^{1}

^{2}

^{1}

^{2}

We construct and analyze a nonlocal global pandemic model that comprises a system of two nonlocal integrodifferential equations (functional differential equations) and an ordinary differential equation. This model was constructed by considering a spherical coordinate transformation of a previously established epidemiology model that can be applied to insect-borne diseases, like yellow fever. This transformation amounts to a nonlocal boundary value problem on the unit sphere and therefore can be interpreted as a global pandemic model for insect-borne diseases. We ultimately show that a weak solution to the weak formulation of this model exists using a fixed point argument, which calls upon the construction of a weak formulation and the existence and uniqueness of an auxiliary problem.

In [

The following homogeneous extension of model (

Upon the completion of [

We next give an overview of this model, which is based on [

In this paper, we consider an extension of model (

Finally, our interest here is to show that a solution exists to (

Several papers have recently been published regarding the mathematical analysis of pandemics, as well as the construction of mathematical models comprising a system of differential equations applicable to such biological phenomena. Several of these papers influenced our current work and it is therefore worth briefly describing them before we further discuss (

For example, in [

As a precaution to a potential H5N1 pandemic, the authors in [

Other fairly recent papers related to the modeling of pandemics are worth mentioning here as well. For example, in [

Our current model is quite different than the aforementioned ones, since it comprises two nonlocal integrodifferential equations, also often referred to as functional differential equations, and an ordinary differential equation. Moreover, in order to establish the existence of the weak solution to the weak formulation of (

This paper is organized as follows. In Section

We begin by multiplying each equation in (

In considering a weak solution to (

Now let

We replace

Since the integrals involving the

Now, define

Then, by Gronwall’s lemma we see that

From there, we see that

Next, we take (

Upon considering the energy inequality for parabolic equations (cf. [

As the auxiliary problem possesses a unique weak solution for each continuous

For the continuity of

It therefore follows from (

To show that

From (

See the preceding analysis.

We leave open for consideration the establishment of a unique solution to the weak formulation developed in this paper. This will entail a numerous amount of estimates, which are too tedious to be included in the present work. We refer the interested reader to Chapter 3, Section 2, of [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank the referee for the timely insights, suggestions, and inquiries. In addition, they appreciate the assistance of Frank B. Jones (Rice University) for sharing with them his unpublished lecture notes pertaining to the heat equation on the unit sphere.