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Optimal Vaccination Strategies for a new Dengue Model with two Serotypes
Proceedings of the 9th Vienna International Conference on Proceedings of the 9th Vienna International Conference on Proceedings ofModelling the 9th Vienna International Conference on Mathematical Mathematical Proceedings ofModelling the 9th Vienna International Conference on Mathematical Modelling Vienna, Austria, February 21-23, 2018 Available online at www.sciencedirect.com Vienna, Austria, February 21-23, 2018 Mathematical Modelling Vienna, Austria, February 21-23, 2018 Vienna, Austria, February 21-23, 2018
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Optimal Vaccination Strategies for a new Optimal Vaccination Strategies for a new Optimal Vaccination Strategies for a Optimal Vaccination Strategies for a new new Dengue Model with two Serotypes Dengue Model with two Serotypes Dengue Model with two Serotypes Dengue Model with two Serotypes ∗ ∗∗
Kurt Chudej ∗ ,, Anne Fischer ∗∗ Kurt ∗∗ Kurt Chudej Chudej ∗∗∗ , Anne Anne Fischer Fischer ∗∗ Kurt Chudej , Anne Fischer ∗∗ ∗ u r Wissenschaftliches a t Bayreuth, ∗ Lehrstuhl f¨ u Wissenschaftliches Rechnen, Rechnen, Universit¨ Universit¨ a ∗ Lehrstuhl f¨ ∗ f¨ urrGermany Universit¨ att Bayreuth, Bayreuth, Wissenschaftliches Rechnen, Bayreuth, (e-mail: [email protected]) ∗ Lehrstuhl f¨ urGermany [email protected]) Rechnen, Universit¨ at Bayreuth, Bayreuth, ∗∗ Lehrstuhl Bayreuth, Germany (e-mail: (e-mail: [email protected]) f¨ u r Ingenieurmathematik, Universit¨ a t Bayreuth, ∗∗ Lehrstuhl Bayreuth, Germany (e-mail: [email protected]) f¨ ur Ingenieurmathematik, Universit¨ at Bayreuth, Bayreuth, Bayreuth, ∗∗ ∗∗ Lehrstuhl f¨ u r Ingenieurmathematik, Universit¨ a Bayreuth, Germany; current address: Witt Gruppe, Weiden, Germany ∗∗ Lehrstuhl Lehrstuhl f¨ ur current Ingenieurmathematik, Universit¨ att Bayreuth, Bayreuth, Bayreuth, Germany; address: Witt Gruppe, Weiden, Germany Germany; current address: Witt Gruppe, Weiden, Germany Germany; current address: Witt Gruppe, Weiden, Germany Abstract: Dengue fever is a virus infection transmitted by asian tiger mosquitos which affects Abstract: Dengue fever is virus transmitted by asian tiger mosquitos which affects Abstract: Dengue fever is aa population. virus infection infection transmitted bycases asian tiger mosquitos which affects more than half of the world’s Previous dengue in Europe could be traced back Abstract: Dengue fever is a population. virus infection transmitted bycases asian tiger mosquitos which affects more than half of the world’s Previous dengue in Europe could be traced back more than half of the world’s population. Previous dengue cases in Europe could be traced back to an infection outside of Europe, so called non autochthonous cases. Unfortunately, mosquitos more than half outside of the world’s population. Previous dengue casescases. in Europe could be traced back to an infection of Europe, so called non autochthonous Unfortunately, mosquitos to an infection outside of Europe, so called non autochthonous cases. Unfortunately, mosquitos capable of are (re-)invading Europe Germany. Thereto an infection outside ofdengue Europe, called non autochthonous cases.including Unfortunately, mosquitos capable of transmitting transmitting dengue aresocurrently currently (re-)invading Europe including Germany. Therecapable of dengue are currently (re-)invading Europe including Germany. Therefore, possible outbreaks of Europe seem in future. possibility is capable of transmitting transmitting are in currently Germany. Therefore, possible outbreaks dengue of dengue dengue in Europe(re-)invading seem possible possibleEurope in the theincluding future. This This possibility is fore, possible outbreaks of dengue in Europe seem possible in the future. This possibility is emphasized by the outbreaks of Chikungunya fever (transmitted by the same mosquitos) in fore, possiblebyoutbreaks of dengue in Europe seem possible in thebyfuture. Thismosquitos) possibility in is emphasized the outbreaks of Chikungunya fever (transmitted the same emphasized by the outbreaks of Chikungunya fever (transmitted by the same mosquitos) in France and Italy in the year 2007. The dengue virus comes in four different serotypes. Extensive emphasized by the outbreaks of Chikungunya fevercomes (transmitted by the serotypes. same mosquitos) in France and Italy in the year 2007. The dengue virus in four different Extensive France and Italy in the year 2007. The dengue virus comes in four differentdengue serotypes. Extensive research was performed previously for several variants of serotype model. In France and in the year 2007. The comes four differentdengue serotypes. Extensive research wasItaly performed previously for dengue several virus variants of aainone one serotype model. In this this research was performed previously for several variants of a one serotype dengue model. In this paper we consider a new version of an optimal control problem for a vaccination strategy research was performed previously for several variants of a one serotype dengue model. Inwith this paper we consider a new version of an optimal control problem for a vaccination strategy with paper we consider a new version of an optimal control problem for a vaccination strategy with two dengue serotypes. paper we consider a new version of an optimal control problem for a vaccination strategy with two dengue serotypes. two dengue serotypes. two dengue © 2018, IFACserotypes. (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Dengue, vaccination, two serotypes, coinfection, vector, optimal control, nonlinear Keywords: Dengue, vaccination, Keywords: Dengue, vaccination, two two serotypes, serotypes, coinfection, coinfection, vector, vector, optimal optimal control, control, nonlinear nonlinear programming. Keywords: Dengue, vaccination, two serotypes, coinfection, vector, optimal control, nonlinear programming. programming. programming. 1. aa new 1. INTRODUCTION INTRODUCTION new two two serotype serotype model. model. Moreover Moreover in in certain certain regions regions 1. INTRODUCTION a new two serotype model. Moreover in certain regions currently none, one or only two different serotypes are 1. INTRODUCTION a new twonone, serotype model. Moreover in certain regions currently one or only two different serotypes currently none, one or only two different serotypes are are present. currently none, one or only two different serotypes are present. Dengue (and Chikungunya) fever is transmitted via vecDengue (and Chikungunya) fever is transmitted via vecpresent. Dengue (and Chikungunya) fever is transmitted viacalled vec- present. tors, i.e. (female) mosquitos the species, Dengue Chikungunya) is transmitted viacalled vec2. WITH tors, i.e. (and (female) mosquitos of offever the Aedes Aedes species, so so 2. NEW NEW TWO TWO SEROTYPE SEROTYPE DENGUE DENGUE MODEL MODEL WITH WITH tors, i.e. (female) mosquitos of the Aedes species, so called asian tiger mosquitos, see Moutailler al. A 2. NEW TWO SEROTYPE DENGUE MODEL tors, i.e. (female) mosquitos the Aedeset species, so called VACCINATION asian tiger mosquitos, see of Moutailler et al. (2009). (2009). A 2. NEW TWO SEROTYPE DENGUE MODEL WITH VACCINATION asian tiger mosquitos, see Moutailler et al. (2009). A human to human transmission of dengue is not possible. VACCINATION asian tiger mosquitos, see Moutailler A human to human transmission of dengueet isal. not(2009). possible. VACCINATION human transmission of dengue is Currently, the mosquito (e.g. Aedes albopictus) is reThe following new two serotype dengue model, see Fig. 1, human to to human human transmission of dengue is not not possible. possible. Currently, the mosquito (e.g. Aedes albopictus) is reThe following new two dengue model, see 1, Currently, the mosquito (e.g. Aedes albopictus) is reinvading Europe, see et (2017, 2014); al. The following new two serotype serotype dengue model, see Fig. Fig. 1, pediatric vaccination and imperfect random mass Currently, the mosquito invading Europe, see Becker Becker(e.g. et al. al.Aedes (2017, albopictus) 2014); Frank Frankiset et real. with The following new two serotype dengue model, see Fig. 1, with pediatric vaccination and imperfect random mass invading Europe, see Becker et al. (2017, 2014); Frank et al. (2014); Benedict et al. (2007); Werner and Kampen (2015); with pediatric vaccination and imperfect random mass vaccination with waning immunity is a generalization of invading Europe, see Becker et al. (2017, 2014); Frank et al. (2014); Benedict et al. (2007); Werner and Kampen (2015); vaccination with pediatric and imperfect random mass withvaccination waning immunity is a generalization of (2014); Benedict et al. (2007); Werner and Kampen (2015); Scholte et al. (2008); Takken and Knols (2007). The invawith waning immunity is of the one of et (2014); Benedict et al.Takken (2007); and Werner and(2007). Kampen (2015); Scholte et al. (2008); Knols The inva- vaccination vaccination withdengue waningmodel immunity is aa generalization generalization of the one serotype serotype dengue model of Rodrigues Rodrigues et al. al. (2013b), (2013b), Scholte et Takken and (2007). The invasion is further favoured by global warming Thomas al. the one serotype dengue model of Rodrigues et al. (2013b), see Remark 1 later. We use the following colours: SusScholte et al. al. (2008); (2008); Takken and Knols Knols (2007). The et invasion is further favoured by global warming Thomas et al. the one serotype dengue model of Rodrigues et al. (2013b), see Remark 1 later. We use the following colours: Sussion is further favoured by global warming Thomas et al. (2014, 2012). dengue virus, serologically belonging see Remark 1 later. Weinuse the(imperfectly) following colours: Suscompartiments blue; vaccinated sion is further favoured by global Thomas et al. ceptible (2014, 2012). The The dengue virus, warming serologically belonging see Remark 1 later. Weinuse the(imperfectly) following colours: Susceptible compartiments blue; vaccinated (2014, 2012). The dengue virus, serologically belonging to the group of flaviviruses, consists of four serotypes, ceptible compartiments in blue; (imperfectly) vaccinated humans in cyan; with serotype 1/2 infected in red/orange; (2014, 2012). The dengue virus, serologically belonging to the group of flaviviruses, consists of four serotypes, humans ceptible in compartiments in blue; (imperfectly) vaccinated cyan; with serotype 1/2 infected in red/orange; to the group of flaviviruses, consists of four serotypes, DENV-1, DENV-2, DENV-3 and DENV-4 (WHO (2017)). in with 1/2 in healthy in green. The human population to the group of flaviviruses, consists of (WHO four serotypes, DENV-1, DENV-2, DENV-3 and DENV-4 (2017)). humans humans aquatic in cyan; cyan;mosquitos with serotype serotype 1/2 infected infected in red/orange; red/orange; healthy mosquitos in The human DENV-1, DENV-2, DENV-3 and DENV-4 (WHO (2017)). After an infection one of the serotypes, one is lifelong healthy aquatic aquatic mosquitos in green. green. Thefollowing human population population (subscript h) is subdivided into the compartDENV-1, DENV-2,with DENV-3 and DENV-4 (WHO (2017)). After an infection with one of the serotypes, one is lifelong healthy aquatic mosquitos in green. Thefollowing human population (subscript h) is subdivided into the compartAfter an against infection with one of the serotypes, one isnolifelong immune this serotype. However, there are cross(subscript h) is subdivided into the following compartsusceptible humans, V vaccinated humans, S After an against infection with one of the serotypes, immune this serotype. However, thereone areisnolifelong cross- ments: h h (subscript is subdivided into Vthe following compartsusceptible humans, vaccinated humans, ments: Sh h) h immune against this serotype. However, there are no crossreactivities: An initial infection with one of the dengue 1 2 susceptible humans, V vaccinated humans,2 ments: S h h I with serotype 1 infected humans, I with serotype immune against this serotype. However, there are no crossreactivities: An initial infection with one of the dengue Iments: h susceptible humans, Vh vaccinated 1 2 h h humans,2 S with serotype 1 infected humans, with serotype h h Ih 1 2 reactivities: An initial infection with one of the dengue 1 2 serotypes does not confer immunity against any of the 1 h Ihh1 with serotype 1 h1infected humans, Iserotype with serotype 2 infected humans, R previously with 1 infected reactivities:does An not initial infection with against one of the dengue 2 h serotypes confer immunity any of the h I with serotype 1 infected humans, I with serotype 2 infected humans, R previously with serotype 1 infected serotypes does not confer immunity against any of the h h other three. Therefore, one could fall ill up to four humans, Rhhh111 previously withresistant serotype 1 serotype infected humans which recovered and are now to serotypes does not confer immunity against any oftimes the infected other three. Therefore, one could fall ill up to four times infected humans, R previously with serotype 1 infected humans which recovered and are now resistant to serotype other three. Therefore, one could fall ill up to four times h with four different serotypes of 2 which recovered and are 2now resistant to serotype 1, R with serotype infected humans which other the three. Therefore, could fall ill up to An fourimportimes humans with the four different one serotypes of dengue. dengue. An impor2 previously h humans which recovered and are 2now resistant to 1, R with serotype infected humans which 2 with the four different serotypes of dengue. An impor2 previously tant theory in connection with the occurrence of several 12 serotype h 1, R previously with serotype 2 infected humans which recovered and are now resistant to serotype 2, I with the four different serotypes of dengue. An impor2 h tant theory in connection with the occurrence of several recovered 12 humans h previously h 1, R with serotype 2 infected humans which and are now resistant to serotype 2, I 12 humans h tant theory in connection with the occurrence of several 12 serotypes is that of the infection-enhancing antibody, in h recovered and are now resistant to serotype 2, Ihh12 humans which were firstly infected with serotype 1, recovered, and tant theory in connection with the occurrence of several serotypes is that of the infection-enhancing antibody, in recovered and are now resistant to serotype 2, I humans which were firstly infected with serotype 1, recovered, and serotypes is that of the infection-enhancing antibody, in h which it is in the case of a second infection, 21 were firstlywith infected with 2, serotype 1, recovered, and are now infected serotype I humans which were serotypes isassumed that of that, the infection-enhancing antibody, in which which it is assumed that, in the case of a second infection, 21 h which were firstly infected with serotype 1, recovered, and are now infected with serotype 2, I humans which were which it is assumed that, in the case of a second infection, 21 21 humans which were the already existing antibodies form an antigen-antibody h are now infected with serotype 2, I firstly are now which it is assumed in the form case of second infection, are the already existing that, antibodies ana antigen-antibody nowinfected infectedwith withserotype serotype2, 2, recovered, Ihhh21 humansand which were firstly infected with serotype 2, recovered, and are now the already existing antibodies form an antigen-antibody complex with the newly added virus, which is difficult to infected with serotype 2, recovered, and are now ˜˜ h humans the already existing antibodies form an antigen-antibody complex with the newly added virus, which is difficult to firstly infected with serotype 1, R which are resistant firstly infected with serotype 2, recovered, and are now infected with serotype 1, which are resistant complex with the newly added virus, which is to control the body, see Gubler (1998). Secondary infec˜ hh humans infected with serotype 1, R R humans which are resistant complexby with the newly added virus, which is difficult difficult to to control by the body, see Gubler (1998). Secondary infecserotype 1 and 2. h humans which are resistant ˜ infected with serotype 1, R control by the body, see Gubler (1998). Secondary infech tions potentially more than to serotype serotype 11 and and 2. 2. controlare the body, see Gubler (1998). Secondary infec- to tions arebythus thus potentially more dangerous dangerous than primary primary to serotype 1 mosquito and 2. population (subscript m) is subditions are thus potentially more dangerous than primary The (female) infections. This theory would, among other things, explain tions are thus more dangerous than primary mosquito population (subscript m) is subdiinfections. Thispotentially theory would, among other things, explain The The (female) (female) mosquito population (subscript m) is larvae, subdiinfections. This theory among other explain into four compartments: A of eggs, why an variant of fever occurs m amount The (female) mosquito population (subscript m) is subdiinfections. This amniotic theory would, would, among other things, things, vided into four compartments: A why an acute acute amniotic variant of dengue dengue fever explain occurs vided amount of eggs, larvae, m 1 vided into four compartments: Am amount of eggs, larvae, why an acute amniotic variant of dengue fever occurs and pupae, S susceptible adult mosquitos, I mainly in secondary infections. Mathematical models of adult m 1 m m vided into four compartments: A amount of eggs, larvae, why an inacute amniotic variant Mathematical of dengue fever occurs mainly secondary infections. models of and pupae, S susceptible adult mosquitos, I m 1 1 adult 2 and pupae,infected Sm susceptible adult1,mosquitos, Im mainly fever in secondary infections. Mathematical modelse.g. of mosquitos m dengue with one serotype models were presented with serotype I adult mosquitos 1 adult m m 2 m m and pupae,infected Sm susceptible adult1,mosquitos, Im adult mainly fever in secondary Mathematical modelse.g. of mosquitos dengue with one infections. serotype models were presented with serotype mosquitos 2 2 adult mosquitos infected with serotype 1, IIm adult mosquitos dengue fever with one models were presented e.g. by Esteva and Vargas (1999), Rodrigues al. with serotype 2. For have assumed 2 we m mosquitos infected with 1, Im adult dengue fever with one serotype serotype wereet e.g. infected by Esteva and Vargas (1999), models Rodrigues etpresented al. (2013a), (2013a), infected with serotype 2. serotype For simplicity simplicity havemosquitos assumed m we by Esteva and Vargas (1999), Rodrigues et al. (2013a), infected with serotype 2. For simplicity we have assumed Rodrigues et al. (2014). In an intermediate step towards a that a mosquito cannot carry both dengue viruses of infected with serotype 2. For simplicity we have assumed by Esteva and Vargas (1999), Rodrigues et al. (2013a), Rodrigues et al. (2014). In an intermediate step towards a that a mosquito cannot carry both dengue viruses of Rodrigues et al. (2014). In an intermediate step towards a that a mosquito cannot carry both dengue viruses mathematical four serotype model of dengue, we introduce serotype 1 and 2. cannot carry both dengue viruses of of Rodrigues et al. (2014). In an intermediate step towards a that a mosquito mathematical serotype 11 and and 2. 2. mathematical four four serotype serotype model model of of dengue, dengue, we we introduce introduce serotype mathematical four serotype model of dengue, we introduce serotype 1 and 2.
Copyright © 2018, 2018 IFAC 1 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018 IFAC 1 Copyright ©under 2018 responsibility IFAC 1 Control. Peer review of International Federation of Automatic Copyright © 2018 IFAC 1 10.1016/j.ifacol.2018.03.003
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dRh1 I2 = ηh Ih1 − (δ1 Bβmh m + µh )Rh1 , dt Nh I1 dRh2 2 = ηh Ih − (δ2 Bβmh m + µh )Rh2 , dt Nh 2 Im dIh12 1 = δ1 Bβmh R − (µh + ηh )Ih12 , dt Nh h I1 dIh21 = δ2 Bβmh m Rh2 − (µh + ηh )Ih21 , dt Nh ˜h dR ˜h, = ηh (Ih12 + Ih21 ) − µh R dt and the mosquito population
The following general assumptions hold: Both mosquitos (=vectors) and humans are neither born infected nor resistant. The size of the human population (Nh ) is constant at any time. Furthermore the size of the human population is sufficiently large, such that the state variables can be assumed to be continuous real functions of time. Neither emigration nor immigration are considered. The passing of a certain percentage of the population is modeled through the proportionality factor µh , with 1/µh denoting an average lifespan. The same holds for the adult mosquito population with factor µm and the mosquitos in the aquatic phase with factor µA . Humans recover from the disease (of any serotype) at a rate ηh . A mosquito in the aquatic phase requires 1/ηA days to grow up. Mosquitos from compartment Am are removed with proportionality factor ηA and join the compartment Sm . Every (female) adult mosqito leaves ϕ eggs at one breeding place per day. The reproduction of the mosquito population is modeled using logistic growth. This considers the possible amount of eggs a (human-build) breeding place can hold. As the capacity bound kNh is chosen. Homogeneity between the populations is assumed, thus, with the same probability every female mosquito bites every human host. Per day, one female mosquito, averagely, bites B-times and human individuals become infected with the probability βmh . The product Bβmh represents the contact rate between infected female mosquitos and susceptible humans. The contact rate of infected humans and susceptible female mosquitos is modeled analogously. This model does not consider an impact of the disease on lifespans.
Am dAm 1 2 = ϕ(1 − )(Sm + Im (t) + Im )− dt kNh − (ηA + µA )Am , dSm = η A Am − dt I 1 + Ih2 + Ih12 + Ih21 + µm )Sm , (2) − (Bβhm h Nh 1 I 1 + Ih21 dIm 1 = Bβhm h Sm − µ m I m , dt Nh 2 2 12 I + Ih dIm 2 = Bβhm h Sm − µ m I m . dt Nh Remark 1: The specialization δ1 = δ2 = 0 and Ih := Ih1 + 1 2 Ih2 , Rh := Rh1 +Rh2 , Im := Im +Im , together with the initial 1 values fulfilling Ih (0) = Ih (0) + Ih2 (0), Ih12 (0) = Ih21 (0) = ˜ h ≡ 0 the one ˜ h (0) = 0 yields due to I 12 ≡ I 21 ≡ R R h h serotype dengue model with vaccination of Rodrigues et al. (2013b):
Only susceptible human subjects are vaccinated with a vaccine that acts against both serotypes, but infected humans are not vaccinated here, even after an initial infection. Depending on the vector they come into contact with, humans either change to the compartment Ih1 or Ih2 . After a primary infection, the same holds for the compartments Ih12 , Ih21 .
dSh dt dVh dt dIh dt dRh dt
The vaccination model includes a pediatric vaccination with a time-dependent control p(t) ∈ [0, 1] and an imperfect random mass vaccination with a time-dependent control ψ(t) ∈ [0, 1]. Two important constants appear: The efficiency of the vaccine is modeled by 1 − σ ∈ [0, 1]. σ = 0 denotes a high efficiency of the vaccine, whereas σ = 1 indicates that the vaccine has no effect. If σ > 0 also certain vaccinated individuals can be infected, see Fig. 1. A possible waning imunity is modeled by θ ∈ [0, 1]. In the numerical computations we use σ = 0.2 (efficiency 80%) and θ = 0.05.
= ηh Ih − µh Rh
Am dAm = ϕ(1 − )(Sm + Im ) − (ηA + µA )Am dt kNh dSm Ih (4) = ηA Am − (Bβhm + µm )Sm dt Nh dIm Ih = Bβhm Sm − µ m I m dt Nh 2 ˜ h ≡ 0 yields for The special choice Ih2 ≡ Im ≡ Rh2 ≡ R arbitrary values of δi also the ODE (3), (4) with Ih := Ih1 , 1 Im := Im , Rh := Rh1 .
This results in the ODE for the human population dSh = (1 − p)µh Nh + θVh − dt 2 I 1 + Im − (Bβmh m + ψ + µh )Sh , Nh dVh = pµh Nh + ψSh − dt 2 I 1 + Im − (θ + σBβmh m + µh )Vh , Nh I1 dIh1 = Bβmh m (Sh + σVh ) − (ηh + µh )Ih1 , dt Nh I2 dIh2 = Bβmh m (Sh + σVh ) − (ηh + µh )Ih2 , dt Nh
Im + ψ + µh )Sh Nh Im = pµh Nh + ψSh − (θ + σBβmh + µh )Vh Nh (3) Im = Bβmh (Sh + σVh ) − (ηh + µh )Ih Nh
= (1 − p)µh Nh + θVh − (Bβmh
3. EQUILIBRIA AND THE BASIC REPRODUCTION NUMBER In this section we assume that p and ψ are constants. We consider the relevant set Ω of all non-negative compartments of ODE (3), (4) s.t. Sh + Vh + Ih + Rh ≤ Nh , Am ≤ kNh , and Sm + Im ≤ mNh for the theoretical ˜ of all non-negative analysis. We consider the relevant set Ω compartments of ODE (1), (2) s.t. Sh +Vh +Ih1 +Ih2 +Rh1 + ˜ h ≤ Nh , Am ≤ kNh , and Sm + I 1 + Rh2 + Ih12 + Ih21 + R m 2 Im ≤ mNh for the theoretical analysis.
(1)
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Fig. 1. Compartment model of the two serotype dengue model with vaccination ˜ is positive invariant for the Lemma 2: The set Ω [resp. Ω] ODE (3), (4) [resp. (1), (2)].
disease free equilibrium: only healthy (susceptible or vaccinated) humans, only healthy (but susceptible) mosquitos.
Proof: The ODE can be written as X˙ = A(X)X + F with a Metzler Matrix A(X) and F ≥ 0 and apply Abate et al. (2009); Mitkowski (2008).
˜3 = (S ∗ , V ∗ , I ∗ , 0, R∗ , 0, 0, 0, 0 | A∗ , S ∗ , I ∗ , 0) E m m m h h h h ˜4 = (S ∗ , V ∗ , 0, I ∗ , 0, R∗ , 0, 0, 0 | A∗ , S ∗ , 0, I ∗ ) E m m m h h h h endemic boundary equilibria.
We use the abbreviation M = ϕηA − ηA µm − µA µm .
∗ ∗ , Im in the The lengthy formulas of Sh∗ , Vh∗ , Ih∗ , Rh∗ , A∗m , Sm ˜ ˜ endemic boundary equilibria E3 and E4 are the same as in the endemic equilibrium E3 of Theorem 3.
Theorem 3: The ODE (3), (4) has in Ω (depending on the parameter values) at most the equilibria:
Proof: Remark 1 with Theorem 3.
h +θ) Nh (µh p+ψ) E1 = ( Nh ((1−p)µ , µh +θ+ψ , 0, 0 | 0, 0, 0) µh +θ+ψ trivial equilibrium: only healthy (susceptible or vaccinated) humans, no mosquitos.
Remark 5: Suppose the presented model is sufficiently accurate. For a city without vaccination p = ψ = 0 holds. In a city without asian tiger mosquitos (and no dengue infected travellers) we are currently in the trivial equilibrium E1 . In a city with an asian tiger mosquito population and no dengue infected travellers we are currently in the equilibrium E2 .
h +θ) Nh (µh p+ψ) h M αkNh M , µh +θ+ψ , 0, 0 | kN E2 = ( Nh ((1−p)µ µh +θ+ψ ϕηA , ϕµm , 0) disease free equilibrium: only healthy (susceptible or vaccinated) humans, only healthy (but susceptible) mosquitos.
∗ ∗ E3 = (Sh∗ , Vh∗ , Ih∗ , Rh∗ | A∗m , Sm , Im ) endemic equilibrium.
Question 6: What happens if we introduce a small number of dengue infected travellers into a city with no asian tiger mosquito population?
Proof: See the MAPLE computation in Fischer (2016), the formula for E3 consists of several pages. Theorem 4: The ODE (1), (2) has the following equilib˜ for certain values of the parameters: ria, which are in Ω ˜1 = ( Nh ((1−p)µh +θ) , Nh (µh p+ψ) , 0, . . . , 0 | 0, . . . , 0) E µh +θ+ψ µh +θ+ψ trivial equilibrium: only healthy (susceptible or vaccinated) humans, no mosquitos.
Answer: If we assume, that the presented model is sufficiently accurate (i.e. currently only up to two of the four serotypes are virulent and especially other mosquitos do not transmit the disease), we will again approach the trivial equilibrium E1 . (Proof: Solution of the initial value problem.)
˜2 = ( Nh ((1−p)µh +θ) , Nh (µh p+ψ) , 0, . . . , 0 | E µh +θ+ψ µh +θ+ψ kNh M αkNh M ϕηA , ϕµm , 0, 0)
Question 7: What happens if we introduce a (sufficiently) small number of dengue infected travellers into a city with a previously healthy asian tiger mosquito population?
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humans
humans time [days]
time [days]
Fig. 2. Numerical simulation of ODE (1),(2), humans: Ih1 = Ih2 , Rh1 = Rh2 , Ih12 = Ih21
Fig. 3. Numerical simulation of ODE (1),(2), zoom on infected humans: Ih1 = Ih2 , Ih12 = Ih21
In order to get an answer, we have to do some computation. Theorem 8: The basic reproduction number of the ODE (3), (4) and of the ODE (1), (2) is given by B 2 β β kM((1 − p)µ + θ + σ(µ p + ψ)) 12 hm mh h h . R0 = ϕµ2m (ηh + µh )(µh + θ + ψ) Proof: Application of the next generation approach (van den Driessche and Watmough (2002)), for details see Fischer (2016). Certain special cases of this result for the ODE (3), (4) can be found in Rodrigues et al. (2014). ˜2 are Theorem 9: The disease free equilibria E2 and E locally asymptotically stable if R0 < 1 and instable if R0 > 1.
mosquitos
time [days]
Proof: See van den Driessche and Watmough (2002).
Fig. 4. Numerical simulation of ODE (1),(2), mosquitos: 1 2 Im = Im
Partial answer to Question 7: Assume, Theorem 9 can be ˜2 globally stable. If the model paramestrengthened to E ters fulfill R0 < 1 the solution will approach the disease ˜2 , which is favourable. Otherwise the free equilibrium E disease will become endemic, which is not favourable.
In the human population the total number of infected persons is almost doubled in the event of an outbreak, whereas the increase in the total infected population of the vector is 1.5 times higher compared to the one serotype model. This is a significant and unexpected increase since the initial values for the diseased persons differ from those in the basic model only in that the Ih (0) = 38 infected humans are divided into Ih1 (0) = Ih2 (0) = 19 in each of the two serotypes. But note the change from δ1 = δ2 = 0 in the one serotype model to δ1 = δ2 = 1 in the two serotype model. Another aspect that should be noticed is the number of resistant individuals. At the end of the observation period, not all individuals are immune to both serotypes, but about 7 000 individuals remain in compartment Rh1 and Rh2 . Therefore, some people were not affected by secondary infections. Although the total number of inital infected humans was the same, the presence of another serotype has a considerable effect on the severity of a disease outbreak. In addition, many more risky secondary infections occur. If, thus, infections with dengue are identified, it has to be ascertained as soon as possible whether only one serotype is present or a second serotype has to be dealt with. Since secondary infections also increase the risk of a haemorrhagic course of dengue fever, in the presence of several serotypes effective countermeasures should be initiated as soon as possible.
4. NUMERICAL SIMULATION We consider a numerical simulation with δ1 = δ2 = 1 and without vaccination (p = ψ = 0) for the following initial values Ih1 (0) = Ih2 (0) = 19, Ih12 (0) = Ih21 (0) = 0, Vh (0) = 0, Rh (0) = 0, Sh (0) = Nh − [Ih1 (0) + Ih2 (0) + Ih12 (0) + Ih21 (0)] − 1 Vh (0) − Rh (0), Am (0) = kNh , Sm (0) = mNh , Im (0) = 2 Im (0) = 0 and the following parameters B = 0.8, βmh = 1 , ηh = 13 , µm = 0.1, ϕ = 6, βhm = 0.375, µh = 80·365 µA = 0.25, ηA = 0.08, m = 3, k = 3, Nh = 383 000. With the numerical simulation we obtain approximately the same results as for the one serotype model. There is an outbreak of the disease between the 50th and 100th day, after which no further infections can be detected for a period of one year. The number of individuals with an initial infection is approximately 34 000 for each serotype at the time of the outbreak. There holds: Ih1 ≡ Ih2 and Ih12 ≡ Ih21 and Rh1 = Rh2 . The peak values for each serotype are 27 000, which together reveal an enormous increase in contrast to the numbers in the basic model. The number of infected mosquitos is approximately 167 000 for each serotype at the time of the outbreak. There holds: 1 2 ≡ Im . Im 4
Proceedings of the 9th MATHMOD Vienna, Austria, February 21-23, 2018
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costs 1-serotype 2-serotype 0.0701847 0.221746 0.11339 0.450337 0.0279303 0.0852322
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Table 1. Costs of optimal control in the 3 cases A, B, C
i1h
5. OPTIMAL CONTROL Numerical solutions were computed by a direct transcription of the optimal control problem to a nonlinear programming problem (NLP). The resulting NLP was solved via the modelling language AMPL (Fourer et al. (2003)) and the interior point method IPOPT (W¨achter and Biegler (2006)). In order to stabilize the numerical solution a scaling of the state variables was applied:
time [days]
Fig. 5. Primary infected humans i1h = i2h with optimal vaccination
Sh Vh 1 I1 I2 R1 R2 , vh = , ih = h , i2h = h , rh1 = h , rh2 = h , Nh Nh Nh Nh Nh Nh 12 21 ˜h I I R h i12 , i21 = h , r˜h = , h = Nh h Nh Nh 1 Am Sm 1 I I2 am = , sm = , im = m , i2m = m . kNh mNh mNh mNh This results in a new scaled version of the ODE for the new state variables. We use the same initial values and parameters as in Section 4.
case
sh =
As a cost functional we use tf J= γD · i(t)2 + γV · ψ(t)2 + γC · p(t)2 dt → min
i12 h
time [days] 21 Fig. 6. Secondary infected humans i12 h = ih with optimal vaccination
p,ψ
0
with scaling factors γD , γV , γC . In addition, we have 21 chosen i(t) = i1h (t) + i2h (t) + i12 h (t) + ih (t) as the total number of infected persons in the two serotype model. (In the one serotype model (3), (4) we use i(t) = Ih /Nh .) It is also conceivable to specify individual weights for all compartments containing infected individuals. Thus, it would be possible to place different weights on the reduction of primary and secondary infections. The aim could be to prevent secondary infections as much as possible, since these have a higher probability of inducing amniotic dengue fever. If one of the serotypes shows a higher virulence or aggravation of secondary infections, it is also possible with individual weights to keep the number of occurring fatalities as low as possible for this serotype. However, here no distinction was made: The parameters δi for the secondary infection were both set to 1, which excludes the effect of an initial infection on a secondary infection.
case
time [days]
Fig. 7. Optimal random mass vaccination ψ(t) in the 3 cases A, B, C whereas Case C is the most cost-effective. In contrast to the model with only one serotype, the costs in cases A and C increase to the triple and in case B even to the quadruple. The occurrence of several serotypes therefore has severe implications on the optimal costs of the used cost functional.
We consider three different scenarios. Case A is a general intermediate scenario. In case B a very expensive (hospital) treatment of infected humans is modeled. Case C simulates the case of a very expensive serum.
If we now analyze the optimal vaccination strategies in Fig. 7, we find that the strategy for the application of random mass vaccination is similar to that of the one serotype model. In cases A and C, the curves reach about 1.5 times the value in comparison to the results of the one serotype model. Moreover, in case B the peak is about double in comparison to the results of the one serotype model. In case C with the expensive serum,
In Table 1 the numerical results of the optimization are presented. The one serotype model is chosen in accordance with Remark 1. In the Figures 5-8 the numerical results for the new two serotype model are demonstrated. It is clear from Table 1 that the occurrence of two serotypes causes huge additional costs if countermeasures only include vaccination. Case B is again the most expensive, 5
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Benedict, M.Q., Levine, R.S., Hawley, W.A., and Lounibos, L.P. (2007). Spread of the tiger: global risk of invasion by the mosquito Aedes albopictus. Vector-borne and zoonotic diseases, 7(1), 76–85. Esteva, L. and Vargas, C. (1999). A model for dengue disease with variable human population. Journal of Mathematical Biology, 38, 220–240. Fischer, A. (2016). Optimale Impfstrategien f¨ ur Dengue-Fieber. Masterarbeit, Universit¨ at Bayreuth, Germany. Fourer, R., Gay, D., and Kernighan, B.W. (2003). AMPL: A modeling language for mathematical programming. Duxbury/Thompson, Pacific Grove, CA. Frank, C., Faber, M., Hellenbrand, W., Wilking, H., and Stark, K. (2014). Wichtige, durch Vektoren u ¨bertragene Infektionskrankheiten beim Menschen in Deutschland. Bundesgesundheitsblatt - Gesundheitsforschung - Gesundheitsschutz, 57(5), 557. Gubler, D.J. (1998). Dengue and dengue hemorrhagic fever. Clinical microbiology reviews, 11(3), 480–496. Mitkowski, W. (2008). Dynamical properties of Metzler systems. Bulletin of the Polish Academy of Sciences, Technical Sciences, 56(4), 309–312. Moutailler, S., Barr, H., Vazeille, M., and Failloux, A.B. (2009). Recently introduced Aedes albopictus in Corsica is competent to Chikungunya virus and in a lesser extent to dengue virus. Tropical Medicine & International Health, 14(9), 1105–1109. Rodrigues, H.S., Monteiro, M.T.T., and Torres, D.F. (2013a). Bioeconomic perspectives to an optimal control dengue model. International Journal of Computer Mathematics, 90(10), 2126–2136. Rodrigues, H.S., Monteiro, M.T.T., and Torres, D.F. (2013b). Dengue in Cape Verde: vector control and vaccination. Mathematical Population Studies, 20(4), 208–223. Rodrigues, H.S., Monteiro, M.T.T., and Torres, D.F. (2014). Vaccination models and optimal control strategies to dengue. Mathematical biosciences, 247, 1–12. Scholte, E.J., Dijkstra, E., Blok, H., De Vries, A., Takken, W., Hofhuis, A., Koopmans, M., De Boer, A., and Reusken, C. (2008). Accidental importation of the mosquito Aedes albopictus into the Netherlands: a survey of mosquito distribution and the presence of dengue virus. Medical and veterinary entomology, 22(4), 352–358. Takken, W. and Knols, B.G. (2007). Emerging pests and vectorborne diseases in Europe, volume 1. Wageningen Academic Pub. Thomas, S.M., Obermayr, U., Fischer, D., Kreyling, J., and Beierkuhnlein, C. (2012). Low-temperature threshold for egg survival of a post-diapause and non-diapause European aedine strain, Aedes albopictus (Diptera: Culicidae). Parasites & vectors, 5(1), 100. Thomas, S.M., Tjaden, N.B., van den Bos, S., and Beierkuhnlein, C. (2014). Implementing cargo movement into climate based risk assessment of vector-borne diseases. International journal of environmental research and public health, 11(3), 3360–3374. van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(12), 29–48. W¨ achter, A. and Biegler, L.T. (2006). On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical programming, 106(1), 25–57. Werner, D. and Kampen, H. (2015). Aedes albopictus breeding in southern Germany, 2014. Parasitology research, 114(3), 831–834. WHO (2017). Dengue and severe dengue. URL http://www.who.int/mediacentre/factsheets/.
case
time [days]
Fig. 8. Optimal pediatric vaccination p(t) in the 3 cases A, B, C the optimal use of pediatric vaccination is identical. The random mass vaccination rate is slightly lower in the case of two serotypes, so that a pronounced vaccination occurs at the time of the disease outbreak. In Case A, on the other hand, a continuous vaccination with 1.8-fold compared to before is advisable. Overall, the outbreak of the second infections can only be controlled indirectly by vaccination strategies. However, this is due to the structuring of the model. Comparing the use of vaccination with the simulations makes it clear that the number of primary infections decreases as much as in the model with only one serotype. 6. CONCLUSION The pediatric vaccination is (almost) negligable. Significantly higher infection numbers can be seen in the two serotype model versus the one serotype model, if comparable scenarios (see Remark 1) are used. One reason seems to be the following: In this variant of the model secondary infections can only be controlled indirectly. No random mass vaccination is applied to the compartments Rh1 and Rh2 . On the other hand, secondary infections are potentially more dangerous and cause higher economic costs. As a consequence, it seems to be especially important to vaccinate the infected humans resp. those recovered from an infection with one serotype. In the future, an updated model including an additional vaccination of the compartments Rh1 and Rh2 will be analyzed. ACKNOWLEDGEMENTS Both authors thank Prof. Dr. Hans Josef Pesch for valuable discussions. REFERENCES Abate, A., Tiwari, A., and Sastry, S. (2009). Box invariance in biologically-inspired dynamical systems. Automatica, 45(7), 1601– 1610. Becker, N., Kr¨ uger, A., Kuhn, C., Plenge-B¨ onig, A., Thomas, S., Schmidt-Chanasit, J., and Tannich, E. (2014). Stechm¨ ucken als ¨ Ubertr¨ ager exotischer Krankheitserreger in Deutschland. Bundesgesundheitsblatt - Gesundheitsforschung - Gesundheitsschutz, 57(5), 531. Becker, N., Sch¨ on, S., Klein, A.M., Ferstl, I., Kizgin, A., Tannich, E., Kuhn, C., Pluskota, B., and J¨ ost, A. (2017). First mass development of Aedes albopictus (Diptera: Culicidae)—its surveillance and control in Germany. Parasitology Research, 116(3), 847–858.
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Optimal Vaccination Strategies for a new Optimal Vaccination Strategies for a new Optimal Vaccination Strategies for a Optimal Vaccination Strategies for a new new Dengue Model with two Serotypes Dengue Model with two Serotypes Dengue Model with two Serotypes Dengue Model with two Serotypes ∗ ∗∗
Kurt Chudej ∗ ,, Anne Fischer ∗∗ Kurt ∗∗ Kurt Chudej Chudej ∗∗∗ , Anne Anne Fischer Fischer ∗∗ Kurt Chudej , Anne Fischer ∗∗ ∗ u r Wissenschaftliches a t Bayreuth, ∗ Lehrstuhl f¨ u Wissenschaftliches Rechnen, Rechnen, Universit¨ Universit¨ a ∗ Lehrstuhl f¨ ∗ f¨ urrGermany Universit¨ att Bayreuth, Bayreuth, Wissenschaftliches Rechnen, Bayreuth, (e-mail: [email protected]) ∗ Lehrstuhl f¨ urGermany [email protected]) Rechnen, Universit¨ at Bayreuth, Bayreuth, ∗∗ Lehrstuhl Bayreuth, Germany (e-mail: (e-mail: [email protected]) f¨ u r Ingenieurmathematik, Universit¨ a t Bayreuth, ∗∗ Lehrstuhl Bayreuth, Germany (e-mail: [email protected]) f¨ ur Ingenieurmathematik, Universit¨ at Bayreuth, Bayreuth, Bayreuth, ∗∗ ∗∗ Lehrstuhl f¨ u r Ingenieurmathematik, Universit¨ a Bayreuth, Germany; current address: Witt Gruppe, Weiden, Germany ∗∗ Lehrstuhl Lehrstuhl f¨ ur current Ingenieurmathematik, Universit¨ att Bayreuth, Bayreuth, Bayreuth, Germany; address: Witt Gruppe, Weiden, Germany Germany; current address: Witt Gruppe, Weiden, Germany Germany; current address: Witt Gruppe, Weiden, Germany Abstract: Dengue fever is a virus infection transmitted by asian tiger mosquitos which affects Abstract: Dengue fever is virus transmitted by asian tiger mosquitos which affects Abstract: Dengue fever is aa population. virus infection infection transmitted bycases asian tiger mosquitos which affects more than half of the world’s Previous dengue in Europe could be traced back Abstract: Dengue fever is a population. virus infection transmitted bycases asian tiger mosquitos which affects more than half of the world’s Previous dengue in Europe could be traced back more than half of the world’s population. Previous dengue cases in Europe could be traced back to an infection outside of Europe, so called non autochthonous cases. Unfortunately, mosquitos more than half outside of the world’s population. Previous dengue casescases. in Europe could be traced back to an infection of Europe, so called non autochthonous Unfortunately, mosquitos to an infection outside of Europe, so called non autochthonous cases. Unfortunately, mosquitos capable of are (re-)invading Europe Germany. Thereto an infection outside ofdengue Europe, called non autochthonous cases.including Unfortunately, mosquitos capable of transmitting transmitting dengue aresocurrently currently (re-)invading Europe including Germany. Therecapable of dengue are currently (re-)invading Europe including Germany. Therefore, possible outbreaks of Europe seem in future. possibility is capable of transmitting transmitting are in currently Germany. Therefore, possible outbreaks dengue of dengue dengue in Europe(re-)invading seem possible possibleEurope in the theincluding future. This This possibility is fore, possible outbreaks of dengue in Europe seem possible in the future. This possibility is emphasized by the outbreaks of Chikungunya fever (transmitted by the same mosquitos) in fore, possiblebyoutbreaks of dengue in Europe seem possible in thebyfuture. Thismosquitos) possibility in is emphasized the outbreaks of Chikungunya fever (transmitted the same emphasized by the outbreaks of Chikungunya fever (transmitted by the same mosquitos) in France and Italy in the year 2007. The dengue virus comes in four different serotypes. Extensive emphasized by the outbreaks of Chikungunya fevercomes (transmitted by the serotypes. same mosquitos) in France and Italy in the year 2007. The dengue virus in four different Extensive France and Italy in the year 2007. The dengue virus comes in four differentdengue serotypes. Extensive research was performed previously for several variants of serotype model. In France and in the year 2007. The comes four differentdengue serotypes. Extensive research wasItaly performed previously for dengue several virus variants of aainone one serotype model. In this this research was performed previously for several variants of a one serotype dengue model. In this paper we consider a new version of an optimal control problem for a vaccination strategy research was performed previously for several variants of a one serotype dengue model. Inwith this paper we consider a new version of an optimal control problem for a vaccination strategy with paper we consider a new version of an optimal control problem for a vaccination strategy with two dengue serotypes. paper we consider a new version of an optimal control problem for a vaccination strategy with two dengue serotypes. two dengue serotypes. two dengue © 2018, IFACserotypes. (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Dengue, vaccination, two serotypes, coinfection, vector, optimal control, nonlinear Keywords: Dengue, vaccination, Keywords: Dengue, vaccination, two two serotypes, serotypes, coinfection, coinfection, vector, vector, optimal optimal control, control, nonlinear nonlinear programming. Keywords: Dengue, vaccination, two serotypes, coinfection, vector, optimal control, nonlinear programming. programming. programming. 1. aa new 1. INTRODUCTION INTRODUCTION new two two serotype serotype model. model. Moreover Moreover in in certain certain regions regions 1. INTRODUCTION a new two serotype model. Moreover in certain regions currently none, one or only two different serotypes are 1. INTRODUCTION a new twonone, serotype model. Moreover in certain regions currently one or only two different serotypes currently none, one or only two different serotypes are are present. currently none, one or only two different serotypes are present. Dengue (and Chikungunya) fever is transmitted via vecDengue (and Chikungunya) fever is transmitted via vecpresent. Dengue (and Chikungunya) fever is transmitted viacalled vec- present. tors, i.e. (female) mosquitos the species, Dengue Chikungunya) is transmitted viacalled vec2. WITH tors, i.e. (and (female) mosquitos of offever the Aedes Aedes species, so so 2. NEW NEW TWO TWO SEROTYPE SEROTYPE DENGUE DENGUE MODEL MODEL WITH WITH tors, i.e. (female) mosquitos of the Aedes species, so called asian tiger mosquitos, see Moutailler al. A 2. NEW TWO SEROTYPE DENGUE MODEL tors, i.e. (female) mosquitos the Aedeset species, so called VACCINATION asian tiger mosquitos, see of Moutailler et al. (2009). (2009). A 2. NEW TWO SEROTYPE DENGUE MODEL WITH VACCINATION asian tiger mosquitos, see Moutailler et al. (2009). A human to human transmission of dengue is not possible. VACCINATION asian tiger mosquitos, see Moutailler A human to human transmission of dengueet isal. not(2009). possible. VACCINATION human transmission of dengue is Currently, the mosquito (e.g. Aedes albopictus) is reThe following new two serotype dengue model, see Fig. 1, human to to human human transmission of dengue is not not possible. possible. Currently, the mosquito (e.g. Aedes albopictus) is reThe following new two dengue model, see 1, Currently, the mosquito (e.g. Aedes albopictus) is reinvading Europe, see et (2017, 2014); al. The following new two serotype serotype dengue model, see Fig. Fig. 1, pediatric vaccination and imperfect random mass Currently, the mosquito invading Europe, see Becker Becker(e.g. et al. al.Aedes (2017, albopictus) 2014); Frank Frankiset et real. with The following new two serotype dengue model, see Fig. 1, with pediatric vaccination and imperfect random mass invading Europe, see Becker et al. (2017, 2014); Frank et al. (2014); Benedict et al. (2007); Werner and Kampen (2015); with pediatric vaccination and imperfect random mass vaccination with waning immunity is a generalization of invading Europe, see Becker et al. (2017, 2014); Frank et al. (2014); Benedict et al. (2007); Werner and Kampen (2015); vaccination with pediatric and imperfect random mass withvaccination waning immunity is a generalization of (2014); Benedict et al. (2007); Werner and Kampen (2015); Scholte et al. (2008); Takken and Knols (2007). The invawith waning immunity is of the one of et (2014); Benedict et al.Takken (2007); and Werner and(2007). Kampen (2015); Scholte et al. (2008); Knols The inva- vaccination vaccination withdengue waningmodel immunity is aa generalization generalization of the one serotype serotype dengue model of Rodrigues Rodrigues et al. al. (2013b), (2013b), Scholte et Takken and (2007). The invasion is further favoured by global warming Thomas al. the one serotype dengue model of Rodrigues et al. (2013b), see Remark 1 later. We use the following colours: SusScholte et al. al. (2008); (2008); Takken and Knols Knols (2007). The et invasion is further favoured by global warming Thomas et al. the one serotype dengue model of Rodrigues et al. (2013b), see Remark 1 later. We use the following colours: Sussion is further favoured by global warming Thomas et al. (2014, 2012). dengue virus, serologically belonging see Remark 1 later. Weinuse the(imperfectly) following colours: Suscompartiments blue; vaccinated sion is further favoured by global Thomas et al. ceptible (2014, 2012). The The dengue virus, warming serologically belonging see Remark 1 later. Weinuse the(imperfectly) following colours: Susceptible compartiments blue; vaccinated (2014, 2012). The dengue virus, serologically belonging to the group of flaviviruses, consists of four serotypes, ceptible compartiments in blue; (imperfectly) vaccinated humans in cyan; with serotype 1/2 infected in red/orange; (2014, 2012). The dengue virus, serologically belonging to the group of flaviviruses, consists of four serotypes, humans ceptible in compartiments in blue; (imperfectly) vaccinated cyan; with serotype 1/2 infected in red/orange; to the group of flaviviruses, consists of four serotypes, DENV-1, DENV-2, DENV-3 and DENV-4 (WHO (2017)). in with 1/2 in healthy in green. The human population to the group of flaviviruses, consists of (WHO four serotypes, DENV-1, DENV-2, DENV-3 and DENV-4 (2017)). humans humans aquatic in cyan; cyan;mosquitos with serotype serotype 1/2 infected infected in red/orange; red/orange; healthy mosquitos in The human DENV-1, DENV-2, DENV-3 and DENV-4 (WHO (2017)). After an infection one of the serotypes, one is lifelong healthy aquatic aquatic mosquitos in green. green. Thefollowing human population population (subscript h) is subdivided into the compartDENV-1, DENV-2,with DENV-3 and DENV-4 (WHO (2017)). After an infection with one of the serotypes, one is lifelong healthy aquatic mosquitos in green. Thefollowing human population (subscript h) is subdivided into the compartAfter an against infection with one of the serotypes, one isnolifelong immune this serotype. However, there are cross(subscript h) is subdivided into the following compartsusceptible humans, V vaccinated humans, S After an against infection with one of the serotypes, immune this serotype. However, thereone areisnolifelong cross- ments: h h (subscript is subdivided into Vthe following compartsusceptible humans, vaccinated humans, ments: Sh h) h immune against this serotype. However, there are no crossreactivities: An initial infection with one of the dengue 1 2 susceptible humans, V vaccinated humans,2 ments: S h h I with serotype 1 infected humans, I with serotype immune against this serotype. However, there are no crossreactivities: An initial infection with one of the dengue Iments: h susceptible humans, Vh vaccinated 1 2 h h humans,2 S with serotype 1 infected humans, with serotype h h Ih 1 2 reactivities: An initial infection with one of the dengue 1 2 serotypes does not confer immunity against any of the 1 h Ihh1 with serotype 1 h1infected humans, Iserotype with serotype 2 infected humans, R previously with 1 infected reactivities:does An not initial infection with against one of the dengue 2 h serotypes confer immunity any of the h I with serotype 1 infected humans, I with serotype 2 infected humans, R previously with serotype 1 infected serotypes does not confer immunity against any of the h h other three. Therefore, one could fall ill up to four humans, Rhhh111 previously withresistant serotype 1 serotype infected humans which recovered and are now to serotypes does not confer immunity against any oftimes the infected other three. Therefore, one could fall ill up to four times infected humans, R previously with serotype 1 infected humans which recovered and are now resistant to serotype other three. Therefore, one could fall ill up to four times h with four different serotypes of 2 which recovered and are 2now resistant to serotype 1, R with serotype infected humans which other the three. Therefore, could fall ill up to An fourimportimes humans with the four different one serotypes of dengue. dengue. An impor2 previously h humans which recovered and are 2now resistant to 1, R with serotype infected humans which 2 with the four different serotypes of dengue. An impor2 previously tant theory in connection with the occurrence of several 12 serotype h 1, R previously with serotype 2 infected humans which recovered and are now resistant to serotype 2, I with the four different serotypes of dengue. An impor2 h tant theory in connection with the occurrence of several recovered 12 humans h previously h 1, R with serotype 2 infected humans which and are now resistant to serotype 2, I 12 humans h tant theory in connection with the occurrence of several 12 serotypes is that of the infection-enhancing antibody, in h recovered and are now resistant to serotype 2, Ihh12 humans which were firstly infected with serotype 1, recovered, and tant theory in connection with the occurrence of several serotypes is that of the infection-enhancing antibody, in recovered and are now resistant to serotype 2, I humans which were firstly infected with serotype 1, recovered, and serotypes is that of the infection-enhancing antibody, in h which it is in the case of a second infection, 21 were firstlywith infected with 2, serotype 1, recovered, and are now infected serotype I humans which were serotypes isassumed that of that, the infection-enhancing antibody, in which which it is assumed that, in the case of a second infection, 21 h which were firstly infected with serotype 1, recovered, and are now infected with serotype 2, I humans which were which it is assumed that, in the case of a second infection, 21 21 humans which were the already existing antibodies form an antigen-antibody h are now infected with serotype 2, I firstly are now which it is assumed in the form case of second infection, are the already existing that, antibodies ana antigen-antibody nowinfected infectedwith withserotype serotype2, 2, recovered, Ihhh21 humansand which were firstly infected with serotype 2, recovered, and are now the already existing antibodies form an antigen-antibody complex with the newly added virus, which is difficult to infected with serotype 2, recovered, and are now ˜˜ h humans the already existing antibodies form an antigen-antibody complex with the newly added virus, which is difficult to firstly infected with serotype 1, R which are resistant firstly infected with serotype 2, recovered, and are now infected with serotype 1, which are resistant complex with the newly added virus, which is to control the body, see Gubler (1998). Secondary infec˜ hh humans infected with serotype 1, R R humans which are resistant complexby with the newly added virus, which is difficult difficult to to control by the body, see Gubler (1998). Secondary infecserotype 1 and 2. h humans which are resistant ˜ infected with serotype 1, R control by the body, see Gubler (1998). Secondary infech tions potentially more than to serotype serotype 11 and and 2. 2. controlare the body, see Gubler (1998). Secondary infec- to tions arebythus thus potentially more dangerous dangerous than primary primary to serotype 1 mosquito and 2. population (subscript m) is subditions are thus potentially more dangerous than primary The (female) infections. This theory would, among other things, explain tions are thus more dangerous than primary mosquito population (subscript m) is subdiinfections. Thispotentially theory would, among other things, explain The The (female) (female) mosquito population (subscript m) is larvae, subdiinfections. This theory among other explain into four compartments: A of eggs, why an variant of fever occurs m amount The (female) mosquito population (subscript m) is subdiinfections. This amniotic theory would, would, among other things, things, vided into four compartments: A why an acute acute amniotic variant of dengue dengue fever explain occurs vided amount of eggs, larvae, m 1 vided into four compartments: Am amount of eggs, larvae, why an acute amniotic variant of dengue fever occurs and pupae, S susceptible adult mosquitos, I mainly in secondary infections. Mathematical models of adult m 1 m m vided into four compartments: A amount of eggs, larvae, why an inacute amniotic variant Mathematical of dengue fever occurs mainly secondary infections. models of and pupae, S susceptible adult mosquitos, I m 1 1 adult 2 and pupae,infected Sm susceptible adult1,mosquitos, Im mainly fever in secondary infections. Mathematical modelse.g. of mosquitos m dengue with one serotype models were presented with serotype I adult mosquitos 1 adult m m 2 m m and pupae,infected Sm susceptible adult1,mosquitos, Im adult mainly fever in secondary Mathematical modelse.g. of mosquitos dengue with one infections. serotype models were presented with serotype mosquitos 2 2 adult mosquitos infected with serotype 1, IIm adult mosquitos dengue fever with one models were presented e.g. by Esteva and Vargas (1999), Rodrigues al. with serotype 2. For have assumed 2 we m mosquitos infected with 1, Im adult dengue fever with one serotype serotype wereet e.g. infected by Esteva and Vargas (1999), models Rodrigues etpresented al. (2013a), (2013a), infected with serotype 2. serotype For simplicity simplicity havemosquitos assumed m we by Esteva and Vargas (1999), Rodrigues et al. (2013a), infected with serotype 2. For simplicity we have assumed Rodrigues et al. (2014). In an intermediate step towards a that a mosquito cannot carry both dengue viruses of infected with serotype 2. For simplicity we have assumed by Esteva and Vargas (1999), Rodrigues et al. (2013a), Rodrigues et al. (2014). In an intermediate step towards a that a mosquito cannot carry both dengue viruses of Rodrigues et al. (2014). In an intermediate step towards a that a mosquito cannot carry both dengue viruses mathematical four serotype model of dengue, we introduce serotype 1 and 2. cannot carry both dengue viruses of of Rodrigues et al. (2014). In an intermediate step towards a that a mosquito mathematical serotype 11 and and 2. 2. mathematical four four serotype serotype model model of of dengue, dengue, we we introduce introduce serotype mathematical four serotype model of dengue, we introduce serotype 1 and 2.
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dRh1 I2 = ηh Ih1 − (δ1 Bβmh m + µh )Rh1 , dt Nh I1 dRh2 2 = ηh Ih − (δ2 Bβmh m + µh )Rh2 , dt Nh 2 Im dIh12 1 = δ1 Bβmh R − (µh + ηh )Ih12 , dt Nh h I1 dIh21 = δ2 Bβmh m Rh2 − (µh + ηh )Ih21 , dt Nh ˜h dR ˜h, = ηh (Ih12 + Ih21 ) − µh R dt and the mosquito population
The following general assumptions hold: Both mosquitos (=vectors) and humans are neither born infected nor resistant. The size of the human population (Nh ) is constant at any time. Furthermore the size of the human population is sufficiently large, such that the state variables can be assumed to be continuous real functions of time. Neither emigration nor immigration are considered. The passing of a certain percentage of the population is modeled through the proportionality factor µh , with 1/µh denoting an average lifespan. The same holds for the adult mosquito population with factor µm and the mosquitos in the aquatic phase with factor µA . Humans recover from the disease (of any serotype) at a rate ηh . A mosquito in the aquatic phase requires 1/ηA days to grow up. Mosquitos from compartment Am are removed with proportionality factor ηA and join the compartment Sm . Every (female) adult mosqito leaves ϕ eggs at one breeding place per day. The reproduction of the mosquito population is modeled using logistic growth. This considers the possible amount of eggs a (human-build) breeding place can hold. As the capacity bound kNh is chosen. Homogeneity between the populations is assumed, thus, with the same probability every female mosquito bites every human host. Per day, one female mosquito, averagely, bites B-times and human individuals become infected with the probability βmh . The product Bβmh represents the contact rate between infected female mosquitos and susceptible humans. The contact rate of infected humans and susceptible female mosquitos is modeled analogously. This model does not consider an impact of the disease on lifespans.
Am dAm 1 2 = ϕ(1 − )(Sm + Im (t) + Im )− dt kNh − (ηA + µA )Am , dSm = η A Am − dt I 1 + Ih2 + Ih12 + Ih21 + µm )Sm , (2) − (Bβhm h Nh 1 I 1 + Ih21 dIm 1 = Bβhm h Sm − µ m I m , dt Nh 2 2 12 I + Ih dIm 2 = Bβhm h Sm − µ m I m . dt Nh Remark 1: The specialization δ1 = δ2 = 0 and Ih := Ih1 + 1 2 Ih2 , Rh := Rh1 +Rh2 , Im := Im +Im , together with the initial 1 values fulfilling Ih (0) = Ih (0) + Ih2 (0), Ih12 (0) = Ih21 (0) = ˜ h ≡ 0 the one ˜ h (0) = 0 yields due to I 12 ≡ I 21 ≡ R R h h serotype dengue model with vaccination of Rodrigues et al. (2013b):
Only susceptible human subjects are vaccinated with a vaccine that acts against both serotypes, but infected humans are not vaccinated here, even after an initial infection. Depending on the vector they come into contact with, humans either change to the compartment Ih1 or Ih2 . After a primary infection, the same holds for the compartments Ih12 , Ih21 .
dSh dt dVh dt dIh dt dRh dt
The vaccination model includes a pediatric vaccination with a time-dependent control p(t) ∈ [0, 1] and an imperfect random mass vaccination with a time-dependent control ψ(t) ∈ [0, 1]. Two important constants appear: The efficiency of the vaccine is modeled by 1 − σ ∈ [0, 1]. σ = 0 denotes a high efficiency of the vaccine, whereas σ = 1 indicates that the vaccine has no effect. If σ > 0 also certain vaccinated individuals can be infected, see Fig. 1. A possible waning imunity is modeled by θ ∈ [0, 1]. In the numerical computations we use σ = 0.2 (efficiency 80%) and θ = 0.05.
= ηh Ih − µh Rh
Am dAm = ϕ(1 − )(Sm + Im ) − (ηA + µA )Am dt kNh dSm Ih (4) = ηA Am − (Bβhm + µm )Sm dt Nh dIm Ih = Bβhm Sm − µ m I m dt Nh 2 ˜ h ≡ 0 yields for The special choice Ih2 ≡ Im ≡ Rh2 ≡ R arbitrary values of δi also the ODE (3), (4) with Ih := Ih1 , 1 Im := Im , Rh := Rh1 .
This results in the ODE for the human population dSh = (1 − p)µh Nh + θVh − dt 2 I 1 + Im − (Bβmh m + ψ + µh )Sh , Nh dVh = pµh Nh + ψSh − dt 2 I 1 + Im − (θ + σBβmh m + µh )Vh , Nh I1 dIh1 = Bβmh m (Sh + σVh ) − (ηh + µh )Ih1 , dt Nh I2 dIh2 = Bβmh m (Sh + σVh ) − (ηh + µh )Ih2 , dt Nh
Im + ψ + µh )Sh Nh Im = pµh Nh + ψSh − (θ + σBβmh + µh )Vh Nh (3) Im = Bβmh (Sh + σVh ) − (ηh + µh )Ih Nh
= (1 − p)µh Nh + θVh − (Bβmh
3. EQUILIBRIA AND THE BASIC REPRODUCTION NUMBER In this section we assume that p and ψ are constants. We consider the relevant set Ω of all non-negative compartments of ODE (3), (4) s.t. Sh + Vh + Ih + Rh ≤ Nh , Am ≤ kNh , and Sm + Im ≤ mNh for the theoretical ˜ of all non-negative analysis. We consider the relevant set Ω compartments of ODE (1), (2) s.t. Sh +Vh +Ih1 +Ih2 +Rh1 + ˜ h ≤ Nh , Am ≤ kNh , and Sm + I 1 + Rh2 + Ih12 + Ih21 + R m 2 Im ≤ mNh for the theoretical analysis.
(1)
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Fig. 1. Compartment model of the two serotype dengue model with vaccination ˜ is positive invariant for the Lemma 2: The set Ω [resp. Ω] ODE (3), (4) [resp. (1), (2)].
disease free equilibrium: only healthy (susceptible or vaccinated) humans, only healthy (but susceptible) mosquitos.
Proof: The ODE can be written as X˙ = A(X)X + F with a Metzler Matrix A(X) and F ≥ 0 and apply Abate et al. (2009); Mitkowski (2008).
˜3 = (S ∗ , V ∗ , I ∗ , 0, R∗ , 0, 0, 0, 0 | A∗ , S ∗ , I ∗ , 0) E m m m h h h h ˜4 = (S ∗ , V ∗ , 0, I ∗ , 0, R∗ , 0, 0, 0 | A∗ , S ∗ , 0, I ∗ ) E m m m h h h h endemic boundary equilibria.
We use the abbreviation M = ϕηA − ηA µm − µA µm .
∗ ∗ , Im in the The lengthy formulas of Sh∗ , Vh∗ , Ih∗ , Rh∗ , A∗m , Sm ˜ ˜ endemic boundary equilibria E3 and E4 are the same as in the endemic equilibrium E3 of Theorem 3.
Theorem 3: The ODE (3), (4) has in Ω (depending on the parameter values) at most the equilibria:
Proof: Remark 1 with Theorem 3.
h +θ) Nh (µh p+ψ) E1 = ( Nh ((1−p)µ , µh +θ+ψ , 0, 0 | 0, 0, 0) µh +θ+ψ trivial equilibrium: only healthy (susceptible or vaccinated) humans, no mosquitos.
Remark 5: Suppose the presented model is sufficiently accurate. For a city without vaccination p = ψ = 0 holds. In a city without asian tiger mosquitos (and no dengue infected travellers) we are currently in the trivial equilibrium E1 . In a city with an asian tiger mosquito population and no dengue infected travellers we are currently in the equilibrium E2 .
h +θ) Nh (µh p+ψ) h M αkNh M , µh +θ+ψ , 0, 0 | kN E2 = ( Nh ((1−p)µ µh +θ+ψ ϕηA , ϕµm , 0) disease free equilibrium: only healthy (susceptible or vaccinated) humans, only healthy (but susceptible) mosquitos.
∗ ∗ E3 = (Sh∗ , Vh∗ , Ih∗ , Rh∗ | A∗m , Sm , Im ) endemic equilibrium.
Question 6: What happens if we introduce a small number of dengue infected travellers into a city with no asian tiger mosquito population?
Proof: See the MAPLE computation in Fischer (2016), the formula for E3 consists of several pages. Theorem 4: The ODE (1), (2) has the following equilib˜ for certain values of the parameters: ria, which are in Ω ˜1 = ( Nh ((1−p)µh +θ) , Nh (µh p+ψ) , 0, . . . , 0 | 0, . . . , 0) E µh +θ+ψ µh +θ+ψ trivial equilibrium: only healthy (susceptible or vaccinated) humans, no mosquitos.
Answer: If we assume, that the presented model is sufficiently accurate (i.e. currently only up to two of the four serotypes are virulent and especially other mosquitos do not transmit the disease), we will again approach the trivial equilibrium E1 . (Proof: Solution of the initial value problem.)
˜2 = ( Nh ((1−p)µh +θ) , Nh (µh p+ψ) , 0, . . . , 0 | E µh +θ+ψ µh +θ+ψ kNh M αkNh M ϕηA , ϕµm , 0, 0)
Question 7: What happens if we introduce a (sufficiently) small number of dengue infected travellers into a city with a previously healthy asian tiger mosquito population?
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humans
humans time [days]
time [days]
Fig. 2. Numerical simulation of ODE (1),(2), humans: Ih1 = Ih2 , Rh1 = Rh2 , Ih12 = Ih21
Fig. 3. Numerical simulation of ODE (1),(2), zoom on infected humans: Ih1 = Ih2 , Ih12 = Ih21
In order to get an answer, we have to do some computation. Theorem 8: The basic reproduction number of the ODE (3), (4) and of the ODE (1), (2) is given by B 2 β β kM((1 − p)µ + θ + σ(µ p + ψ)) 12 hm mh h h . R0 = ϕµ2m (ηh + µh )(µh + θ + ψ) Proof: Application of the next generation approach (van den Driessche and Watmough (2002)), for details see Fischer (2016). Certain special cases of this result for the ODE (3), (4) can be found in Rodrigues et al. (2014). ˜2 are Theorem 9: The disease free equilibria E2 and E locally asymptotically stable if R0 < 1 and instable if R0 > 1.
mosquitos
time [days]
Proof: See van den Driessche and Watmough (2002).
Fig. 4. Numerical simulation of ODE (1),(2), mosquitos: 1 2 Im = Im
Partial answer to Question 7: Assume, Theorem 9 can be ˜2 globally stable. If the model paramestrengthened to E ters fulfill R0 < 1 the solution will approach the disease ˜2 , which is favourable. Otherwise the free equilibrium E disease will become endemic, which is not favourable.
In the human population the total number of infected persons is almost doubled in the event of an outbreak, whereas the increase in the total infected population of the vector is 1.5 times higher compared to the one serotype model. This is a significant and unexpected increase since the initial values for the diseased persons differ from those in the basic model only in that the Ih (0) = 38 infected humans are divided into Ih1 (0) = Ih2 (0) = 19 in each of the two serotypes. But note the change from δ1 = δ2 = 0 in the one serotype model to δ1 = δ2 = 1 in the two serotype model. Another aspect that should be noticed is the number of resistant individuals. At the end of the observation period, not all individuals are immune to both serotypes, but about 7 000 individuals remain in compartment Rh1 and Rh2 . Therefore, some people were not affected by secondary infections. Although the total number of inital infected humans was the same, the presence of another serotype has a considerable effect on the severity of a disease outbreak. In addition, many more risky secondary infections occur. If, thus, infections with dengue are identified, it has to be ascertained as soon as possible whether only one serotype is present or a second serotype has to be dealt with. Since secondary infections also increase the risk of a haemorrhagic course of dengue fever, in the presence of several serotypes effective countermeasures should be initiated as soon as possible.
4. NUMERICAL SIMULATION We consider a numerical simulation with δ1 = δ2 = 1 and without vaccination (p = ψ = 0) for the following initial values Ih1 (0) = Ih2 (0) = 19, Ih12 (0) = Ih21 (0) = 0, Vh (0) = 0, Rh (0) = 0, Sh (0) = Nh − [Ih1 (0) + Ih2 (0) + Ih12 (0) + Ih21 (0)] − 1 Vh (0) − Rh (0), Am (0) = kNh , Sm (0) = mNh , Im (0) = 2 Im (0) = 0 and the following parameters B = 0.8, βmh = 1 , ηh = 13 , µm = 0.1, ϕ = 6, βhm = 0.375, µh = 80·365 µA = 0.25, ηA = 0.08, m = 3, k = 3, Nh = 383 000. With the numerical simulation we obtain approximately the same results as for the one serotype model. There is an outbreak of the disease between the 50th and 100th day, after which no further infections can be detected for a period of one year. The number of individuals with an initial infection is approximately 34 000 for each serotype at the time of the outbreak. There holds: Ih1 ≡ Ih2 and Ih12 ≡ Ih21 and Rh1 = Rh2 . The peak values for each serotype are 27 000, which together reveal an enormous increase in contrast to the numbers in the basic model. The number of infected mosquitos is approximately 167 000 for each serotype at the time of the outbreak. There holds: 1 2 ≡ Im . Im 4
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γD
A B C
0.3 0.8 0.1
γV γC 0.35 0.1 0.45
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costs 1-serotype 2-serotype 0.0701847 0.221746 0.11339 0.450337 0.0279303 0.0852322
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case
Table 1. Costs of optimal control in the 3 cases A, B, C
i1h
5. OPTIMAL CONTROL Numerical solutions were computed by a direct transcription of the optimal control problem to a nonlinear programming problem (NLP). The resulting NLP was solved via the modelling language AMPL (Fourer et al. (2003)) and the interior point method IPOPT (W¨achter and Biegler (2006)). In order to stabilize the numerical solution a scaling of the state variables was applied:
time [days]
Fig. 5. Primary infected humans i1h = i2h with optimal vaccination
Sh Vh 1 I1 I2 R1 R2 , vh = , ih = h , i2h = h , rh1 = h , rh2 = h , Nh Nh Nh Nh Nh Nh 12 21 ˜h I I R h i12 , i21 = h , r˜h = , h = Nh h Nh Nh 1 Am Sm 1 I I2 am = , sm = , im = m , i2m = m . kNh mNh mNh mNh This results in a new scaled version of the ODE for the new state variables. We use the same initial values and parameters as in Section 4.
case
sh =
As a cost functional we use tf J= γD · i(t)2 + γV · ψ(t)2 + γC · p(t)2 dt → min
i12 h
time [days] 21 Fig. 6. Secondary infected humans i12 h = ih with optimal vaccination
p,ψ
0
with scaling factors γD , γV , γC . In addition, we have 21 chosen i(t) = i1h (t) + i2h (t) + i12 h (t) + ih (t) as the total number of infected persons in the two serotype model. (In the one serotype model (3), (4) we use i(t) = Ih /Nh .) It is also conceivable to specify individual weights for all compartments containing infected individuals. Thus, it would be possible to place different weights on the reduction of primary and secondary infections. The aim could be to prevent secondary infections as much as possible, since these have a higher probability of inducing amniotic dengue fever. If one of the serotypes shows a higher virulence or aggravation of secondary infections, it is also possible with individual weights to keep the number of occurring fatalities as low as possible for this serotype. However, here no distinction was made: The parameters δi for the secondary infection were both set to 1, which excludes the effect of an initial infection on a secondary infection.
case
time [days]
Fig. 7. Optimal random mass vaccination ψ(t) in the 3 cases A, B, C whereas Case C is the most cost-effective. In contrast to the model with only one serotype, the costs in cases A and C increase to the triple and in case B even to the quadruple. The occurrence of several serotypes therefore has severe implications on the optimal costs of the used cost functional.
We consider three different scenarios. Case A is a general intermediate scenario. In case B a very expensive (hospital) treatment of infected humans is modeled. Case C simulates the case of a very expensive serum.
If we now analyze the optimal vaccination strategies in Fig. 7, we find that the strategy for the application of random mass vaccination is similar to that of the one serotype model. In cases A and C, the curves reach about 1.5 times the value in comparison to the results of the one serotype model. Moreover, in case B the peak is about double in comparison to the results of the one serotype model. In case C with the expensive serum,
In Table 1 the numerical results of the optimization are presented. The one serotype model is chosen in accordance with Remark 1. In the Figures 5-8 the numerical results for the new two serotype model are demonstrated. It is clear from Table 1 that the occurrence of two serotypes causes huge additional costs if countermeasures only include vaccination. Case B is again the most expensive, 5
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Benedict, M.Q., Levine, R.S., Hawley, W.A., and Lounibos, L.P. (2007). Spread of the tiger: global risk of invasion by the mosquito Aedes albopictus. Vector-borne and zoonotic diseases, 7(1), 76–85. Esteva, L. and Vargas, C. (1999). A model for dengue disease with variable human population. Journal of Mathematical Biology, 38, 220–240. Fischer, A. (2016). Optimale Impfstrategien f¨ ur Dengue-Fieber. Masterarbeit, Universit¨ at Bayreuth, Germany. Fourer, R., Gay, D., and Kernighan, B.W. (2003). AMPL: A modeling language for mathematical programming. Duxbury/Thompson, Pacific Grove, CA. Frank, C., Faber, M., Hellenbrand, W., Wilking, H., and Stark, K. (2014). Wichtige, durch Vektoren u ¨bertragene Infektionskrankheiten beim Menschen in Deutschland. Bundesgesundheitsblatt - Gesundheitsforschung - Gesundheitsschutz, 57(5), 557. Gubler, D.J. (1998). Dengue and dengue hemorrhagic fever. Clinical microbiology reviews, 11(3), 480–496. Mitkowski, W. (2008). Dynamical properties of Metzler systems. Bulletin of the Polish Academy of Sciences, Technical Sciences, 56(4), 309–312. Moutailler, S., Barr, H., Vazeille, M., and Failloux, A.B. (2009). Recently introduced Aedes albopictus in Corsica is competent to Chikungunya virus and in a lesser extent to dengue virus. Tropical Medicine & International Health, 14(9), 1105–1109. Rodrigues, H.S., Monteiro, M.T.T., and Torres, D.F. (2013a). Bioeconomic perspectives to an optimal control dengue model. International Journal of Computer Mathematics, 90(10), 2126–2136. Rodrigues, H.S., Monteiro, M.T.T., and Torres, D.F. (2013b). Dengue in Cape Verde: vector control and vaccination. Mathematical Population Studies, 20(4), 208–223. Rodrigues, H.S., Monteiro, M.T.T., and Torres, D.F. (2014). Vaccination models and optimal control strategies to dengue. Mathematical biosciences, 247, 1–12. Scholte, E.J., Dijkstra, E., Blok, H., De Vries, A., Takken, W., Hofhuis, A., Koopmans, M., De Boer, A., and Reusken, C. (2008). Accidental importation of the mosquito Aedes albopictus into the Netherlands: a survey of mosquito distribution and the presence of dengue virus. Medical and veterinary entomology, 22(4), 352–358. Takken, W. and Knols, B.G. (2007). Emerging pests and vectorborne diseases in Europe, volume 1. Wageningen Academic Pub. Thomas, S.M., Obermayr, U., Fischer, D., Kreyling, J., and Beierkuhnlein, C. (2012). Low-temperature threshold for egg survival of a post-diapause and non-diapause European aedine strain, Aedes albopictus (Diptera: Culicidae). Parasites & vectors, 5(1), 100. Thomas, S.M., Tjaden, N.B., van den Bos, S., and Beierkuhnlein, C. (2014). Implementing cargo movement into climate based risk assessment of vector-borne diseases. International journal of environmental research and public health, 11(3), 3360–3374. van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(12), 29–48. W¨ achter, A. and Biegler, L.T. (2006). On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical programming, 106(1), 25–57. Werner, D. and Kampen, H. (2015). Aedes albopictus breeding in southern Germany, 2014. Parasitology research, 114(3), 831–834. WHO (2017). Dengue and severe dengue. URL http://www.who.int/mediacentre/factsheets/.
case
time [days]
Fig. 8. Optimal pediatric vaccination p(t) in the 3 cases A, B, C the optimal use of pediatric vaccination is identical. The random mass vaccination rate is slightly lower in the case of two serotypes, so that a pronounced vaccination occurs at the time of the disease outbreak. In Case A, on the other hand, a continuous vaccination with 1.8-fold compared to before is advisable. Overall, the outbreak of the second infections can only be controlled indirectly by vaccination strategies. However, this is due to the structuring of the model. Comparing the use of vaccination with the simulations makes it clear that the number of primary infections decreases as much as in the model with only one serotype. 6. CONCLUSION The pediatric vaccination is (almost) negligable. Significantly higher infection numbers can be seen in the two serotype model versus the one serotype model, if comparable scenarios (see Remark 1) are used. One reason seems to be the following: In this variant of the model secondary infections can only be controlled indirectly. No random mass vaccination is applied to the compartments Rh1 and Rh2 . On the other hand, secondary infections are potentially more dangerous and cause higher economic costs. As a consequence, it seems to be especially important to vaccinate the infected humans resp. those recovered from an infection with one serotype. In the future, an updated model including an additional vaccination of the compartments Rh1 and Rh2 will be analyzed. ACKNOWLEDGEMENTS Both authors thank Prof. Dr. Hans Josef Pesch for valuable discussions. REFERENCES Abate, A., Tiwari, A., and Sastry, S. (2009). Box invariance in biologically-inspired dynamical systems. Automatica, 45(7), 1601– 1610. Becker, N., Kr¨ uger, A., Kuhn, C., Plenge-B¨ onig, A., Thomas, S., Schmidt-Chanasit, J., and Tannich, E. (2014). Stechm¨ ucken als ¨ Ubertr¨ ager exotischer Krankheitserreger in Deutschland. Bundesgesundheitsblatt - Gesundheitsforschung - Gesundheitsschutz, 57(5), 531. Becker, N., Sch¨ on, S., Klein, A.M., Ferstl, I., Kizgin, A., Tannich, E., Kuhn, C., Pluskota, B., and J¨ ost, A. (2017). First mass development of Aedes albopictus (Diptera: Culicidae)—its surveillance and control in Germany. Parasitology Research, 116(3), 847–858.
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