 ) − dm (ml + dl ) 2
(4.9)
Biologically, we can indicate from the previous result that the mosquito population will die out if < < 1, while it grows exponentially if < > 1. Whereas, it oscillate to the positive equilibrium if < = 1.
Figure 4: The stability domain of (4.3).
Fig.4, shows that the (r1 , ) parameter plane (with the parameters values dm = 0.03, mL = 0.068, and dL = 0.8,) is divided into two regions. The first one is asymptotically stable (the amplitude M goes to zero at long times) where < < 1. The second region is unstable (the amplitude M grows exponentially without bound) where < > 1. Between these two zones there are bounded periodic solutions when < = 1. Fig.5 explains these scenarios with different values of and r1 . Fig.5(a) introduces the total number of adult mosquitoes when = 0.8 and r1 has three different values. One of the latter satisfies the equation (4.9); where we have a periodic solution. The second value of r1 is less than the first value; where we obtain asymptotically stable solution. Finally, we have 17
(a) dL = 1.2, mL = 0.68, dm = 0.03 and = 0.8 with different values of r1 .
(b) dL = 0.8, mL = 0.68, dm = 0.03 and r1 = 0.387 with different values of .
Figure 5: Long-term behavior of the total number of adult mosquitoes with different values of and r1 .
unbounded unstable solution if r1 is greater than the first value. Similarly, the same thing occurs in Fig.5(b) but with fixed r1 = 0.387 and three different values of .
4.2
Reproduction number
In what follows, we introduce the basic reproduction number for the model with seasonality. We will follow the work of [42], which is a generalization of the work in [38] to the periodic case. It is easy to see that the system (2.3) (when rm satisfy (4.1) and < = 1) has one ˜ ˜ (t), 0, B˜1 , 0, 0, B˜2 , 0, 0, S, ˜ 0, 0, 0, 0), where M ˜ (t) is disease-free equilibrium E0p = (L(t), 0, M the positive periodic solution of (4.3). Linearizing the system at the disease-free periodic state E0p to obtain F (t) =
0
qrm (t)
0
mL 0 0 0 0 0 0 0 0
0
˜ (t) βm bm M ˜ N
βb bm B˜1 ˜ N
0 βb bm B˜2 ˜ N
0 βh bm S˜ ˜ N
0 0 0
0
0 ν1 0 0 0 0 0 0 18
0 0 0 0 0 0 0 0 0
0 ˜ (t) βm bm M ˜ N
0 0 0 ν2 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0 0 0 0
,
0 0 0 0 0 0 α 0 0
0 0 0 0 0 0 0 γ r
0 0 0 0 0 0 0 0 τ
0 0 0 0 0 0 0 0 0
and V =
dL + mL 0 0 0 0 0 0 0 0 0 0 dm 0 0 0 0 0 0 0 0 0 0 δ1 0 0 0 0 0 0 0 0 0 0 db 0 0 0 0 0 0 0 0 0 0 δ2 0 0 0 0 0 0 0 0 0 0 db 0 0 0 0 0 0 0 0 0 0 (α + dh ) 0 0 0 0 0 0 0 0 0 0 (γ + µl + r + dh ) 0 0 0 0 0 0 0 0 0 0 (µh + τ + dh ) 0 0 0 0 0 0 0 0 0 0 dh
.
Then we can write
dz = (F (t) − V )z(t), dt where z(t) = (Li (t), Mi (t), B1i (t), B1r (t), B2i (t), B2r (t), E(t), I(t), H(t), R(t))T . Assume Y (t, s), t ≥ dy s, is the evolution operator of the linear periodic system = −V y(t). That is, for each dt s ∈ R, the 10 × 10 matrix Y (t, s) satisfies dY (t, s) = −V Y (t, s), dt
Y (s, s) = I,
t ≥ s,
where I is the 10 × 10 identity matrix. Let CT be the Banach space of all T-periodic functions from R to R10 equipped with the maximum norm. Suppose Φ(s) ∈ CT is the initial distribution of infectious individuals in this periodic environment; then F (s)Φ(s) is the rate of new infections produced by the infected individuals who were introduced at time s, and represents the distribution of those infected individuals who were newly infected at time s and remain in the infected compartments at time t for t ≥ s. Thus, Z t Z ∞ Ψ(t) = Y (t, s)F (s)Φ(s)ds = Y (t, t − a)F (t − a)Φ(t − a)da, −∞
0
is the distribution of accumulative new infections at time t produced by all those infected individuals Φ(s) introduced at the previous time. We define the linear operator L : CT −→ CT by Z ∞
Y (t, t − a)F (t − a)Φ(t − a)da ∀t ∈ R,
(LΦ)(t) =
Φ ∈ CT .
0
Following [42], we call L the next infection operator, and define the basic reproduction number as R0p = ρ(L), the spectral radius of L. It should be pointed out that in the special dm case of rm (t) = r1 = m (mL + dL ) ( = 0) we obtain F (t) = F for all t. By [42], we further L 19
obtain the basic reproduction number defined as in (3.1). In the periodic case, we let W (t, λ) be the monodromy matrix of the linear T-periodic system: du 1 = (−V + F (t))u, dt λ with parameter λ ∈ (0, ∞). It is easy to verify that our model with seasonality satisfies assumptions (A1)-(A7) in [42]. Thus, from [42], we have the following results, which will be used in our numerical computation of R0p • R0p = 1 if and only if ρ(ΦF −V (T )) = 1. • R0p < 1 if and only if ρ(ΦF −V (T )) < 1. • R0p > 1 if and only if ρ(ΦF −V (T )) > 1. Thus, the disease-free equilibrium E0p is locally asymptotically stable if R0p < 1, and unstable if R0p > 1.
Figure 6: The graph of the basic reproduction number R0p with respect to . The other parameter values are given in Table 1. By numerical computation, we get the curve of the basic reproduction number R0p (when < = 1) with respect to . In Fig.6, we can see that the basic reproduction number R0p increases with the increase of .
4.3
Simulations of the seasonal impact
Depending on the values of , and without changing the values of the other parameters, the basic reproduction number could remain above 1 or it could drop below 1. To see what could happen, we plotted in Fig.7; choosing the value of the parameters when = 0, 0.1 and 0.3. 20
(a) Infected mosquitoes.
(b) Infected humans.
Figure 7: Time series of (a) infected mosquitoes, (b) infected humans. Values of parameters are = 0 (green) when R0 = 0.989 > R1 = 0.9855, = 0.1 (red) when R0p = 0.995 and = 0.3 (blue) when R0p = 1.15. Solid line with first initial population and dashed line with second initial population.
Fig.7 shows three different cases. The first one, while ignoring the impact of seasonality (i.e = 0), we see the solutions are convergent to both the disease free equilibrium and the endemic equilibrium, depending on the initial sizes of the population. In the second case when = 0.1, we note that the solutions oscillate with small amplitude to both the disease free equilibrium and the endemic equilibrium, depending on the initial sizes of the population. While in the third state when = 0.3, the solutions oscillate to only the endemic equilibrium point at any initial size of the population. Thus, we can conclude that the dynamic behavior of the backward bifurcation state changes when the influence of the seasonal variation becomes stronger. Moreover, Fig.7 shows that when = 0, the infected populations are almost constant to endemic point (as it is expected because the equilibrium is stable), while when increases, the peaks and valleys time series appear. Furthermore, the amplitude of infected populations increases as increases. This is reflected in Fig.8 which shows one season with two different values of : 0.25 and 0.5. We note that when the seasonal variation has high force, it increases the infected cases. Also the highest peak number of infected populations when = 0.25 comes later than when = 0.5. This means that the time of applying the control could depend on the seasonal impact. Finally, from Fig.8(a), and (b) we can see that (t1 , t2 ) (t1 and t2 are the difference in time between highest peaks of infected mosquitoes, and highest peaks of infected birds and humans, respectively) decreases from (18, 35) when = 0.25 to (11, 27) when = 0.5 days. We deduce from numerical analysis that the strength of seasonality increases the number 21
(a) = 0.25.
(b) = 0.5.
Figure 8: Time series and peak time of model (2.3) (infected humans (green); infected birds (blue) and infected mosquitoes (red)) with seasonal impact in two cases = 0.25 (left) and = 0.5 (right).
of infections. This agrees with the studies carried out that show that factors which influence mosquito dynamics such as mean values of temperature and rainfall are strong positive predictors of increased annual WNv incidence [41]. In the next section we investigate the effective optimal control technique for the proposed epidemic model.
5
Optimal control
The goal of this section is to show that it is possible to implement anti-WNv control techniques while minimizing the cost of implementation of such measures. So we formulate an optimal control problem for the transmission dynamics of WNv by extending the model (2.3) in two cases. One model without impact of seasonality and the other one is the modified mathematical model with the effect of seasonal variation (by assuming that the birth rate of mosquitoes satisfies the equation (4.1)). In both cases, for the optimal control problem of the system (2.3), we consider the control variable in the set Γ = {(w1 , w2 , w3 ) : [0, T ] −→ R3 , | 0 ≤ wj ≤ Uj , j = 1, 2, 3} , where all control variables are bounded and Lebesgue measurable and Uj , j = 1, 2, 3 denote the upper bounds of the control variables. In our controls w1 (t) (representing the level of larvicide which means killing mosquito larva) and w2 (t) (representing the level of adulticide which means killing adult mosquitoes) are used for mosquito control administered at mosquito breeding sites. Consequently, the
22
reproduction rate of the mosquito population is reduced by the two factors (1 − w1 (t)) and (1 − w2 (t)). Furthermore, additional mortality rates of larval and adult mosquitoes (susceptible and infected) due to control represented by d0 w1 (t) and d0 w2 (t), where d0 > 0 is a rate constant. In the human population, the associated force of infection is reduced by the factor (1 − w3 (t)) where w3 (t) measures the level of successful prevention (personal protection) efforts. In this part, we seek to minimize the human exposed and infected populations and minimize the total mosquito population. So we suppose that the costs of the control strategies for the proposed system (2.3) are nonlinear and take quadratic form [4]. Thus, the objective (cost) functional is given by Z J= 0
T
3 X 1 bj wj2 dt, (5.1) a1 E(t) + a2 I(t) + a3 Nm (t) + c1 w1 L(t) + c2 w2 M (t) + c3 w3 S(t) + 2 j=1
subject to dLs = rm (Ms + (1 − q)Mi )(1 − w2 ) − (dL + mL (1 − w1 ))Ls − d0 w1 Ls , dt dLi = qrm Mi (1 − w2 ) − (dL + mL (1 − w1 ))Li − d0 w1 Li , dt dMs B1i + B2i Ms − dm Ms − d0 w2 Ms , dt = mL (1 − w1 )Ls − βm bm N dMi B1i + B2i = m Ms − dm Mi − d0 w2 Mi , L (1 − w1 )Li + βm bm dt N dS S(1 − w3 ) = γ − β b Mi − dh S, h h m dt N dE S(1 − w3 ) = β b Mi − αE − dh E, h m dt N
(5.2)
and all the other subpopulation equations are the same as in (2.3). Here, ai , i = 1, 2, 3 are positive constants that represent, respectively, the weight constants of the exposed, infected human and the total mosquito populations. Similarly, bi , i = 1, 2, 3 are also positive constants that represent, the weight constants for the quadratic cost of mosquito control (adult and larval) and personal protection (prevention of mosquito-human contacts), respectively. Also ci , i = 1, 2, 3 are positive constants. The linear part of the cost of each type of control is proportional to the affected population, c1 w1 L(t) + c2 w2 M (t) + c3 w3 S(t). For technical purposes, it is assumed that the cost of larvicide, adulticide and personal protection are given in quadratic form in the cost function (5.1). Using the control variables w1 , w2 and w3 , our main goal here is to minimize the exposed and infected human populations, the total number of mosquitoes and the cost of implementing the control. So the terms b1 w12 , b2 w22 and 23
b3 w32 describe the costs associated with mosquito control and prevention of mosquito-human contacts, respectively. Our purpose is to find an optimal control pair (w1∗ , w2∗ , w3∗ ) such that J(w1∗ , w2∗ , w3∗ ) = min {J(w1 , w2 , w3 ) : (w1 , w2 , w3 ) ∈ Γ} . The existence of optimal control can be proved by using the results in the paper of Fleming and Rishel [17]. It is clear that system of equations given by (5.2) is bounded above by linear system. The boundedness of solution of system (5.2) for finite interval is used to prove the existence of an optimal control. In order to find an optimal solution, first we should find the Hamiltonian of the optimal control problem (5.2) by defining the Hamiltonian ~ as follows: Let Q = (Ls , Li , Ms , Mi , B1s , B1i , B1r , B2s , B2i , B2r , S, E, I, H, R), w = (w1 , w2 , w3 ) and λ = (λi ), i = 1, .., 15 to obtain: 3
15
j=1
j=1
X 1X bj wj + λj fj , (5.3) ~ = a1 E(t) + a2 I(t) + a3 Nm (t) + c1 w1 L(t) + c2 w2 M (t) + c3 w3 S(t) + 2
where fj is the right side of the differential equation of the j-th state variable of (5.2). Based on that, we can demonstrate the next theory. Theorem 5.1. Consider the objective functional J. An optimal control w∗ = (w1∗ , w2∗ , w3∗ ) ∈ Γ exists such that J(w1∗ , w2∗ , w3∗ ) = minwj ,w2 ,w3 )∈Γ J, subject to the control system (5.2). Proof. In this minimizing problem, the necessary convexity of the objective functional in w1 , w2 and w3 is satisfied. The set of controls Γ is also convex and closed by definition. The solutions of the state system is bounded which together with the structure of the system gives the compactness needed for the existence of the optimal control. In addition, the integrand of the objective functional is given by (5.1) on the control set Γ, which completes the existence of an optimal control [17]. The adjoint differential equations and final time conditions and the characterizations of optimal controls can be found using Pontryagin’s Maximum Principle [33], and the details are in the Appendix. Next, we discuss the numerical solutions of the optimality system for the model (5.2), the corresponding optimal control functions, the parameter choices, and the interpretations from various cases.
5.1
Numerical results of the control without the seasonality
In this part we start with an iterative method to obtain results of an optimal control problem for model (2.3) without the effect of the seasonal variation. We use Runge-Kutta fourth order 24
procedure here to solve the optimality system consisting of 30 ordinary differential equations having 15 state equations as well as 15 adjoint equations and boundary conditions. With an initial guess we start for the control variables (w1∗ , w2∗ , w3∗ ) and use Runge-Kutta fourth order forward in time for the state variables Ls , Li , Ms , Mi , B1s , B1i , B1r , B2s , B2i , B2r , S, E, I, H, R. Then using the results from the state equations in the adjoint equations, we apply backward Runge-Kutta fourth order scheme due to transversality conditions. Then the control is updated and we iterate to find new state and adjoint variables [28]. The parameters values used in the simulations are tabulated in Table 1. In choosing upper bounds for the controls, since the control would not be 100% effective, so we chose the upper bound of w1 , w2 to be 0.8 and w3 to be 0.5 [4]. Since reducing the number of exposed and infected humans is important in our goal compared with reducing the total number of mosquitoes, then the weights in the objective functional are taken as a1 = 1, a2 = 1, a3 = 10−4 [4]. The cost associated with w1 and w2 mainly includes ways of eradicating the mosquito breeding (larvicide) and a little labor to spray it (adulticide), while w3 essentially involves the cost of missing work during the infectious time, educating the public and health professionals. This means the cost of lowering the infectivity is higher than the cost of reducing the mosquito population, so we chose b3 = 10 > b2 = b1 = 1. Since we assume that the total numbers of the larval and adult populations are constant, and the change in the human population is small then we suppose that cj = 0, j = 1, 2, 3.
(a) Infected humans.
(b) Infected adult mosquitoes.
Figure 9: Time series of model (5.2) showing impact of A when A = 0, A = in two cases without control (solid) and with control (dashed).
˜ B 20
and A =
˜ B 10
The importance of the parameter A (representing the number of other living organisms that mosquitoes will bite) on the control is considered. Fig.9 illustrates the optimal trajectories of the infectious adult mosquitoes and infectious humans for three different values of A while keeping the other parameters unchanged. Fig.10 shows the three optimal control functions at those values of A. These optimal control functions are designed in such a way 25
(a) w1 .
(b) w2 .
(c) w3 .
Figure 10: Graphical representation of the control functions with different values of A when ˜ ˜ B B A = 0, A = 20 and A = 10 . ˜
˜
B B that they minimize the cost functional J. We can see that when A = 0, 20 and 10 , w1 decreases a little and w3 decreases more while w2 does not change. Moreover, the objective functional value is reduced; J = 8290, 7620 and 6750 respectively. Thus, we can conclude that, if we do not include other mammals, any control impact and objective functional value will be over estimated.
(a) Infected humans.
(b) Infected adult mosquitoes.
Figure 11: Time series of model (5.2) under different optimal control strategies, i.e. (I) (w2 = w3 = 0, w1 6= 0), (II) (w1 = w3 = 0, w2 6= 0) and (III) (w1 = w2 = 0, w3 6= 0).
We investigate the use of one control at a time. Fig.11 illustrates the number of infected mosquitoes and humans under different optimal control strategies: (I) w1 , (w2 = w3 = 0); (II) w2 , (w1 = w3 = 0); (III) w3 , (w1 = w2 = 0). The J value for implementing strategy (I) is much less than that for others strategies (II) or (III). In details, the J value of strategies (I), (II) and (III) are 5282, 5814, and 6700 respectively. Moreover, the total number of infected mosquitoes using strategy (I) is the smallest while those with strategy (III) is the largest. While, the total number of infected humans using strategy (III) is the smallest; on the other 26
hand, those with strategy (II) is the largest. Thus, we can conclude that, the most effective strategy to control the WNv with only one control is by using larvicide during an ongoing epidemic in order to decrease the infected mosquitoes and humans with low cost. This conclusion further concurs the current control strategy used on Ontario: larviciding and not adulticiding is utilized to eradicate mosquitoes.
Figure 12: The optimal value of the objective functional is calculated using only one control function. Showing w1 , w2 and w3 .
5.2
Optimal control with effect of the seasonal variations
Building on Section 4, and using numerical simulations, we carry out numerical experiment to study the impact of seasonal variations on control of WNv. We consider the same objective function (5.1), by assuming that c1 = c2 = 1 and c3 = 10 and use the same values of the other weight factors (similar to the previous section). From the available information on the schedule time of control in Ontario-Canada, it is worth mentioning that the control is usually applied for 50 to 60 days between May and August. Using these above fact, we performed different simulations using various initial populations. We will explore the best time and strategy to apply the control. In Fig.13, we investigated and compared numerical results of control for 50 days at different times of starting the control: case 1 (middle of May); case 2 (very early in July) and case 3 (middle of August). We perceived that, when = 0.5 (see Fig.13(a) and (b)), if we start the control in July (case 2), the number of infected mosquitoes and humans and their highest peaks are lower than that of the other cases. It is worth-noting here that the J value of cases (1), (2) and (3) are 15282, 20814, and 24700 respectively. However, if = 0.25 (see Fig.13(c) and (d)), case 3 (start the control on August) is the best time of starting the control. It is worth-noting here that the J value when = 0.5 will slightly increase 27
(a) Infected humans, = 0.5.
(b) Infected adult mosquitoes, = 0.5.
(c) Infected humans, = 0.25.
(d) Infected adult mosquitoes, = 0.25.
Figure 13: Time series of model (5.2) with seasonal impact (in case of = 0.5 and = 0.25) under different times of optimal control, i.e. without control, and with control in three cases, case 1; case 2; and case 3, with dashed line denoted by the time of applied control.
than the cost of control when = 0.25. Thus, when we focus the control in the course of 50 consecutive days, the optimal period to start depends on change in the temperatures. Whenever seasonal variations become stronger, our results recommend starting the control earlier. In Fig.14, we used the same values of parameters and the same initial population as for Fig.13 (when = 0.5) to compare the results of different optimal control strategies: (I) applying the control at three different times, each for 17 consecutive days (early in May, June and July); (II) applying the control at three different times, each for 17 consecutive days (end of May, June and July); (III) applying the control during 50 consecutive days in July. We can conclude that the best strategy of control depends on occurrence of infected cases of mosquitoes at a higher rate during May. If this takes place, then the best strategy is to apply the control at three different times (each time 17 days). Otherwise, it is best to apply the control one time for 50 days. Also the J values in cases (I) and (II) are almost the same but they are a little higher than the cost in case (III).
28
(a) Infected humans.
(b) Infected adult mosquitoes.
Figure 14: Time series of model (5.2) with seasonal impact (in case of = 0.5) under different strategies of optimal control, i.e. without control, and with control in three cases: (I) 3 different times each one 17 days start early in May; (II) 3 different times each one 17 days start end of May; (III) 50 days time. Dashed line denoted by the time of applied control.
6
Conclusions
This paper presented a comprehensive, deterministic model of differential equations for the transmission dynamics of WNv with and without seasonality. We started by analyzing the model without seasonality and verified the existence of backward bifurcation where the stable disease free equilibrium co-exists with a stable endemic equilibrium. The existence of the backward bifurcation indicated that the spread of the virus when R0 is nearly below unity could depend on the initial sizes of the sub-population of the model. After that, we considered the model with seasonal variations - by assuming that the birth rate of mosquitoes follows a periodic pattern - to study the impact of seasonal variations of the mosquito population on the dynamics of WNv. In this latter model, we proved the existence of periodic solutions under specific conditions using the classical Floquet method, which is based on the calculation of the monodromy matrix and an analysis of its eigenvalues. Moreover, we introduced and calculated the basic reproduction number for this seasonal forced model. Furthermore, we deduced from numerical analysis that the strength of seasonality increases the number of infections. The two models have been extended to assess the impact of some anti-WNv control measures, by reformulating the models as optimal control problems. This entails the use of three control functions: adulticide, larvicide and human protection. The results were analysed to determine the necessary conditions for the existence of an optimal control, using Pontrayagins maximum principle. From our numerical results, we found that Larvicide is the most effective strategy to control an ongoing epidemic in reducing disease cost when we 29
apply only one control. The outcomes further stressed the importance of considering the other animals that could be infected in any region and its effect regarding the cost of control. Finally, the numerical results identified the time of applying the control to achieve the best control strategy. This work strongly justifies the importance of carefully taking into account the impact of the seasonal variation when applying the control.
7
Appendix
In the following theorem, we present the adjoint system and control characterization. Theorem 7.1. For an optimal control variable (w1∗ , w2∗ , w3∗ ) and optimal corresponding state solutions Q∗ , then there exist adjoint variables λj , j = 1, ...., 15, satisfying, λ´1 = −a3 − c1 w1 + λ1 (dL + d0 w1 ) + (λ1 − λ3 )mL (1 − w1 ), λ´2 = −a3 − c1 w1 + λ2 (dL + d0 w1 ) + (λ2 − λ4 )mL (1 − w1 ), λ´3 = −a3 − c2 w2 − λ1 (rm (1 − w2 ) − λ3 (dm + d0 w2 ) + (λ3 − λ4 ) βm bm (BN1i +B2i ) , λ´4 = −a3 − c2 w2 − rm (1 − w2 )λ1 + λ4 (dm + d0 w2 ) + (λ1 − λ2 )qrm (1 − w2 )
λ´5
bm S + (λ5 − λ6 ) βb bmNB1s + (λ8 − λ9 ) βb bmNB2s + (λ11 − λ12 ) βhN (1 − w2 ), βb bm (N −B1s ) 1i +B2i ) = λ5 db + (λ4 − λ3 ) βm bm (B M + (λ − λ ) M s 5 6 i N2 N2
λ´6
+ (λ12 − λ11 ) βhNbm2 S Mi (1 − w3 ) , βm bm (N −(B1i +B2i )) = λ6 δ1 − λ7 ν1 + (λ3 − λ4 ) Ms + (λ6 − λ5 ) N2
λ´7
+ (λ12 − λ11 ) βhNbm2 S Mi (1 − w3 ) , βm bm (B1i +B2i ) βb bm B1s = λ7 db + (λ4 − λ3 ) M + (λ − λ ) M s 6 5 i 2 2 N N
+ (λ9 − λ8 )
+ (λ9 − λ8 )
βb bm B2s Mi N2
βb bm B2s Mi N2
βb bm B1s Mi N2
+ (λ12 − λ11 ) βhNbm2 S Mi (1 − w3 ) , 1i +B2i ) = λ6 db + (λ4 − λ3 ) βm bm (B M + (λ6 − λ5 ) βb bNm2B1s Mi s 2 N −B2s ) βh bm S + (λ8 − λ9 ) βb bm (N M Mi (1 − w3 ) , i + (λ12 − λ11 ) N2 N2 + (λ9 − λ8 )
λ´8
βb bm B2s Mi N2
30
λ´9 = λ9 δ2 − λ10 ν2 + (λ3 − λ4 )
λ´10
λ´12
λ´15
βb bm B1s Mi N2
βb bm B2s Mi N2
βb bm B2s Mi N2
βb bm B2s Mi N2
βb bm B1s Mi N2
βh b m S Mi (1 N2
+ (λ9 − λ8 )
βb bm B2s Mi N2
+ (λ9 − λ8 )
βb bm B2s Mi N2
+ (λ12 − λ11 )
βb bm B1s Mi N2
βh bm S Mi (1 N2
− w3 ) − γ, λ14 − rλ15 , βm bm (B1i +B2i ) βb bm B1s + (λ − λ ) = (µh + τ + dh )λ14 − τ λ15 + (λ4 − λ3 ) M M 6 5 s i 2 2 N N
+ (λ9 − λ8 ) λ´14
+ (λ6 − λ5 )
− w3 ) , 1i +B2i ) = −a2 + (γ + µl + r + dh )λ13 (λ4 − λ3 ) βm bm (B M s + (λ6 − λ5 ) N2
+ (λ9 − λ8 ) λ´13
+ (λ12 − λ11 ) βhNbm2 S Mi (1 − w3 ) , βm bm (B1i +B2i ) βb bm B1s = λ11 dh + (λ4 − λ3 ) M + (λ − λ ) M s 6 5 i 2 2 N N βh bm (N −S) βb bm B2s Mi + (λ11 − λ12 ) Mi (1 − w3 ) , + (λ9 − λ8 ) N2 N2 βb bm B1s 1i +B2i ) M Mi + (λ12 − λ13 )α = −a1 + λ12 dh + (λ4 − λ3 ) βm bm (B s + (λ6 − λ5 ) N2 N2 + (λ9 − λ8 )
λ´11
βm bm (N −(B1i +B2i )) Ms N2
+ (λ12 − λ11 ) βhNbm2 S Mi (1 − w3 ) , 1i +B2i ) = −c3 w3 + λ10 db + (λ2 − λ1 ) βm bm (B M s + (λ6 − λ5 ) N2 + (λ9 − λ8 )
βb bm B2s Mi N2
+ (λ10 − λ9 )
+ (λ12 − λ11 ) βhNbm2 S Mi (1 − w3 ) , βm bm (B1i +B2i ) βb bm B1s = dh λ15 + (λ4 − λ3 ) M + (λ − λ ) M s 6 5 i 2 2 N N βh b m S Mi (1 N2
+ (λ12 − λ11 )
− w3 ) ,
with transversality conditions (or final time conditions) λi (T ) = 0,
i = 1, ....., 15.
(7.1)
Furthermore, optimal control functions are given as follows: 1 ∗ w1 = max 0, min U1 , ((λ3 − λ1 )mL Ls + (λ4 − λ2 )mL Li + d0 Ls λ1 + d0 Li λ2 − c1 L) , b1 1 ∗ w2 = max 0, min U2 , (rm M λ1 + (λ2 − λ1 )qrm Mi + λ3 d0 Ms + λ4 d0 Mi − c2 M ) , b2 w3∗
1 βh bm S = max 0, min U3 , ((λ12 − λ11 ) Mi − c3 S) . b3 N
Proof. The adjoint system results from Pontryagin’s Principle [33], ∂~ λ´1 = − , ∂Ls
∂~ λ´2 = − , ∂Li 31
∂~ ............ λ´15 = − . ∂R
The optimality conditions (characterization of the optimal control) given by ∂~ = 0, ∂w1
∂~ = 0, ∂w2
∂~ = 0, ∂w3
on the interior of the control set. Using the bounds on the controls, we obtain the desired characterization.
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