Optimal media reporting intensity on mitigating spread of an emerging infectious disease

Mass media reports can induce individual behaviour change during a disease outbreak, which has been found to be useful as it reduces the force of infection. We propose a compartmental model by including a new compartment of the intensity of the media reports, which extends existing models by considering a novel media function, which is dependent both on the number of infected individuals and on the intensity of mass media. The existence and stability of the equilibria are analyzed and an optimal control problem of minimizing the total number of cases and total cost is considered, using reduction or enhancement in the media reporting rate as the control. With the help of Pontryagin’s Maximum Principle, we obtain the optimal media reporting intensity. Through parameterization of the model with the 2009 A/H1N1 influenza outbreak data in the 8th Hospital of Xi’an in Shaanxi Province of China, we obtain the basic reproduction number for the formulated model with two particular media functions. The optimal media reporting intensity obtained here indicates that during the early stage of an epidemic we should quickly enhance media reporting intensity, and keep it at a maximum level until it can finally weaken when epidemic cases have decreased significantly. Numerical simulations show that media impact reduces the number of cases during an epidemic, but that the number of cases is further mitigated under the optimal reporting intensity. Sensitivity analysis implies that the outbreak severity is more sensitive to the weight α1 (weight of media effect sensitive to infected individuals) than weight α2 (weight of media effect sensitive to media items).

(1) can be shown that D is a positively invariant and attracting region.

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Stability of the disease free equilibrium

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Let V = E + σ+µ σ I, then if R 0 ≤ 1, the derivative of V along a solution of (1) is since S ≤ Λ µ , I ≥ 0 and 0 ≤ f (I, M ) ≤ 1. V = 0 holds true only when I = 0. Thus we have that the disease free 20 equilibrium E 0 is globally asymptotically stable when R 0 ≤ 1.

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Stability of the endemic equilibrium 22 When R 0 > 1, the unique endemic equilibrium E * exists and the disease free equilibrium E 0 is unstable. The

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Jacobian matrix of the reduced model at E * is and the corresponding characteristic equation is from which we have a 1 a 2 − a 3 > 0, and a 3 (a 1 a 2 − a 3 ) − a 2 1 a 4 > 0. Thus, by the Hurwitz stability criterion, we 26 know that the real parts of the eigenvalues of J(E * ) are negative, indicating the local stability of the endemic 27 equilibrium E * .

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When we prove the local stability of the endemic state we have the following detailed calculation.
Appendix B: Calculation of the optimal control 32 Consider the optimal control problem to minimize the objective functional Define the Hamiltonian H for the control problem: 3/5 then the adjoint equations and transversality conditions are given: By the optimality conditions, we have Note that the boundness are placed on the control variable u(t), 0 ≤ u(t) ≤ u max , then the optimality condition 38 is changed to This can be rewritten in compact notation which is the optimal control. Model 1: Model 2: Model 3: Model 4: 4/5 In the first two models, the transmission rate is modified by a media function only related to the number of 51 infected individuals. In the latter two models, which are the two particular models proposed in our paper, the 52 dynamics of media reports is considered and the media function is a decreasing function with respect to the 53 number of infected individuals (I) and the media reports (M ). We use the data from 3rd September to 21st 54 September and the Least Square method to estimate the parameter values and AIC values, the data fitting for 55 four different models are shown in the following Figure S1 and Table S1.  We see that the model without the M compartment has a slightly better fit. However, as we are interested in 57 study the effects of M and I in later stages of the outbreak, we have elected to include an in-depth study of the 58 model with M compartment and media functions f 1 and f 2 .