Epidemiological Models with Seasonality - Mathematical Methods and Models in Biomedicine - page 395

on the right-hand side of the equation for I 1 , noting that the parameters in Eqs.
( 46a )-( 46d ) should be such that d N / d t ≤ 0 (since HIV reduces life expectancy).

We finally need to prove that if I 3 ( t ) ≥ η for 0 < t < Λ then I 1 ( t ) ≥ c 6 η for

t 0 < 0. This, however, we can only establish if we modify the system

( 46a )-( 46d ) by adding a term

t < Λ and c 6 >

S
N I 3 to the right-hand side of the first equation
and subtracting it from the right-hand side of the last equation. We conclude that
Theorem 4.2 holds for this modified system.

β 3 ( t )

9 Conclusion and Discussion

The basic reproduction number R 0 provides important but limited information
on the spread of an infectious disease. It informs whether initial small infection
leads to endemic disease (which is the case if R 0 >

1) or whether the infection

will die out (which is the case if R 0 < 1). When dealing with a disease that

is seasonality(periodically) dependent, it is clearly more difficult to make the
same determination on the course of an initial infection; this is also reflected
mathematically by the difficulty in computing R 0 . The aim of this chapter was to
determine, for some disease models, a procedure to compute when R 0 <

1andwhen

R 0 > 1 for a disease with seasonality. For clarity, we developed this method for

water-dependent diseases, such as cholera, and then explained in the Sect. 8 ,how
this method can be extended to other models.
We also proved that, in case R 0 >

1, there exists a periodic endemic solution. The
case of a malaria model was developed earlier, in [ 3 ].
One immediate question is whether there is just one periodic solution, and, if not,
how many. The answer may be important, since knowing the course of the disease
progression may help in the development of a strategy to contain it.
The inequality R 0 > 1 tells us that the disease will become endemic. We expect

that the larger R 0 is the larger by the size of infected compartment, at any future time.
But such a result has not been proved and remains an interesting open problem.

References

1. Bacaer, N.: Approximation of the basic reproduction number R 0 for waterborne diseases with
seasonality. Bull. Math. Biol. 69 , 1067-1091 (2007)
2. Bacaer, N., Guernaoui, S.: The epidemic threshold of vector-borne disease with seasonality.
J. Math. Biol. 53 , 421-436 (2006)
3. Dembele, B., Friedman, A., Yakubu, A.A.: Malaria model with periodic mosquito birth and
death rates. J. Biol. Dyn. 3 , 430-445 (2009)
4. Greenhalgh, D., Moneim, I.A.: SIRS epidemic model and simulations using different types of
seasonal contact rate. Syst. Anal. Model. Simul. 43 , 573-600 (2003)
5. Horn, W.A.: Some fixed point theorems for compact maps and flows in Banach spaces. Trans.
Am. Math. Soc. 149 , 391-402 (1970)