key: cord-0988667-osx45zma authors: Jia, Peiqi; Yang, Junyuan; Li, Xuezhi title: Optimal control and cost-effective analysis of an age-structured emerging infectious disease model date: 2021-12-25 journal: Infect Dis Model DOI: 10.1016/j.idm.2021.12.004 sha: 5b31a235e199fae6799d9e5db0b05f96302cc8f2 doc_id: 988667 cord_uid: osx45zma Emerging infectious diseases are one of the global public health problems which may lead to widespread epidemics and potentially life-threatening infection. Integrated vaccination and physical distancing interventions are two elementary methods for preventing infectious diseases transmission. In this paper, we construct a continuous age-structured model for investigating the transmission dynamics of an emerging infection disease during a short period. We derive the basic regeneration number [Formula: see text] , the spectral radius of the next generation operator [Formula: see text] , which determines the disease outbreak or not. Furthermore, we propose an optimal control problem to take account for the cost-effectiveness of social distancing intervention and vaccination. We rigorously obtain sufficient conditions for a L(1) control problem. Numerical simulations show that coupling integrated vaccination and physical distancing intervention could effectively eliminate the infection, and such control strategy is more sensitive for people aged 10–39 and over 60. Emerging infectious diseases are infections that have recently appeared within a population or those whose incidence or geographic range is rapidly increasing or threatens to increase in the near future (Jones et al., 2008) . Since 1940, there has been reported more than 335 emerging infectious diseases caused by pathogens, including newly evolved strains of pathogens (muti-strain resistent tuberculosis and chloroquine-resistent malaria), novel pathogens , and historically existing and recently increasing pathogens (Lyme and Ebola). The continuous outbreaks of emerging infectious diseases have led to some extremely serious impacts on global public health and economics. As reported, about 575,400 people worldwide died from H1N1 infection in 2009, SARS firstly reported in 2003 has spread more than 29 countries with 8096 cases and with 774 deaths, Ebola periodically appears and leads to more than 27,000 infectives and more than 11,000 deaths. Especially, COVID-19 has caused by a new coronavirus, and SARS-CoV-2 has resulted in more than 200 million cases and with 2.5 million deaths and it plausibly survives together with mankind. Hence, it is an ungent need to find suiTable quantitative and qualitative theories and approaches for exemplifying key principles and providing pertinent control measures. The mathematical modeling of infectious diseases is an useful tool to understand the mechanisms of infectious diseases, which can anticipate their future development trend and evaluate control measures (Martcheva, 2015a) . Pandemic H1N1 is transmitted by person-to-person (Fraser et al., 2009) ; Middle East Respiratory Syndrome-associated coronavirus (MERS-CoV) is driven by zoonotic infections from camels ; Ebola is spread through direct with serious sick or dead individuals (Agua-Agum et al., 2016) ; Zika virus has proved through mosquito-to-human (Majumder et al., 2018) ; COVID-19 can be transmitted by both person-to-person and environment-to-human (Eslami & Jalili, 2020) . These identified mechanisms are in favor of understanding the principles of the disease dynamics and designing corresponding control strategies. To evaluate the seriousness of emerging infectious diseases, forecasting future epidemic trajectory and potential numbers of unobserved cases become an ungent issue. Bayesian probability models are usually employed to predict the incidence cases and correct the underestimations for fatality rate (Ghani et al., 2005) . Facing an emerging infectious disease outbreak, one commonsense view is to develop a instant vaccine and exploit antiviral therapy, as well as improve the early detection, isolate the diagnosed cases and quarantine those individuals close contacting with confirmed cases for reducing the prevalence. Antimicrobial use and antiviral prophylaxis have been illustrated effectively to develop resistance and delay the spread of influenza pandemic (Glass et al., 2006) and it also involves social distancing measures such as isolation of suspected cases, school closures, travel restrictions and cancellation of mass gathering (Gojovic and SanderFisman, 2009) . Massive vaccination has been extensively programmed to prevent the transmissions of COVID-19, HBV and HPV (Beutels, 1988; Lazarus et al., 2021) . Most of epidemic models have been assumed to be well-mixed with homogeneous disease infection between susceptibles and infectives. Generally, such homogeneous assumptions are suiTable to study the diseases well established in a population and it means that the transmission equally happens among humans, regardless of their age or any other behavior traits. However, some evidences and serological surveys have shown that the frequency of immunity to disease increases with age (Anderson & May 1974) . The gradual increase of age-dependent variation in severity has been observed (Gao et al., 2016) . The influenza mortality rate ranks children, young adults and the elderly (Oei & Nishiura, 2012) . About 31e59% of the most severe symptomatic cases and particularly fatal cases of COVID-19 caused by SARS-Cov-2 have age from 75 to 84 (Ahmed et al., 2020) . Understanding the role of age in transmission and disease severity is significant for determining the likely impact of social distancing interventions and vaccination programmes. Age-structured epidemic models have been proposed to characterize the heterogeneous transmission such as variable infectivity and variable susceptibility to infection (Li et al., 2020; Metz, 1978) . Response to an emerging infectious disease, policymakers and public health governments instantly take quarantine, isolation, treatment and vaccination for curbing the disease dispersion. Limitation to the knowledge of the disease, various resource production would likely be insufficient (Matrajt et al., 2021) and then effective disease controls will become a critical problem. The optimal control theory (Lev Semenovich Pontryagin, 1987; Wendell & Rishel, 2012 ) is a practical theory in epidemic control (Blayneh et al., 2009; Lenhart & Workman, 2007) with goals of reducing the deaths of infected individuals to project cost-effective intervention measures. Most of optimal control problems described by ordinary differential equations can be resolved by Pontryagins Maximum Principle, however, it is not always useful for studying such problems in L 1 space due to lack of the regularity of the solution of the state equations. As the aforementioned matter, age-heterogeneity is a key factor affecting the emerging infectious diseases spread. Hence, using time-dependent and age-specific control functions to measure the effectiveness of strategies is a feature of emerging infection disease model . Kwon et al. investigated the optimal treatment strategies with an age-structured model of HIV infection, and numerical simulations have demonstrated that the dynamic treatment strategy delays the peak arrival time of viral loads and reduces the sizes of that load (Kwon et al., 2012) . Demasse et al. proposed three control strategies, such as minimizing immunization of young adults, the reduction of perinatal infection and treatment of HBV symptomatic infections, to optimize HBV-related deaths through the cost-effective analysis. They showed that mass vaccination in infants might be not enough to eradicate the virus, and an optimal control strategy is a combination of immunization of young adults and treatment of HBV symptomatic infections (Demasse et al., 2016) . The main contributions of this paper contain two aspects. One is that we construct a mathematical model to characterize the age-heterogeneity of an emerging infectious disease model and furthermore, we propose two age-time varying control measures-reducing social distance and mass vaccination to reduce the deaths of infected cases. To overcome the regularity problem of a L 1 control problem, we rigorously prove the existence of the optimal control and clarify the specific conditions of such existence by Ekerland variational principle. The structure of this paper is organized as follows. In Section 2, we propose an age-structured model, calculate the basic reproduction number and completely study the global stability of the disease-free steady state. In Section 3, we take account for an optimal control problem dealing with mass vaccination efforts and reducing physical distancing. In this part, we take a great effort to prove the sufficient and necessary conditions of the existence of the optimal control pairs. Section 4 is conduct to the numerical experiments for identifying the optimal control strategies. The paper ends with a brief discussion in the last section. In this paper, we focus on infections recently appeared within a population in a short period. In this case, we ignore the demography of the population, and consider the age heterogeneity. Facing an emerging disease, vaccination is one of the effective control strategies for combating it. We divide the total population into six disjoint classes: susceptible, the latent, the infected, the hospitalized, the recovered and the vaccinated. S(t, a) denotes the density of susceptibles of age a at time t; E(t, a) denotes the density of the exposed of age a at time t; I(t, a) denotes the density of infectives of age a at time t; H(t, a) denotes the density of the hospitalized of age a at time t; V(t, a) denotes the density of the vaccinated of age a at time t; R(t, a) denotes the density of the recovered of age a at time t. Susceptibles with age a can be infected by infected individuals of age b at rate b(a, b) and move to the exposed class. A susceptible is vaccinated at rate j(a) and then becomes a vaccinator. The exposed transfer to the infected at rate a(a). Infectious individuals have been confirmed to be hospitalized individuals at rate h(a), and both infected and hospitalized individuals are recovered at rate g(a) and die at rate d(a). The vaccinated individuals can be infected at a reduced coefficient d compared with an original infection. To address the above mechanisms, we propose an age-structured emerging disease model as follows: (2.1) where u is the maximal active age for a person. System (2.1) has the following initial and boundary conditions: Sð0; aÞ ¼ S 0 ðaÞ; Eð0; aÞ ¼ E 0 ðaÞ; Ið0; aÞ ¼ I 0 ðaÞ; Hð0; aÞ ¼ H 0 ðaÞ; Vð0; aÞ ¼ V 0 ðaÞ; Rð0; aÞ ¼ R 0 ðaÞ: To study the dynamics of model (2.1), we make following assumptions on the model parameters from biological and mathematical view of points. Assumption 2.1. (1) The functions b2C BU ðR þ  R þ ; R þ Þ, where C BU is the set of all bounded and uniformly continuous functions from R þ  R þ to R þ (2) jð,Þ; að,Þ; hð,Þ; dð,Þ; gð,Þ2L ∞ þ ðR þ Þ, L ∞ þ ðR þ Þ is the positive cone of L ∞ ðR þ Þ. Obviously, system (2.1) always has the following disease-free equilibrium E 0 : (1) If j(a) ¼ 0, system (2.1) has a disease-free equilibrium E 01 ¼ (S 0 (a), 0, 0, 0, 0, 0); (2) If j(a) s 0, system (2.1) has a disease-free equilibrium E 02 ¼ (0, 0, 0, 0, V 0 (a), 0). Without loss of generation, let us assume E 0 (a) ¼ 0 for any a2R þ . Hence, the inverse of V is calculated by hðaÞ þ gðaÞ þ dðaÞ : Therefore, the next generation operator K is defined by K ½4ðaÞ ¼ FV À1 ½4ðaÞ ¼ S 0 ðaÞ Z u 0 bða; bÞ4ðbÞ N 0 ðbÞðhðbÞ þ gðbÞ þ dðbÞÞ db: (2.5) Thus, the reproduction number R 01 is defined by (2.6) where r denotes the spectral radius. Similarly, let us proceed the above process around the other disease-free equilibrium E 02 . We obtain the reproduction number for Case (2) as follows Theorem 2.1. Let R 0j ðj ¼ 1; 2Þ be defined in equations (2.6) and (2.7). If R 0j < 1ðj ¼ 1; 2Þ, then the disease-free equilibrium E 0j (j ¼ 1, 2) is locally asymptotically stable. Proof ðl þ ðhðaÞ þ gðaÞ þ dðaÞÞzðaÞ ¼ aðaÞyðaÞ; ðl þ ðgðaÞ þ dðaÞÞlðaÞ ¼ hðaÞzðaÞ: ), one has that zðaÞ ¼ aðaÞyðaÞ l þ hðaÞ þ gðaÞ þ dðaÞ : Replacing z(a) in the second equation of (2.9) leads to ðl þ aðaÞÞyðaÞ ¼ S 0 ðaÞ Z u 0 bða; bÞ aðbÞyðbÞ ðl þ hðbÞ þ g þ dðbÞÞN 0 ðbÞ dbdK ½ayðaÞ: (2.10) Suppose by argument, provided (2.10) has an eigenvalue l with non-negative real parts, then the left modulus of (2.10) is larger than kað,Þyð,Þk L 1 . On the other hand, the right side of (2.10) is less kað,Þyð,Þk L 1 if R 0 < 1. This leads to a contradiction. Hence, the roots of equation (2.10) have only negative real parts. , then one has that y(a) ¼ 0 for a.e. a2R þ . From the second and the fourth equations of (2.9), it follows that zðaÞ ¼ lðaÞ ¼ 0; a:e:a2R þ : Substituting these quantities into the first equation of (2.9), one must concludes that x(a) ¼ 0, for a.e. a2R þ , or l ¼ 0. The first case is impossible since (x,y,z,l) T is an eigenvector associated an eigenvalue l. For the last case, for ε > 0 small enough, then for any R u 0 js 0 ðaÞ À xðaÞjda < ε=2, Therefore, for any case, the disease-free equilibrium E 01 is locally asymptotically sTable if R 01 < 1. Proceeding the similar process, we can conclude the local stability of E 02 if R 02 < 1. This completes the Proof. Theorem 2.2. Suppose d(a) ¼ 0 for all a2R þ . If R 0j < 1ðj ¼ 1; 2Þ, then the disease-free equilibria E 0j is globally asymptotically stable. Proof. Provided j(a) ¼ 0 for any a2R þ , then we have from the first and fifth equations known that for a2R þ , Sðt; aÞ S 0 ðaÞ; lim t/þ∞ Vðt; aÞ ¼ 0: If d(a) ¼ 0, adding all the equations of (2.1) yields to vN t (t, a) ¼ 0, and so that N(t, a) ¼ N 0 (a) for each pair ðt;aÞ2R 2 þ . From Assumption 2.1, we have known that K 1 has a positive left eigenvector U(a) associated with R 01 such that UðaÞK ½4ðaÞ ¼ R 01 4ðaÞUðaÞ: Now, we construct a Lyapunov functional as follows UðaÞ½Eðt; aÞ þ Iðt; aÞda: Taking the derivative of L along the trajectory of (2.1), one arrives at UðaÞðK ½ðgðaÞ þ hðaÞ ÞI ðt; aÞ À ðgðaÞ þ hðaÞ ÞIðt; aÞ Þda þ εu UðaÞðR 01 À 1ÞðgðaÞ þ hðaÞ ÞIðt; aÞda þ εu: (2.12) From the arbitrary of ε, we have that vL t 0 and the equality holds if and only if I(t, a) ¼ 0 for ðt; aÞ2R 2 þ . Therefore, the largest invariant set M ¼ f4 2X _ L ¼ 0g contains a singleton point E 01 . By LaSalle invariant principle, we have that the diseasefree equilibrium E 01 is globally attractive when R 01 < 1 and d(a) ¼ 0. Similarly, we conclude that the disease-free equilibrium E 02 is also globally asymptotically sTable if R 02 < 1. 3. An optimal control problem As mentioned in Introduction, we introduce such two strategies: reducing physical distance and taking mass vaccination to find an optimal intervention through minimizing the disease-related deaths and the costs for control implementation. Then, an age-structured epidemic model with two kinds of intervention strategies is described by the following system: (3.1) where u 1 and u 2 denote the two control variables of age a and time t. The parameters and state variables remain the same meaning as model (2.1). Setting x(t, ,) ¼ (S(t,,); E(t,,); I(t,,); H(t,,); V(t,,); R(t,,)) T , system (3.1) becomes 8 < : v t xðt; aÞ ¼ f ðt; a; xðt; aÞ; u 1 ðt; aÞ; u 2 ðt; aÞÞdf ðt; aÞ; xðt; 0Þ ¼ 0; xð0; ,Þ ¼ x 0 ð,ÞdðSð0; ,Þ; Eð0; ,Þ; Ið0; ,Þ; Hð0; ,Þ; Vð0; ,ÞÞ; gðaÞHðt; aÞ þ gðaÞIðt; aÞ 2. An optimal control question As stated above, our goal is to find an optimal control strategy corresponding to age a and time t that minimizes both the deaths due to disease and the costs of epidemic controls. We apply L 1 optimal control theory to determine the "best" vaccination regime and physical distancing interventions. To achieve this goal, we define the objective function by such two interventions as follows: J½u 1 ðt; aÞ; u 2 ðt; aÞ ¼ Mðt; a; x; u 1 ; u 2 Þ ¼ B½dðaÞIðt; aÞ þ dðaÞHðt; aÞ þ A 1 2 u 2 1 ðt; aÞ þ A 2 2 u 2 2 ðt; aÞ þ A 3 jðaÞSðt; aÞ; where B, A 1 , A 2 , A 3 are balancing coefficients and T f counts the final time. The first sum in the integral is the cost of diseaserelated deaths and the last three terms represent costs of vaccination and physical distancing interventions. To derive the necessary optimality condition, we introduce a time-varying and age-varying Lagrange multiplier vector zðt; aÞ ¼ ðl S ðt; aÞ; l E ðt; aÞ; l I ðt; aÞ; l H ðt; aÞ; l V ðt; aÞ; l R ðt; aÞ Þ T , whose elements are called the adjoint variables of system (3.1). Next, we define a generalized Hamiltonian function as follows: 3 . Epidemic curves of infected individuals in different age groups under strategy C. The red solid curves represent the infected number I j (t), j ¼ 1, 2, …, 6 without control, the blue lines denote the infectives I j (t), j ¼ 1, 2, …, 6, with strategy C. The parameters are fixed at (3.4) where , represents an inner product. Let △ x denotes the differentiation with respect to variable x. The adjoint system is given by ( Àv t zðt; aÞ ¼ △ x Mðt; aÞ þ zðt; aÞ,△ x f ðt; aÞ; z T f ; a ¼ 0; zðt; uÞ ¼ 0; (3.5) that is From the set of the admissible control, the optimal control pair is characterized by the following formula: u * 1 ¼ minfmaxf0;û 1 g; 1g; u * 2 ¼ minfmaxf0;û 2 g; 1g: In this subsection, we search for control pairs ðu * 1 ; u * 2 Þ definitely satisfying (3.9) on the control set We define the norm k ,k L 1 ðX Þ for X dQ 5 as follow: jx i ðt; aÞj; for each xdðx 1 ; …; x 5 Þ2X . Accordingly, we can define the norms k ,k L 1 ðU Þ , k ,k L ∞ ðU Þ similarly. Then, we embed the optimal problem in space L 1 (Q) by Before proving the existence of the optimal control, we introduce the following lemma. Lemma 1. Setting T f be sufficiently small, the following statements are true. (1) The map u2U /x u 2L 1 ðX Þ satisfies Lipschitz properties: for all u; n2U . (2) For u2U , adjoint system has a weak solution l u in L ∞ ðX Þ, such that for all u; n2U . (3) The functional J ðuÞ is lower semi-continuous with respect to L 1 ðL Þ convergence. Do the similar process, we have known that kG u ða; tÞ À G v ða; tÞk L 1 ðX Þ C 1 G x u ðt; aÞ À x v ðt; aÞk L 1 ðX Þ þ u 1 ðt; aÞ À v 1 ðt; aÞk L 1 ðU Þ : From the first equation of (3.1), we integrate from 0 to t and obtain Collecting all the mentioned above, one arrives at kx u ðt; aÞ À x v ðt; aÞk L ∞ ðX Þ C ∞ x kuðt; aÞ À vðt; aÞk L ∞ ðU Þ : Replacing L ∞ norm by L 1 norm, we repeat the above programs to get kx u ðt; aÞ À x v ðt; aÞk L 1 ðX Þ C 1 x kuðt; aÞ À vðt; aÞk L 1 ðU Þ : For Case (2), we observe that vNðt; aÞ vt ¼ ÀdðaÞðIðt; aÞ þ Hðt; aÞÞ ! ÀdðaÞNðt; aÞ; which suggests that N 0 (a)e Àd(a)t N(t, a) N 0 (a) for all ðt; aÞ2R 2 þ . vLðt; aÞ vt C L ; L ¼ S; E; I; H; V; (3.11) vl L ðt; aÞ vt (3.12) Observing that for any 0 < u j < u jmax , j ¼ 1, 2, one can easily derive the following equation For the convenience, let us denote vl L vu j ¼ f Lu j ; L ¼ S; E; I; H; V; j ¼ 1; 2: x u ðt; aÞ À x v ðt; aÞk L ∞ ðX Þ þ u 2 ðt; aÞ À v 2 ðt; aÞk L ∞ ðU Þ : P. Jia, J. Yang and X. Li Infectious Disease Modelling 7 (2022) 149e169 From formulas (3.7) and (3.8), it follows that kf Luj k L ∞ ðQ Þ Ĉ Luj ;L ¼ S;E;I;H;V;j ¼ 1;2. If u j 0 or u j ! u jmax , j ¼ 1, 2, we have from continuous dependence of solutions l L (L ¼ S, E, I, H, V) on parameters u j (j ¼ 1, 2) that kf Luj k L ∞ ðQ Þ C Luj ;L ¼ S;E;I;H;V;j ¼ 1; From what has been discussed, one concludes that for every u, kf Lu j k L ∞ ðQ Þ C Lu j ; where C Luj ¼ maxfC̄L uj ;C Luj g, L ¼ S, E, I, H, V, j ¼ 1, 2. Therefore, for any ðt; aÞ2R 2 þ , kl u ðt; aÞ À l v ðt; aÞk L ∞ ðX Þ C ∞ l kuðt; aÞ À vðt; aÞk L ∞ ðU Þ ; where C ∞ l d max L¼S;E;I;H;V;j¼1;2 fC Luj g is proportional to T f . Now, we are concerned with Case (3). Assume that either u n d(u 1n , u 2n ) / u ¼ (u 1 , u 2 ) in ðL 1 ðQ ÞÞ 2 or there exists a subsequence, u 2 n /u 2 a.e. on Q. Using Lebesgue's dominated convergence theorem, it comes lim n/∞ k u 2 n k L 1 ðQ Þ ¼ ku 2 k L 1 ðQ Þ . The similar arguments is true for k n 2 k L 1 ðQ Þ . Then, we address the convergence of one term in the functional, Hence, k J ðu n Þ À J ðuÞ k C 1 J k u n À uk L 1 ðQ Þ : Moreover, we have that J ðuÞ lim inf n/∞ J ðu n Þ. We notice that J ðuÞ is lower semi-continuous with respect to strong L 1 convergence. However, it is not associated with weak L 1 convergence. Therefore, it generally does not attain its infimum on L . To overcome this default, Ekerland variational principle will be employed to handles this situation (Ekeland, 1974) . For e > 0, there exists u e in L 1 ðU Þ such that k u e À uk L 1 ðU Þ : (3.14) Note that, by the second expression of (3.14), the perturbed functional J e ðuÞ ¼ J ðuÞ þ ffiffi e p k u e À uk L 1 ðU Þ attains its infimum at u e . By the same argument as in subsection 3.2 and using the projection map L on U , we detect a pair of optimal control ðû 1 ;û 2 Þ such that u ε minimizes L . Lemma 2. If u e is an optimal control minimizing the functional J e ðuÞ,then Combining results of Lemma 2 and Theorem 3.1, the following Theorem gives a sufficient condition of the existence of a unique optimal control. is sufficiently small, there exists a unique optimal control u* in U minimizing J ðuÞ. Proof. Define G : U /U by G ðuÞ ¼ L ðû 1 ðx ue ; l ue Þ;û 2 ðx ue ; l ue ÞÞ; where x u and l u are state and adjoint solutions associated with control u. By using Lipschitz properties of x u and l u , for every u; n2U , one reaches In the same way, we also have ( 3.15) where the constant C L depends on the L ∞ bounds on the state and adjoint solutions, and Lipschitz constants. The functional G has a unique fixed point u* when C L 1 A1 þ 1 A2 : < 1. Then, the key is to prove that the fixed point is an optimal control. By using the approximate minimizers u e by Ekerland variational principle, and combining Lemma 2 with the scaling of G , we find that k G ðu e Þ À u e k L ∞ ðX Þ ¼ k L ðu 1 ðx ue ; l ue Þ; u 2 ðx ue ; l ue Þ Þ À L (3.16) With the help of inequalities (3.15) and (3.16), we have is sufficiently small. Hence, it comes that Thus, an approximate minimiter u e converges to the fixed point u*, namely u e / u* in L ∞ ðU Þ, and by using Ekelands's principle (as e / 0) To visibly observe the control effects, we are conducted to numerical experiments to capture them. To achieve this goal, we employ the forward-backward sweep approach (Lenhart & Workman, 2007) . Actually, a forward fourth-order Runge-Kutta method is done for the state variables and a backward fourth-order Runge-Kutta method is designed for the adjoint variables (Martcheva, 2015b) . For suppressing an emerging infectious disease, we introduce two elementary controls: physical distancing u 1 and vaccination u 2 . To take account effects for every feasible control, we classify the following four strategies: Strategy A measures to reduce physical distancing; Strategy B measures to increase vaccination rate; Strategy C measures to take coupled controls; Strategy D measures no control strategy(blank group). In order to ensure the authenticity and validity of the investigation, we select data from the initial COVID-19 outbreak in Sao Paulo, Brazil for numerical simulation. According to the transmission characteristics of COVID-19 infection in Brazil, the total population is divided into six age groups: 0e9 years old, 10e39 years old, 40e49 years old, 50e59 years old, 60e69 years old and over 70 years old. The transmission rate b(a, b) is taken in form of bða; bÞ ¼ b  cða; bÞ; (4.1) where b denotes the effective transmission probability and c(a, b) represents the number of physical contacts per day between susceptibles with age a and infectives with age b. For the convenience, we define the contacts during each group are constants and so it generates a contact matrix (see Fig. 1 ). Other parameters used in the simulation are taken in Table 1 , as well as the initial values of state variables for each age group are selected from Table 2 (Rocha Filho et al., 2020) . To evaluate the control effects, judiciously controlling cost of the strategy is an important index, which needs to mitigate financial burden without jeopardizing the most vulnerable members of the population (Zhao & Feng, 2020) . Since the major purpose of optimal control questions is to balance revenue and costs of the controls, we take three typical cases to reflect their effects: (1) low cost, that is, A 1 ¼ A 2 ¼ 1; (2) moderate cost, that is, A 1 ¼ A 2 ¼ 100; (3) high cost, that is, A 1 ¼ A 2 ¼ 1000. Now we are going to investigate the total number of infected persons in six age groups. Fig. 2 shows the total number of infected cases under strategy A À D. It is clear that each control strategy reduces the size of the prevalence, shortens the epidemic duration, advances the peak arrived time and deduces the peak values. Conversely, the decreasing trends rank: low-cost control, median-cost control, high-cost control. Therefore, the control cost is the key to disease prevention and control. Next, we will analyze the question in a quantitative perspective, that is focusing on the values of the objective function at different costs under different strategies (see Table 3 ). A clear observation is that under the same strategy, the more the cost is, the bigger the corresponding to objective function value will be, which means that the higher the control cost is, the worse the control effect will be. Under the same cost, the objective function value correspondingly strategy C is smaller than that of other strategies, that is, strategy C is optimal with the least cost. We further analyze the specific effects of strategy C for different age groups. As observed in Fig. 3 , compared with no control, infection peaks in all age groups decrease and the peak arrived times reach earlier after the implementation of strategy C. The prevalence decreases the fastest for the group aged 10e39 and reduces the lowest for the group over 70 years old. In the comparative analysis, we find that the difference between strategy A and strategy C is subtle. To further distinguish the effects of the implementation of two strategies, we are conducted to quantitative evaluation, which could be globally analyzed from three aspects: the total number of infected cases in 0 À T period, the number of persons who avoids becoming infected as a result of the implementation of the control in the 0 À T period and objective function value. According to Table 5 , compared with other strategies, strategy C (i.e. the combination strategy) had the lowest number of infections within 0 À T, and more people avoided becoming infected. At the same time, the objective function value of strategy C is the smallest, which indicates that strategy C can save manpower and material resources and prevent more susceptible individuals from becoming infected persons. Therefore, the mixed measure simultaneously increasing social distancing and vaccination is a dominant strategy. In order to quantify the effects of strategy C on each age group, we define an Infection Averted Rate(IAR) (Okosun et al., 2013) where the number of persons who avoided infection is the number of infected persons caused without control minus the number of infected persons with control. Table 4 shows that the rank of the IAR values are people aged 60e69, aged over 70, aged 10e39, aged 0e9, aged 40e49 and aged 50e59. Clearly, strategy C takes better significant effects on people aged 10e39 and over 60 (see Table 4 ). Step two From (Arnold et al., 2021), we have known that vaccination is a highly cost-effective action than that taking social distancing measures for control COVID-19. Then, we set weight coefficients A 1 ¼ 1, A 2 ¼ 1000, A 3 ¼ 1, B ¼ 1, evaluate effects of feasible control measures and then look for an optimal control measure associated with each age group. Fig. 4 displays that the implementation of strategy A, B and C deduces peaks values and advance the peak arrived time. Further analysis shows that compared with other strategies, enlarging physical distancing enables the prevalence peak arrives faster, and the slope of decline after the peak is sharper, which suggests that physical distancing measure, such as limiting gathering, closing building, working at home, etc., plays a significant role in preventing the infections. For each strategy, the prevalence are lowest for people aged over 60, which indicates that the control effects are beneficial for people aged over 60. Table 6 and Fig. 5 give the durations of strong control measures and the optimal controls associated each age group. Under the optimal control strategy C, the durations of strongly increasing social distancing and strengthening vaccination for aged 10e39 need 27.24 and 57.18 days, respectively. Conversely, such durations are obviously longer than the durations of other age groups. On the other hand, the duration of intensive vaccination control is almost double than that duration of increasing social distancing in the same age group. This finding has an important implication for the actual allocation of medical resources for the disease prevention and control. In this case, the policymakers and the public health governments should pay more attention on controlling payoffs in the group aged 10e39. Especially, they should markedly note that the effective control impacts of taking vaccination should be a longer patience. Age heterogeneity is a noteworthy features of many emerging infectious diseases, which has well-known impact on the magnitudes and intensity of such diseases. In this paper, we introduced a continuous age-structured model to study the transmission dynamics of emerging infectious diseases in a short period or in the initial outbreak. Since some emerging infectious diseases are highly sensitive due to a higher social activity in certain age groups. We tried to identify the age groups to target for effectively slowing down the disease infection. In this paper, we calculated the basic reproduction number R 0 , which is a spectral radius of the next generation operator K . The value of R 0 determines the stability of the disease-free equilibrium E 0 . If it is less than 1, then E 0 is globally asymptotically stable by constructing a suitable Lyapunov function. In such construction, we employed the left eigenvector of K as a kernel function to completely address the global stability of E 0 , which provides a new approach for dealing with the global stability of continuous age-structured models. Facing a novel emerging diseases, maintaining social distance and projecting vaccination are two main control measures for curtaining the diseases infection. However, because of the economic and social burdens and limitations of medical resources, identifying optimal control measures has become an urgent issue. In this paper, we tried to evaluate the two control strategies and found the optimal control strategies by cost-effective analysis. Moreover, we derived the adjoint system and optimal control pairs by constructing a generalized Hamilton functional. As the continuous age-structured model is characterized by a series of first order partial differential equations, the optimal problems are quietly different from the problems describing by a second order PDE system, due to the lack of the regularity of the state solutions. To overcome this default, we employed Ekeland Principle's to analyze the existence of the optimal control, such methods have been extensively used by many scholars (Barbu, 1994; Barbu & Iannelli, 1999; Numfor et al., 2014; Yang et al., 2019) . We pointed out that if the weight parameters are sufficiently large or the control target time T is small enough, such optimal control exists from mathematical view of points. In order to illustrate our theoretical results, we picked up an initial outbreak of COVID-19 in Sao Paulo, Brazil. We proposed three different control measures such as single reducing physical distance, single vaccination and mixed the two measures to take account for the impact on the control COVID-19. We found that reducing physical distance is necessary for the initial outbreak of COVID-19 and vaccination is most effective for a long-term outbreak. Conversely, the effect of the mixed strategy is more significant under the low cost and this control strategy is most effective for old peoples and the young age group. Overall, the combination of enlarging social distance and the mass vaccination is the most cost-effective. Meanwhile, the primary target of the two control variables is the group aged 10e39, which means that the Brazil government should take more investments on people aged 10e39 for the prevention and control COVID-19 infection. For the age-structured models, model parameters are generally assumed to change with the activity of people in different ages such as death rate due to diseases, latent period, recovered period etc. As it reported, environment-to-human has become a non-negligible transmission route of COVID-19. Furthermore, pathogenic mutation, human behaviors and other factors have great effects on the transmission of emerging infectious diseases, we will leave these work for the future. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 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