Multi-Strain Age-Structured Dengue Transmission Model: Analysis and Optimal Control
MMulti-Strain Age-Structured Dengue
Transmission Model: Analysis andOptimal Control ∗ Michelle N. Raza and Randy L. Caga-anan Department of Mathematics and StatisticsCollege of Science and MathematicsMSU-Iligan Institute of TechnologyTibanga, Iligan City [email protected] [email protected] Abstract
Dengue is a serious health problem in the Philippines. In 2016, theDepartment of Health of the country launched a dengue vaccinationcampaign using Dengvaxia. However, the campaign was mired withcontroversy and the use of Dengvaxia was banned in the country. Thisstudy proposes a mathematical model that represents the dynamics ofthe transmission of dengue with its four strains. Considering that theDengvaxia vaccine was intended to be given only to people aging from9 to 45 years old, the human population is divided into two age groups:from 9-45 years old and the rest of the population. Using this modeland optimal control theory we simulate what could have been the effectof Dengvaxia in the number of dengue cases and then compare thiswith the other usual dengue intervention strategies. Results show thatthe best implementation of the usual strategies is better than that ofDengvaxia.
AMS Subject Classification (2010): 92D30, 37N25, 49N90Keywords: dengue, dengvaxia, SIR, optimal control ∗ The first author was funded by the Department of Science and Technology (DOST) ofthe Philippines and the second author by the Premier Research Institute of Science andMathematics (PRISM) of MSU-IIT. a r X i v : . [ q - b i o . P E ] A ug ulti-Strain Age-Structured Dengue Transmission Model Dengue is a a mosquito-borne viral infection for which there is no specific anti-viral treatment. Dengue virus is transmitted by female mosquitoes mainlyof the species
Aedes aegypti and, to a lesser extent,
Aedes albopictus . Thereare four dengue virus serotypes, namely DEN-1, DEN-2, DEN-3, and DEN-4.Infection by one serotype confers life-long immunity to that serotype, but thereis no cross-protective immunity to the other serotypes.An effective technique used to detect dengue virus and the specific serotypehas been developed. This laboratory test involves taking clinical samples andanalyzing it through the use of Polymerase Chain Reaction (PCR) [9]. Thoughthe method is very efficient, the laboratory test costs from 4,500 to 7,000 pesoswhich is expensive for an average Filipino [2]. Data shows that there were atotal of 69,088 of dengue cases reported nationwide from January 1 to July 28,2018 but only 301 cases were confirmed via PCR [8].According to the World health Organization, 40 percent of the global pop-ulation is estimated to be at risk of dengue fever [29]. Hence, there is a growingpublic health need for effective preventive interventions against dengue. Deng-vaxia is the first dengue vaccine to be licensed. The vaccine was intended toprevent all four dengue types in individuals from 9 to 45 years of age livingin endemic areas. In 2017, the manufacturer recommended that the vaccineonly be used in people who have previously had a dengue infection, as out-comes may be worsened in those who have not been previously infected.Thishas caused a scandal in the Philippines where more than 733,000 childrenwere vaccinated regardless of serostatus [30]. News have exposed deaths dueto Dengvaxia and because of this the use of the vaccine is now permanentlybanned in the country.In this paper, we propose a compartmental model describing the dynam-ics of dengue transmission considering its four strains. The model is age-structured taking into account the age requirement for the implementation ofthe Dengvaxia vaccine. Structuring our model this way helps us easily incorpo-rate the control variable for Dengvaxia. There are existing models consideringthe four strains of dengue but our proposed model have the advantage of fo-cusing only on the number of strains a person have been infected to and donot need any information on what specific strain(s) the person have been in-fected to. This is mathematically advantageous because the model now needsfewer equations than when one needs to be specific about the strains. Thisstudy aims to know how effective Dengvaxia could have been if it was imple-mented succesfully. This could be done by using optimal control theory on ourmodel. Since Dengvaxia is now banned in the country, we also investigate onthe other usual alternatives, like vector control and seeking early medical helpand compare these with Dengvaxia. ulti-Strain Age-Structured Dengue Transmission Model The proposed model considers eight susceptible classes depending on how manytimes a person was infected with dengue. The researcher considers primary,secondary, tertiary and quaternary infection considering the four serotypes ofdengue. But unlike the model in [1], the specific kind of strain that infectsthe person will not matter anymore. The model divides the human populationinto two compartments: 9-45 years old human population and the rest ofthe human population taking into account the age implemetantation of theDengvaxia vaccine. Pandey et al. [25] compared SIR model and vector-hostmodel in dengue transmission and finds it that explicitly incorporating themosquito population may not be necessary in modeling dengue transmissionfor some populations.The total population at time t , denoted by N ( t ), is subdivided into eighteencompartments. Age group a consist of individuals of age less than 9 years orgreater than 45 years. Age group b consist of individuals of age 9 up to 45years. Let i = 1 , , , j = a, b . We denote by S ij ( t ) the number ofindividuals of age group j who are susceptible to i strain(s) at time t ; I ij ( t )the number of individuals of age group j who were infected with any i − t ) infected with a different strain;and R j is the number of individuals of age group j who had been infected byall the strains and has now (at time t ) completely recovered. It is assumedthat all individuals are born susceptible to any of the dengue strains and thusenter the class S a and a recovered individual from a strain of dengue willhave complete immunity to that strain. Humans leave the population throughnatural death rate and through per-capita death rate due to infection. Humansof age less than 9 years in the susceptible class ( S ia ) leave the population andmove to infected class ( I ia ) or grow older to susceptible class S ib . Furthermore,humans of age greater than 45 years in the susceptible class ( S ib ) leave thepopulation and move to infected class ( I ib ) or grow older to susceptible class S ia . In general, any member of the human population remains susceptiblewith the four strains of dengue and stays in the class S j for a certain periodunless bitten and infected with one strain. Once in the infective class I j ,an individual may die or recover from that strain. If recovered from the firstinfection, susceptibility remains to the other 3 strains and then transfered to S j . If the same individual is infected with another strain, then he will bemoved to the class I j . Once recovered from that strain, he is still susceptibleto the other two strains and now belongs to the class S j . The cycle repeatsuntil the individual gets infected four times and identified into the class I j . Ifever the person survived the fourth infection, he then completely recovers andremains in the class R j .The parameters used are transmission coefficient ( α ij ), recovery rate ( β ij ), ulti-Strain Age-Structured Dengue Transmission Model µ ), natural death rate ( δ ), per-capita death rate through infection( γ ij ), rate of progression from age group a to b ( (cid:15) a ), and rate of progressionfrom age group b to a ( (cid:15) b ). The following figure gives the flow diagram of thedengue transmission model.Figure 1: Flow chart of the modelThe model can be mathematically described by the following system of 18 ulti-Strain Age-Structured Dengue Transmission Model dS a dt = µN + (cid:15) b S b − ( (cid:15) a + δ ) S a − α a S a INdS ia dt = (cid:15) b S ib + β (4 − i ) a I (4 − i ) a − ( (cid:15) a + δ ) S ia − α (5 − i ) a S ia IN , for i = 1 , , dI ia dt = α ia S (5 − i ) a IN + (cid:15) b I ib − ( (cid:15) a + δ + γ ia + β ia ) I ia , for i = 1 , , , dR a dt = β a I a + (cid:15) b R b − ( δ + (cid:15) a ) R a (1) dS b dt = (cid:15) a S a − ( (cid:15) b + δ ) S b − α b S b INdS ib dt = (cid:15) a S ia + β (4 − i ) b I (4 − i ) b − ( (cid:15) b + δ ) S ib − α (5 − i ) b S ib IN , for i = 1 , , dI ib dt = α ib S (5 − i ) b IN + (cid:15) a I ia − ( (cid:15) b + δ + γ ib + β ib ) I ib , for i = 1 , , , dR b dt = β b I b + (cid:15) a R a − ( (cid:15) b + δ ) R b where I = (cid:88) i =1 ( I ia + I ib ) and N = (cid:88) i =1 ( S ia + S ib + I ia + I ib ) + R a + R b . Notethat dNdt = µN − δN − (cid:88) i =1 ( γ ia I ia + γ ib I ib ) . To analyze the model, fractionalquantities will be used and this could be done by scaling the population ofeach class with the total population. The new variables will be as follows: S iA = S ia N , I iA = I ia N , S iB = S ib N , I iB = I ib N , R A = R a N , and R B = R b N , for i = 1 , , ,
4. Note that capital letters are used as subscripts to denote thescaled quantities. Furthermore, observe that (cid:88) ( S iA + S iB + I iA + I iB ) + R A + R B = 1or S A = 1 − (cid:32) (cid:88) i =1 ( S iA ) + (cid:88) i =1 ( S iB + I iA + I iB ) + R A + R B (cid:33) . Using this and differentiating each fractional quantities with respect to time, ulti-Strain Age-Structured Dengue Transmission Model dI A dt = α a I N x + (cid:15) b I B − ( (cid:15) a + µ + γ a + β a ) I A + I A JdS iA dt = (cid:15) b S iB + β (4 − i ) a I (4 − i ) A − ( (cid:15) a + µ ) S iA − α (5 − i ) a S iA I N + S iA J, for i = 1 , , dI iA dt = α ia S (5 − i ) A I N + (cid:15) b I iB − ( (cid:15) a + µ + γ ia + β ia ) I iA + I iA J, , for i = 2 , , dR A dt = β a I A + (cid:15) b R B − ( (cid:15) a + µ ) R A + R A JdS B dt = (cid:15) a x − ( (cid:15) b + µ ) S B − α b S B I N + S B J (2) dI iB dt = α b S (5 − i ) B I N + (cid:15) a I iA − ( (cid:15) b + µ + γ ib + β ib ) I iB + I iB J, for i = 1 , , , dS iB dt = (cid:15) a S iA + β (4 − i ) b I (4 − i ) B − ( (cid:15) b + µ ) S iB − α (5 − i ) b S iB I N + S iB J, for i = 1 , , dR B dt = β B I B + (cid:15) a R A − ( (cid:15) b + µ ) R B + R B J where I N = (cid:88) i =1 ( I iA + I iB ) and J = (cid:80) i =1 ( γ ia I iA + γ ib I iB ).This system of equations is epidemiologically and mathematically well-posed on the domain D = (cid:26) ( S A , S A , S A , I A , I A , I A , I A , R A , S B , S B , S B , S B , I B , I B , I B ,I B , R B ) ∈ R (cid:12)(cid:12)(cid:12)(cid:12) S iA ≥ , I ia ≥ , R A ≥ , S iA ≥ , I iA ≥ ,R B ≥ , where i = 1 , , , (cid:32) (cid:88) i =1 ( S iA + S iB ) + S B + (cid:88) i =1 ( I iA + I iB ) + R a + R b (cid:33) ≤ (cid:27) . The space R denotes the positive orthant in R . In this section, we obtain the disease-free equilibrium point and the repro-ductive number of the model. A disease-free equilibrium (DFE) is a steadystate solution of an epidemic model with all infected variables equal to zero[6]. Over time, we want to achieve a disease-free state. A threshold that de-termines whether a disease-free state is achievable or not is the reproductivenumber R . The reproductive number is the expected number of individu-als infected by a single infected individual over the duration of the infectiousperiod in a population, which is entirely susceptible [11]. The reproductionnumber also gives an idea of which parameters in our model may be significant ulti-Strain Age-Structured Dengue Transmission Model R . Theorem 2.1
Assuming that the initial conditions lie in D , the system ofequations (4) has a unique solution that exists and remains in D for all time t ≥ .Proof : Note that the right-hand side of the system of equations (2) is con-tinuous with continuous partial derivatives in D , by Cauchy-Lipschitz The-orem, the system of equations (2) has a unique solution. To show that D is forward invariant, note that if I iA = 0 then dI iA dt ≥ i = 1 , , , S iA = 0 then dS iA dt ≥ i = 1 , ,
3. If S iB = 0 then dS iB dt ≥ i = 1 , , ,
4. If I iB = 0 then dI iB dt ≥ i = 1 , , ,
4. Note also that if (cid:88) i =1 ( S iA + S iB ) + S B + (cid:88) i =1 ( I iA + I iB ) + R A + R B = 1, it follows that (cid:88) i =1 dS iA dt + (cid:88) i =1 dI iA dt + (cid:88) i =1 dS iB dt + (cid:88) i =1 dI iB dt + dR A dt + dR B dt < . Hence, none of the orbits can leave D and a unique solution exists for all time. (cid:4) Theorem 2.2
The disease-free equilibrium point of the epidemic model (2) is (cid:18) , , , , , , , , (cid:15) a (cid:15) a + (cid:15) b + µ , , , , , , , , (cid:19) . Proof : Set the right-hand side of system (2) to zero and let I iA = I iB = 0 for i = 1 , , ,
4. Since µ > D , we have S B = S A = S B = S A = S B = S A = R A = R B = 0. With these values, we are left with (cid:15) (1 − S B ) − ( (cid:15) + µ ) S B = 0. Thus, we have S B = (cid:15) (cid:15) + (cid:15) + µ . (cid:4) Theorem 2.3
The reproductive number for system (2) is R = α a (cid:18) (cid:15) + µ(cid:15) + (cid:15) + µ (cid:19) (cid:18) ( (cid:15) + µ + γ b + β b + (cid:15) )( (cid:15) + µ + γ a + β a )( (cid:15) + µ + γ b + β b ) − (cid:15) (cid:15) (cid:19) + α b (cid:18) (cid:15) (cid:15) + (cid:15) + µ (cid:19) (cid:18) ( (cid:15) + µ + γ a + β a + (cid:15) )( (cid:15) + µ + γ a + β a )( (cid:15) + µ + γ b + β b ) − (cid:15) (cid:15) (cid:19) , and the system is asymptotically stable if R < and unstable if R > . ulti-Strain Age-Structured Dengue Transmission Model Proof : We form the matrices F and V where F is the matrix of the rates ofappearance of new infections and V is the matrix of the rates of transfer ofindividuals out of the compartments. Let X be the vector of infected classesand Y be the vector of susceptible and recoverd classes.Let F ( X, Y ) be the vector of new infection rates (flows from Y to X ).Let V ( X, Y ) be the vector of all other rates (not new infection). These ratesinclude flows from X to Y (for instance, recovery rates), flows within X andflows leaving from the system (for instance, death rates). The next generationoperator formed is K = F V − where F = (cid:20) ∂ F ∂X (cid:21) , V = (cid:20) ∂ V ∂X (cid:21) . Evaluatedat the disease-free equilibrium (0 , , , , , , , , S ∗ B , , , , , , , ,
0) where S ∗ B = (cid:15) a (cid:15) a + (cid:15) b + µ , this becomes F ( I A , I A , . . . , I B ) = α a x α a x α a x α a x α a x α a x α a x α a x α b S ∗ B α b S ∗ B α b S ∗ B α b S ∗ B α b S ∗ B α b S ∗ B α b S ∗ B α b S ∗ B where x = 1 − (cid:15) a (cid:15) a + (cid:15) b + µ = (cid:15) b + µ(cid:15) a + (cid:15) b + µ . We also have, V ( I A , I A , . . . , I B ) = v (cid:48) − (cid:15) b v (cid:48) − (cid:15) b v (cid:48) − (cid:15) b
00 0 0 v (cid:48) − (cid:15) b − (cid:15) a v (cid:48) − (cid:15) a v (cid:48) − (cid:15) a v (cid:48)
00 0 0 − (cid:15) a v (cid:48) where v (cid:48) = (cid:15) a + µ + γ a + β a v (cid:48) = (cid:15) b + µ + γ b + β b v (cid:48) = (cid:15) a + µ + γ a + β a v (cid:48) = (cid:15) b + µ + γ b + β b v (cid:48) = (cid:15) a + µ + γ a + β a v (cid:48) = (cid:15) b + µ + γ b + β b v (cid:48) = (cid:15) a + µ + γ a + β a v (cid:48) = (cid:15) b + µ + γ b + β b ulti-Strain Age-Structured Dengue Transmission Model V − ( I A , I A , . . . , I B ) = d b d b d b
00 0 0 d b c d c d c d
00 0 0 c d where d = v (cid:48) v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , d = v (cid:48) v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , d = v (cid:48) v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , d = v (cid:48) v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , d = v (cid:48) v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , d = − v (cid:48) v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , d = v (cid:48) v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , d = v (cid:48) v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , c = (cid:15) a v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , c = (cid:15) a v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , c = (cid:15) a v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , c = (cid:15) a v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , b = (cid:15) b v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , b = (cid:15) b v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , b = (cid:15) b v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , b = (cid:15) b v (cid:48) v (cid:48) − (cid:15) a (cid:15) b .Now, the next generation matrix K is formed as K = F ( I A , I A , . . . , I B ) V − ( I A , I A , . . . , I B ) . Using cofactor expansion to get the eigenvalue of K , we have | K − λI | = ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) α a (1 − S ∗ B )( d + c ) − λ α a (1 − S ∗ B )( d + b ) α b S ∗ B ( d + c ) α b S ∗ B ( d + b ) − λ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) = ( λ ) (cid:2) − λα a (1 − S ∗ B )( d + c ) − λα b S ∗ B ( d + b ) + λ (cid:3) = ( λ ) [ − α a x ( d + c ) − α b S ∗ b ( d + b ) + λ ]Hence we have, λ = α a x ( d + c ) + α b S ∗ b ( d + b ) and λ = 0 of multiplicity 7.Taking the dominant eigenvalue of matrix K , we get R = α a (1 − S ∗ B )( d + c ) + α b S ∗ B ( d + b ) . Substituting the values S ∗ B = (cid:15) a (cid:15) a + (cid:15) b + µ , x = (cid:15) b + µ(cid:15) a + (cid:15) b + µ , d = v (cid:48) v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , d = v (cid:48) v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , b = (cid:15) b v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , c = (cid:15) a v (cid:48) v (cid:48) − (cid:15) a (cid:15) b , v (cid:48) = (cid:15) a + µ + γ a + β a ,and v (cid:48) = (cid:15) b + µ + γ b + β b , we finally have R = α a (cid:18) (cid:15) b + µ(cid:15) a + (cid:15) b + µ (cid:19) (cid:18) ( (cid:15) b + µ + γ b + β b + (cid:15) a )( (cid:15) a + µ + γ a + β a )( (cid:15) b + µ + γ b + β b ) − (cid:15) a (cid:15) b (cid:19) + α b (cid:18) (cid:15) a (cid:15) a + (cid:15) b + µ (cid:19) (cid:18) ( (cid:15) a + µ + γ a + β a + (cid:15) b )( (cid:15) a + µ + γ a + β a )( (cid:15) b + µ + γ b + β b ) − (cid:15) a (cid:15) b (cid:19) . (cid:4) ulti-Strain Age-Structured Dengue Transmission Model To mitigate the spread of dengue in the Philippines, three control interventionstrategies u i ( t ), i = 1 , , transmission reduc-tion conrol u ( t ). This consists of holistic and effective methods in fightingmosquito at every stage of their life cycle, such as searching and eliminatingbreeding grounds, larvicide treatment, using adult mosquito trap, insecticidesand thermal fogging during outbreaks. This also encompasses environmen-tal management and sources reduction, self-protection measures which is juststrengthening the capability of a person to avoid dengue and health educa-tion to provide awareness for their protection against dengue. Next strategy is proper medical care control u ( t ). This means an effort for patients to seekearly medication and immediate reporting to the nearest health care facilityas symptoms are observed. With this control, chances of recovery increasesspecially from severe dengue. Last control is the Dengvaxia vaccine u ( t ).This controversial Dengvaxia vaccine could have been a major breakthroughfor dengue prevention in the country, but, the Philippines holds a lot of pend-ing issues against its implementation. Moreover, it has been banned and notallowed for marketing in the Philippines. In this paper, we would like to knowwhat could be the best effect of Dengvaxia (number of infected individuals areminimized, while the intervention costs are kept low) if it was properly imple-mented, that is, if it had been given only to susceptible population from 9-45years old and to those who have had previous infection. We also investigate onthe best possible impact of the other intervention strategies to compare it withDengvaxia. This study is influenced by a similar study done for tuberculosisin the Philippines in [17].The dengue state dynamics incorporating the different controls can now be ulti-Strain Age-Structured Dengue Transmission Model dS a dt = µN + (cid:15) b S b − ( (cid:15) a + δ ) S a − (1 − u ( t )) α a S a INdS ia dt = (cid:15) b S ib + (1 + u ( t )) β (4 − i ) a I (4 − i ) a − ( (cid:15) a + δ ) S ia − (1 − u ( t )) α (5 − i ) a S ia IN , for i = 1 , , dI ia dt = (1 − u ( t )) α ia S (5 − i ) a IN + (cid:15) b I ib − ( (cid:15) a + δ + γ ia ) I ia − (1 + u ( t )) β ia I ia , for i = 1 , , , dR a dt = (1 + u ( t )) β a I a + (cid:15) b R b − ( δ + (cid:15) a ) R a dS b dt = (cid:15) a S a − ( (cid:15) b + δ ) S b − (1 − u ( t )) α b S b IN (3) dS ib dt = (cid:15) a S ia + (1 + u ( t )) β (4 − i ) b I (4 − i ) b − ( (cid:15) b + δ ) S ib − (1 − u ( t )) α (5 − i ) b S ib IN − u S ib , for i = 1 , , dI ib dt = (1 − u ( t )) α ib S (5 − i ) b IN + (cid:15) a I ia − ( (cid:15) b + δ + γ ib ) I ib − (1 + u ( t )) β ib I ib , for i = 1 , , dR b dt = (1 + u ( t )) β b I b + (cid:15) a R a − ( (cid:15) b + δ ) R b dNdt = µN − δN − (cid:88) i =1 ( γ ia I ia + γ ib I ib ) − u ( S b + S b + S b )Transmission reduction control u ( t ) is applied on all compartments havingthe transmission rates and proper medical control u ( t ) is incorporated on allcompartments involving recovery rates. The Dengvaxia control u ( t ) is onlyapplied on age group b since it has restrictions on its implementation to peopleof age 9-45. Moreover, it is not incorporated to the compartment S b since itmust only be applied to people who have had previous infection.The controls range is the interval (0 , u i ( t ) ≡ u i ( t ) ≡ u ( t ) , u ( t ) and u ( t ), the objective functional to be minimized is ulti-Strain Age-Structured Dengue Transmission Model J ( u , u , u ) = (cid:90) T T (cid:20) I ( t ) + B u ( t ) + B u ( t ) + B u ( t ) (cid:21) dt where I ( t ) = (cid:88) i =1 ( I ia ( t ) + I ib ( t )). The bounds T and T are taken as 2020and 2040, respectively, to consider a 20-year dengue intervention program and B i ’s, i = 1 , ,
3, represent the weight constants associated to the relative costsof implementating the respective control strategy. These constants balancedthe size and importance of each term in the integrand. In this work, we seekto find optimal controls u ∗ , u ∗ and u ∗ satisfying J ( u ∗ , u ∗ , u ∗ ) = min Ω J ( u , u , u ) , where Ω = (cid:8) ( u , u , u ) (cid:12)(cid:12) a ≤ u i ( t ) ≤ b, u i ∈ L (2020 , , i = 1 , , (cid:9) . Pa-rameters a and b are the upper and lower bounds of the controls and areassumed to be 0.05 and 0.95, respectively. The weight parameters are set as B = B = B = 10 . The goal is to minimize the number of infected individuals and correspondingcosts. From Pontryagin’s Maximum principle, optimal controls should satisfythe necessary conditions. Pontryagin’s Maximum Principle changes into aproblem that minimize pointwise a Hamiltonian H , with respect to the control.We have H = I ( t ) + B u ( t ) + B u ( t ) + B u ( t ) + (cid:88) i =1 λ i g i , where g i is theright hand side of the differential equation of the i th state variable. ApplyingPontryagin’s Maximum Principle, we obtain the following theorem. Theorem 3.1
There exist optimal controls u ∗ ( t ) , u ∗ ( t ) and u ∗ ( t ) minimizingthe objective functional Ω = (cid:8) ( u , u , u ) (cid:12)(cid:12) a ≤ u i ( t ) ≤ b, u i ∈ L (2020 , , i = 1 , , (cid:9) . Given these optimal solutions, there exist adjoint variables, λ ( t ) , λ ( t ) , . . . , λ ( t ) ,which satisfy ulti-Strain Age-Structured Dengue Transmission Model dλ dt = λ (cid:20) (1 − u ( t )) α a IN + (cid:15) + δ (cid:21) − λ (cid:20) (1 − u ( t )) α a IN (cid:21) − λ (cid:15) dλ dt = − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ γ a − λ (cid:20) (1 − u ( t )) α a S a N − ( (cid:15) + δ ) − γ a − (1 + u ( t )) β a (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N − (1 + u ( t )) β a (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N + (cid:15) (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) dλ dt = λ (cid:20) (1 − u ( t )) α a IN + (cid:15) + δ (cid:21) − λ (cid:20) (1 − u ( t )) α a IN (cid:21) − λ (cid:15) dλ dt = − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ γ a − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N − ( (cid:15) + δ ) − γ a − (1 + u ( t )) β a (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N − (1 + u ( t )) β a (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N + (cid:15) (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) dλ dt = λ (cid:20) (1 − u ( t )) α a IN + (cid:15) + δ (cid:21) − λ (cid:20) (1 − u ( t )) α a IN (cid:21) − λ (cid:15) dλ dt = − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ γ a − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N − ( (cid:15) + δ ) − γ a − (1 + u ( t )) β a (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N − (1 + u ( t )) β a (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N + (cid:15) (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) dλ dt = λ (cid:20) (1 − u ( t )) α a IN + (cid:15) + δ (cid:21) − λ (cid:20) (1 − u ( t )) α a IN (cid:21) − λ (cid:15) dλ dt = − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ [(1 + u ( t )) β a ] − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N − ( (cid:15) + δ ) − γ a − (1 + u ( t )) β a (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N + (cid:15) (cid:21) + λ γ a dλ dt = λ ( (cid:15) + δ ) − λ (cid:15) dλ dt = λ (cid:20) (1 − u ( t )) α b IN + (cid:15) + δ (cid:21) − λ (cid:20) (1 − u ( t )) α b IN (cid:21) − λ (cid:15) ulti-Strain Age-Structured Dengue Transmission Model dλ dt = − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N + (cid:15) (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N − ( (cid:15) + δ ) − γ b − (1 + u ( t )) β b (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N − (1 + u ( t )) β b (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ γ b dλ dt = λ (cid:20) (1 − u ( t )) α b IN + (cid:15) + δ + u (cid:21) − λ (cid:20) (1 − u ( t )) α b IN (cid:21) − λ (cid:15) + λ u dλ dt = − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N + (cid:15) (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N − ( (cid:15) + δ ) − γ b − (1 + u ( t )) β b (cid:21) + λ γ b + λ (cid:20) (1 − u ( t )) α b S b N − (1 + u ( t )) β b (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) dλ dt = λ (cid:20) (1 − u ( t )) α b IN + (cid:15) + δ + u (cid:21) − λ (cid:20) (1 − u ( t )) α b IN (cid:21) − λ (cid:15) + λ u dλ dt = − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N + (cid:15) (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N − ( (cid:15) + δ ) − γ b − (1 + u ( t )) β b (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N − (1 + u ( t )) β b (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ γ b dλ dt = λ (cid:20) (1 − u ( t )) α b IN + (cid:15) + δ + u (cid:21) − λ (cid:20) (1 − u ( t )) α a IN (cid:21) − λ (cid:15) + λ u dλ dt = − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ [(1 + u ( t )) β b ] − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N (cid:21) + λ (cid:20) (1 − u ( t )) α a S a N (cid:21) − λ (cid:20) (1 − u ( t )) α a S a N + (cid:15) (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N (cid:21) + λ [(1 + u ( t )) β b ]+ λ (cid:20) (1 − u ( t )) α b S b N (cid:21) − λ (cid:20) (1 − u ( t )) α b S b N − ( (cid:15) + δ ) − γ b − (1 + u ( t )) β b (cid:21) + λ γ b dλ dt = λ ( (cid:15) + δ ) − λ (cid:15) ulti-Strain Age-Structured Dengue Transmission Model dλ dt = − λ (cid:20) µ + (1 − u ( t )) α a S a IN (cid:21) + λ (cid:20) (1 − u ( t )) α a S a IN (cid:21) − λ (cid:20) (1 − u ( t )) α a S a IN (cid:21) + λ (cid:20) (1 − u ( t )) α a S a IN (cid:21) − λ (cid:20) (1 − u ( t )) α a S a IN (cid:21) + λ (cid:20) (1 − u ( t )) α a S a IN (cid:21) − λ (cid:20) (1 − u ( t )) α a S a IN (cid:21) + λ (cid:20) (1 − u ( t )) α a S a IN (cid:21) − λ (cid:20) (1 − u ( t )) α b S b IN (cid:21) + λ (cid:20) (1 − u ( t )) α b S b IN (cid:21) − λ (cid:20) (1 − u ( t )) α b S b IN (cid:21) + λ (cid:20) (1 − u ( t )) α b S b IN (cid:21) + λ [ µ − δ ] − λ (cid:20) (1 − u ( t )) α b S b IN (cid:21) + λ (cid:20) (1 − u ( t )) α b S b IN (cid:21) − λ (cid:20) (1 − u ( t )) α b S b IN (cid:21) + λ (cid:20) (1 − u ( t )) α b S b IN (cid:21) with transversality conditions λ i ( t f ) = 0 , for i = 1 , , . . . , . Furthermore, u ∗ ( t ) = min (cid:18) b, max (cid:18) a, − ZB (cid:19)(cid:19) ,u ∗ ( t ) = min (cid:18) b, max (cid:18) a, − XB (cid:19)(cid:19) ,u ∗ ( t ) = min (cid:18) b, max (cid:18) a, λ S b + λ S b + λ S b + λ [ S b + S b + S b ] B (cid:19)(cid:19) where Z = λ α a S a IN − λ (cid:18) α a S a IN (cid:19) + λ α a S a IN − λ (cid:18) α a S a IN (cid:19) λ α a S a IN − λ (cid:18) α a S a IN (cid:19) + λ α a S a IN − λ (cid:18) α a S a IN (cid:19) λ α b S b IN − λ (cid:18) α b S b IN (cid:19) + λ α b S b IN − λ (cid:18) α b S b IN (cid:19) λ α b S b IN − λ (cid:18) α b S b IN (cid:19) + λ α b S b IN − λ (cid:18) α b S b IN (cid:19) and X = ( λ − λ ) β a I a + ( λ − λ ) β a I a + ( λ − λ ) β a I a ( λ − λ ) β a I a + ( λ − λ ) β b I b + ( λ − λ ) β b I b + ( λ − λ ) β b I b + ( λ − λ ) β b I b . Proof : The existence of optimal controls u ∗ ( t ) , u ∗ ( t ) , and u ∗ ( t ) such that J ( u ∗ ( t ) , u ∗ ( t ) , u ∗ ( t )) = min Ω ( u , u , u ), with state system (3) is given by theconvexity of the objective functional integrand. By Pontryagin’s MaximumPrinciple, the adjoint equations and transversality conditions can be obtained.Differentiation of the Hamiltonian H with respect to the state variable givesthe following system: dλ dt = − ∂H∂S a , dλ dt = − ∂H∂I a , dλ dt = − ∂H∂S a , dλ dt = − ∂H∂I a ,dλ dt = − ∂H∂S a , dλ dt = − ∂H∂I a , dλ dt = − ∂H∂S a , dλ dt = − ∂H∂I a ,dλ dt = − ∂H∂R a , dλ dt = − ∂H∂S b , dλ dt = − ∂H∂I b , dλ dt = − ∂H∂S b , ulti-Strain Age-Structured Dengue Transmission Model dλ dt = − ∂H∂I b , dλ dt = − ∂H∂S b , dλ dt = − ∂H∂I b , dλ dt = − ∂H∂S b ,dλ dt = − ∂H∂I b , dλ dt = − ∂H∂R b , dλ dt = − ∂H∂N , with λ i ( t f ) = 0, for i = 1 , , . . . , . Optimal controls u ∗ ( t ) , u ∗ ( t ) , and u ∗ ( t ) are derived by the following opti-mality conditions: ∂H∂u = B u + (cid:20) λ α a S a IN − λ (cid:18) α a S a IN (cid:19) + λ α a S a IN − λ (cid:18) α b S b IN (cid:19) − λ (cid:18) α a S a IN (cid:19) + λ α a S a IN − λ (cid:18) α a S a IN (cid:19) + λ α a S a IN − λ (cid:18) α a S a IN (cid:19) + λ α b S b IN − λ (cid:18) α b S b IN (cid:19) + λ α b S b IN − λ (cid:18) α b S b IN (cid:19) + λ α b S b IN − λ (cid:18) α b S b IN (cid:19) + λ α b S b IN = 0 ∂H∂u = B u + ( λ − λ ) β a I a + ( λ − λ ) β a I a + ( λ − λ ) β a I a ( λ − λ ) β a I a + ( λ − λ ) β b I b + ( λ − λ ) β b I b + ( λ − λ ) β b I b + ( λ − λ ) β b I b = 0 ∂H∂u = B u − λ S b − λ S b − λ S b − λ ∗ ( S b + S b + S b ) = 0 at u ∗ ( t ) , u ∗ ( t ) , and u ∗ ( t ) on the set Ω. On this set u ∗ ( t ) = − ZB u ∗ ( t ) = − XB u ∗ ( t ) = 1 B [ λ S b + λ S b + λ S b + λ ( S b + S b + S b )]where Z = λ α a S a IN − λ (cid:18) α a S a IN + γ a I a (cid:19) + λ α a S a IN − λ (cid:18) α a S a IN + γ a I a (cid:19) λ α a S a IN − λ (cid:18) α a S a IN + γ a I a (cid:19) + λ α a S a IN − λ (cid:18) α a S a IN + γ a I a (cid:19) λ α b S b IN − λ (cid:18) α b S b IN + γ b I b (cid:19) + λ α b S b IN − λ (cid:18) α b S b IN + γ b I b (cid:19) λ α b S b IN − λ (cid:18) α b S b IN + γ b I b (cid:19) + λ α b S b IN − λ (cid:18) α b S b IN + γ b (cid:19) + λ (cid:32) (cid:88) i =1 ( γ ia I ia + γ ib I ib ) (cid:33) , ulti-Strain Age-Structured Dengue Transmission Model X = ( λ − λ ) β a I a + ( λ − λ ) β a I a + ( λ − λ ) β a I a ( λ − λ ) β a I a + ( λ − λ ) β b I b + ( λ − λ ) β b I b + ( λ − λ ) β b I b + ( λ − λ ) β b I b . Taking into account the bounds on controls, we obtain the characterization of u ∗ ( t ) , u ∗ ( t ) , and u ∗ ( t ). We have u ∗ ( t ) = min (cid:18) b, max (cid:18) a, − ZB (cid:19)(cid:19) ,u ∗ ( t ) = min (cid:18) b, max (cid:18) a, − XB (cid:19)(cid:19) ,u ∗ ( t ) = min (cid:18) b, max (cid:18) a, B [ λ S b − λ S b − λ S b − λ ∗ ( S b + S b + S b )] (cid:19)(cid:19) . (cid:4) Pandey et al. [25] compared the SIR models and the vector-host models fordengue transmission and found that explicitly incorporating the mosquito pop-ulation may not be necessary in modeling dengue transmission for some pop-ulations. In their paper, by comparing the equilibria of the vector host modeland the SIR model they obtained the transmission coefficient β in terms of theparameters of the vector host model: β ≈ mc β H β V µ V where m is the number ofmosquitos per person, c is the biting rate, β H is the mosquito-to-human trans-mission probability, β V is the human-to-mosquito transmission probability, µ V is the mosquito mortality rate and β is the composite human to human trans-mission rate. Note that in our paper, we use α as our transmission coefficient.From [7], we can get the following values: biting rate c = 1, mosquito-to-human transmission probability β H = 0 . β V = 0 .
75, mosquito mortality rate µ V = 0 .
1. From [21], we alsoget the value of the number of mosquito per person m =1. Substituting theseparameter values to α , we have mc β H β V µ V = 1(1) (0 . . . . . This now becomes the value for α a and α b .Data from [8] reveals that 54 % of the dengue cases is caused by DENV3, 25 % is caused by DENV 1, 18 % is caused by DENV 2 and 3 % of the ulti-Strain Age-Structured Dengue Transmission Model α a = 0 . × α a , α a = 0 . × α a , α a = 0 . × α a , α b =0 . × α b , α b = 0 . × α b , and α b = 0 . × α b . Thus, we have α a = 1 . α a = 0 . α a = 0 . α b = 1 . α b = 0 . α b = 0 . β a should be lower compared to β b . We estimate β b = 0 . β a = 0 .
30. However, dengue can become severe in the next infectionsthus we may have β b = 0 . β b = 0 . β b = 0 . β a = 0 . β a = 0 .
15, and β a = 0 .
15. From the same literature,we also obtain the value for the death rate through infection given by γ =0 . γ a should be higher compared to γ b . We let γ b = 2 . × − and γ b = 3 . × − . In the next infections, dengue can become severe and maylead to more number of deaths . Thus, we let γ b = γ b = γ b = 4 . × − and γ a = γ a = γ a = 6 . × − .From [7], we have the birth rate µ equal to 8 . × − and death rate δ equal to 4 . × − . Furthermore, we estimate the growth rate from age group a to age group b to be (cid:15) = 6 . × − and the growth rate from age group b to age group a to be (cid:15) = 6 . × − . For our numerical simulations, we use the following initial values: S a = 2 . × , I a = 7 . × , S a = 8 . × , I a = 4 . × , S a = 4 . × , I a = 2 . × , S a = 1 . × , I a = 3 . × , R a = 1 . × , S b = 2 . × , I b = 1 . × , S b = 1 . × , I b = 6 . × , S b = 4 . × , I b = 2 . × , S b = 1 . × , I b = 6 . × , and R b = 3 . × .With the estimated parameters above and the set of initial conditions, andbeing assured of the existence of the optimal controls u ∗ ( t ) , u ∗ ( t ) , and u ∗ ( t )by Theorem 4.1.1, we performed the numerical simulations. The results aregiven by the following figures. On each of the figures, the graph on the leftshows how the control should be implemented to have the minimum value ofthe objective functional and the graph on the right shows the correspondingeffect of the implementation on the total number of infected individuals. Theblack graph represents the total number of infected individuals over time when ulti-Strain Age-Structured Dengue Transmission Model ulti-Strain Age-Structured Dengue Transmission Model As expected, we can see a significant decrease in the total number of infectedindividuals if Dengvaxia is implemented with the optimal strategy. However,one can clearly see that the optimal strategies for the other alternative controlshave better impacts than that of Dengvaxia. The best impact if only onecontrol is to be used can be seen in the optimal strategy for the transmissionreduction control. These results is telling us that even without Dengvaxia wecan still reduce the number of infected individuals and the reduction could be ulti-Strain Age-Structured Dengue Transmission Model
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