Biodiversity is the opposite of monoculture, and we know that one possible consequence of monoculture is disease transmission -- bananas being the modern poster child for monoculture leading to disease spread.
The concept seems to have been most clearly presented in
Schmidt, K.A. & Ostfeld, R.S. Biodiversity and the dilution effect in disease
ecology. Ecology 82: 609–619 (2001)
A community of abundant alternative hosts with reduced competence mitigates Lyme disease risk in this system by decreasing the average infection prevalence of ticks. When alternative hosts have net negative impacts on the most competent reservoirs, via competition and predation, the dilution effect is compounded. We believe this dilution effect to be a general phenomenon of disease ecology because the majority of vector-borne zoonoses are characterized by the fea- tures necessary for it to occur.
There are many articles following up on this one, such as Declining ecosystem health and the dilution effect:
The “dilution effect” implies that where species vary in susceptibility to infection by a pathogen, higher diversity often leads to lower infection prevalence in hosts. For directly transmitted pathogens, non-host species may “dilute” infection directly (1) and indirectly (2). Competitors and predators may (1) alter host behavior to reduce pathogen transmission or (2) reduce host density. ... Our study provides further evidence for dilution effect ...
However, not all agree that this is a general or even a real effect. For example:
How changes in biodiversity alter the transmission of infectious diseases is presently under debate. Epidemiologists and ecologists have put a lot of effort to understand the mechanism behind biodiversity–disease relationship. Two important mechanisms, i.e. dilution and amplification theories have in some manner made it clear that biodiversity and disease outcome have an intimate relationship.... We suggest a multidimensional approach whereby the same disease system needs to be studied in different ecological zones and then the effect of biodiversity on disease outcome needs to be ascertained. Nonetheless, caution is to be taken while jumping to any conclusion as biodiversity–disease relationship is a multifactorial process.
--Does alteration in biodiversity really affect disease outcome? – A debate is brewing
The book seems to be making the specific claim that biodiversity reduces transmission of avian influenza, and without a reference I can't evaluate that, but I couldn't find any articles supporting the claim. I'd be skeptical about that claim without some good evidence, because avian influenza viruses are exceptionally good at crossing species barriers, so biodiversity should be much less of an obstacle for them than more species-constrained pathogens.