IMPACT OF BEHAVIORAL CHANGE ON THE EPIDEMIC CHARACTERISTICS OF AN EPIDEMIC MODEL WITHOUT VITAL DYNAMICS

. The epidemic characteristics of an epidemic model with behavioral change in [V. Capasso, G. Serio, A generalizaition of the Kermack-McKendrick deterministic epidemic model, Math. Bios., 42 (1978), 43-61] are investigated, including the epidemic size, peak and turning point. The conditions on the appearance of the peak state and turning point are represented clearly, and the expressions determining the corresponding time for the peak state and turning point are described explicitly. Moreover, the impact of behavioral change on the characteristics is discussed.


1.
Introduction. Dynamical models for epidemic spread have made a great contribution to understanding transmission mechanism of the infection and controlling the spread. In 1927, Kermack and McKendrick [5] established the following simple SIR epidemic model to investigate the outbreak of the Great Plague lasting from 1665 to 1666 in London        S = −βSI, where the population is divided into three classes, susceptible (S), infective (I) and removed (R); S(t), I(t) and R(t) denote their numbers respectively at time t; β is the infection rate coefficient, and α is the removal rate coefficient. A threshold theorem of epidemic spread was found by Kermack and McKendrick [5] for system (1). Since then a lot of epidemic models are established based on model (1) [9,2] and references therein.
In 1978, after a study of the cholera epidemic spread in Bari in 1973, Capasso and Serio [3] proposed a saturation incidence rate βSI/(1 + εI) to measure the inhibition effect due to behavioral change (e.g. reduction of contact rate, strengthening of protection measures, etc.) of the susceptible individuals when the number of infective individuals increases, where ε is referred to as inhibition parameter reflecting the intensity of behavioral change of susceptible individuals during the disease spread. The replacement of the bilinear incidence in model (1) with the saturation incidence yields the following model with behavioral change For model (1), the epidemic final size (i.e. the number of individuals who are infected over the course of the epidemic) can be determined easily by dividing the first two equations of the model and then integrating it [1,6,8,10], and the epidemic peak (i.e. the largest number of real-time infected individuals in the population (not cumulative cases)) can also be found directly from the first integral and the second equation of model (1) [6].
In [3], Capasso and Serio compared the two models (1) and (2) in a qualitative way, and extended the threshold theorem for model (1) by replacing the threshold line of model (1) with the threshold curve of model (2). But with respect to the epidemic final size and peak state of model (2), there is not an investigation in detail. Especially, the role of the inhibition parameter ε for the epidemic characteristics has not been discussed.
During an epidemic outbreak, for the local public health department to control the spread of the disease, while concerning about the peak state of disease spread and the epidemic final size, the turning point and the associated state of population are also the important characteristics that need to be paid attention too. The turning point denotes the time at which the rate of cumulative cases changes from increasing to decreasing or vice versa [4]. Recently, we theoretically investigated the epidemic characteristics including the epidemic final size, the peak and the turning point of some simple epidemic models without vita dynamics including model (1), and analyzed the dependence of the related quantities on the initial conditions [6,7,11]. However, in the preceding models considered by us, no behavioral intervention was involved. In this paper, our aim is to investigate the impact of behavioral change on the epidemic characteristics for model (2). Based on some fundamental and elegant mathematical deductions, the dependence of the epidemic characteristics on the initial condition and the inhibition parameter is established.
2. Formulation of model and preliminary. Since the variable R does not appear in the first two equations of (2), and the system (called as SI model) consisting of the first two equations of (2) can determine its dynamics and epidemic characteristics, we consider the reduced model with the initial condition S(0) = S 0 > 0 and I(0) = I 0 > 0, where α may represent the sum of the recovery and disease-induced death rates. Obviously, the initial condition implies that S(t) > 0 and I(t) > 0 for t > 0. Moreover, S(t) is decreasing since S (t) < 0 for t > 0. Thus S(t) ≤ S 0 for t ≥ 0.
Further, for σ = 1 and σ = 1, (5) and (6) can be rewritten as and respectively. In particular, when σ Additionally, we state the following lemmas and inequalities, which will be used in our later inferences.
for u > 0, and the equality holds if and only if u = 1.
It is easy to prove the above lemmas and inequalities by applying the fundamental knowledge of differential calculus, so we omit them.
3. Analysis of epidemic characteristics. In this section, we analyze the epidemic characteristics of SI model (3) including the final state, the peak state and the turning point by means of Lemmas and the relation between variable S and I obtained in Section 2.

Epidemic final state.
Epidemic final size is the number of the cumulative cases. According to the character of SI model (3), there is no reinfection for the model. Then the size can be obtained by subtracting the number of the individuals, who have not been infected when the spread of the disease terminates, from the initial number of susceptible individuals. The termination of infection is indicated by the fact that there is no infected individuals.
Additionally, with respect to x ∞ , we have the following statement which will be used later.
Summarizing the above inference, the conditions on the existence of the turning point can be unified as the expression (εI 0 + σ)(εI 0 + 1) < εS 0 .
Since functions h i (x) (i = 1, 2) are transcendental functions, the value of x corresponding to the inflection point of C(t) could not expressed explicitly. If it exists, denoted by x t , then, similar to the determination of time corresponding to the peak state, the time of the turning point of epidemic spread can be found by the following expression and the corresponding state is (S t , I t ), where S t = S 0 x t and I t = f i (x t )/ε (i = 1, 2.) 4. Conclusion and discussion. In Section 3, we have discussed the epidemic characteristics of model (3) with behavioral change, obtained the condition on the existence of the peak state, σ(1 + εI 0 ) < εS 0 , and the condition on the appearance of the turning point of epidemic spread, (εI 0 + σ)(1 + εI 0 ) < εS 0 , and provided the methods or expressions determining the associated quantities.
In order to make the dependence of the related conditions on the parameters and the initial conditions more clear, we replace the parameter σ with αε/β in the associated expressions. Correspondingly, the condition on the existence of the peak state is that ε < (βS 0 /α − 1) /I 0 , and the condition on the appearance of the turning point is that ε < [βS 0 /(α + βI 0 ) − 1]/I 0 . Conversely, when ε ≥ (βS 0 /α − 1) /I 0 , the peak can not appear; when ε ≥ [βS 0 /(α + βI 0 ) − 1]/I 0 , there is no turning point. Note that βS 0 /(α + βI 0 ) < βS 0 /α. Then, when ε < [βS 0 /(α + βI 0 ) − 1]/I 0 , both the peak and the turning point can appear; when [βS 0 /(α + βI 0 ) − 1]/I 0 ≤ ε < (βS 0 /α − 1) /I 0 , there is a peak but no turning point; if ε ≥ (βS 0 /α − 1) /I 0 , both the two characteristics could not exist. These statements show the impact of the behavioral change on the existence of the two characteristics. From another point of view, when the turning point can appear, there must be the peak state. But there may not be the turning point if the peak state exists. The inequalities above also provide the threshold condition on the appearance of the peak and/or the turning point. The results obtained here would be useful to make the effective control strategy for disease spread.
In order to visually show the impact of behavioral change on the peak state and turning point of disease spread, we plot a set of graphes (Figures 1, 2  According to the obtained results, there are both the peak and the turning point if ε < 7.8333, there is the peak and no turning point if 7.8333 ≤ ε < 9.8, and there is neither peak and no turning point for ε ≥ 9.5. These theoretic results are verified by Figures 1, 2 and 3. In Figure 1 for ε = 0.08, the peak I = 75.53400 achieves at t = 0.57065, and the turning point is (0.16853, 37.07000). In Figure 2 for ε = 8, the peak is I = 2.29549 at t = 1.36570 and there is no turning point. For Figure 3 for ε = 9.8, I = I(t) is decreasing, and C = C(t) is convex upwards. That is, both the peak and the turning point do not appear.