Another argument against these transfers comes from the fact that we can compute that if *N* = 6, *x*(*t*) = *y*(*t*)/2 and the ratio of infected individuals changes from *x*(*t* ~0~)/*N* = 0.15 to 0.3, then the total number of infected individuals at the time *t* ~1~ changes to *I* ~1~ = 0.1+0.15*y*(*t* ~1~)/2 = 0.22. With *x*(*t* ~0~)/*N* = 0.15, the transmission time of disease was also *t* ~1~ = 0.25 years. Now consider the case where *y*(*t* ~1~)/*N* = 0.18 and the ratio of infected individuals changes from 0.15 to 0.25. The total number of infected individuals changes to *I* ~1~ = 0.12+0.18*y*(*t* ~1~)/2 = 0.2, which is smaller than *I* ~1~ = 0.22 and it shows that the transmission time of disease is earlier than before (*t* ~1~−*t* ~0~ = 0.1--0.15*y*(*t* ~1~)/*N* = −0.05--0.12/0.15 = −0.4 years). Therefore, the new epidemic curve is very close to the old one as shown in [Fig. 1(d)](#pone-0047459-g001){ref-type="fig"}. These analyses can be summarized as follows: an increase in the prevalence causes the number of newly infected individuals to be larger and a decrease in the prevalence causes the number of newly infected individuals to be smaller. Infected individuals {#s3c} -------------------- From the previous subsection, we expect that changes in the prevalence changes the value of the total number of infected individuals. Let us also consider the relationship between transmission rate and the total number of infected individuals. For the basic SIR model, let us assume that there is no time delay between *x*(*t*) and *y*(*t*). In this case, we obtain a simple relationship between the transmission rate and the total number of infected individuals as *y*(*t*) = *a*(*t*)*I* (*a*(*t*) is the transmission rate of disease, *I* is the number of infected individuals). Considering this relationship, we can rewrite Eqs. (8) and (9) as Eqs. (11) and (12), respectively. As a result, we obtain the following equation: . That is, an increase in *a*(*t*) causes an increase in the total number of infected individuals. Since *x*(*t*)/*N* = 0.09 at *t* = 1.0, we can estimate the change in the number of infected individuals by computing the change in the number of infected individuals for *x*(*t*)/*N* = 0.05 at *t* = 1.1, as follows. As a result, we obtain the relationship *I* = 0.05/(1−0.095) = 0.12 = 1.2−1.0 = 0.2 for *x*(*t*)/*N* = 0.05 and *y*(*t*)/*N* = 0.12. Similarly, as a result, we obtain the relationship *I* = 0.09/(1−0.095) = 0.19 = 1.2--1.1 = 0.1 for *x*(*t*)/*N* = 0.1 and *y*(*t*)/*N* = 0.19. As a result, we obtain the relationship . As is explained in the previous subsection, an increase in the prevalence causes an increase in *I* and a decrease in *I* causes an increase in the prevalence, and thus the change in *I* is in general greater than that in *x*(*t*). We can derive another relationship if the time delay of transmission is included. Let us compute the values of *I* that are obtained by including the effect of the time delay for *x*(*t*)/*N* = 0.05 at *t* = 1.1 and *x*(*t*)/*N* = 0.1 at *t* = 1.2, respectively. If we focus on the change in *I* by inclusion of the time delay, the change in *x*(*t*)/*N* is not included, and we need to consider the fact that the time delay of *y*(*t*) can have impact on the number of newly infected individuals as well. This consideration is very important, because changes in the prevalence also have a small impact on the number of newly infected individuals. In this case, we obtain the following equation: . As a result, we can show that the change in *I* is larger than that in *x*(*t*) and also larger than the change in *y*(*t*), as shown in [Fig. 3](#pone-0047459-g003){ref-type="fig"}. {#pone-0047459-g003} Discussion {#s4} ========== In this study, we proposed a mathematical model to show that the total number of infected individuals in a population is sensitive to changes in the prevalence of a disease in the population. We demonstrated that the total number of infected individuals increases if the prevalence increases and decreases if the prevalence decreases. This result demonstrates that the size of the epidemic will be greater if the prevalence increases. In this analysis, we included only four typical examples of the time-delay model, *a*(*t*) and *d*(*t*) are linear, which are simple but it would be worth understanding what would happen in a more realistic model. In other words, we considered a simple nonlinear case of *d*(*t*), which was designed so as to explain the trend of an epidemic curve. This nonlinear term has a significant effect on our results in that it increases the speed of decline in the number of infected individuals. We conclude that *I* changes faster in comparison to *x*(*t*) and *y*(*t*). This change in the number of infected individuals has a significant impact on the dynamics of epidemic. We expect that the dynamics of an epidemic in an epidemic-prone region will change drastically if a large scale of outbreak occurs. An increase in *a*(*t*) and a decrease in *d*(*t*) cause the total number of infected individuals to increase to a greater level than before, and this may lead to a rapid increase in the number of infected individuals. This is why the rapid spread of the pandemic influenza A (H1N1) has caused major problems for health care systems. This is why it is important to find and determine the time delay *d*(*t*) for future pandemics. In the present model, we considered only those individuals who can transmit infection to others. In reality, an individual who cannot transmit infection to others can also become infected due to infection. Such individuals have been included in the model recently [@pone.0047459-Blyuss1]. That is, we have modified the model as *d*(*t*) = *a*(*t*)*N*−*d*(*t*−*T* ~1~)*I* ~1~+*d*(*t*−*T* ~2~)*I* ~2~. We considered that the time delay is *T* ~1~ for infection of the first individual and *T* ~2~ for infection of the second individual. The total number of infected individuals can be defined as *I* ~1~+*I* ~2~. As shown in [Fig. 4](#pone-0047459-g004){ref-type="fig"}, the size of the epidemic increases in the early stage of the epidemic. If the value of *I* ~1~+*I* ~2~ is small at *t* = 0, then it increases at *t* = 1, 2, etc., and finally it becomes constant. In the early stage of the epidemic, the change in the total number of infected individuals increases. However, if *d*(*t*) can be small in the final stage, *I* ~1~+*I* ~2~ can be small in the final stage. {#pone-0047459-g004} The results for simple infection with transmission delay are worth comparing to those for a model with complex transmission delay. The mathematical model for multiple infections with simple transmission delay is as follows. Let us consider the number of infected individuals, *x*(*t*). The transmission rate in this model is given by *a*(*t*)*I*(*t*), and therefore we can obtain the relationship *x*(*t*) = *a*(*t*)*I*(*t*). However, as shown